CHAPTER 2

The Basic Theory of Interest

Principal and Interest

 Interest is the time value of money.

 If you invest $ 𝐴(principal amount), then at the end of the year the amount will
grow to 𝐴 × (1 + 𝑟), 𝑟 being the interest rate.

Simple Interest

 If an Amount 𝐴 is left in an account at simple interest 𝑟, the total value after 𝑡 years
is 𝑉 = (1 + 𝑟𝑡)𝐴.
 The account grows linearly with time.

Compound Interest

 If interest is compounded yearly, then after 1 year, the first year’s interest is added
to the original principal getting a larger principal base for second year. Thus,
during the second year, the account earns interest on interest.
 This means, under yearly compounded, the account will grow to:

(1 + 𝑟)times after 1 year,

(1 + 𝑟) times after 2 years,

(1 + 𝑟) times after n years.

It exhibits geometric growth.

Doubling

 Under compounding, the value doubles in about 7 years.

 Seven-ten rule. Money invested at 7% p.a. doubles in approx. 10 years. Also,
money invested at 10% p.a. doubles in approx. 7 years.

Compounding at Various Intervals

 Compounding can be done on more frequent basis like semi-annually, quarterly,

monthly, or even daily.
 If interest rate is quoted on a yearly basis then for frequent compounding
appropriate proportion of that interest rate is applied over each compounding
period.
 Effective Interest Rate (EIR). The equivalent yearly interest rate that would
produce the same result after one year without compounding.

EIR = [1 + 𝑟/𝑚] − 1; where, 𝑚 is the no. of periods

.
Example: 𝑟 = 8% compounded quarterly, will produce 1 + 4 = 1.0824. This
means 𝐸𝐼𝑅 = 1.0824 – 1 = 0.0824, i.e., 8.24%.
 Nominal Rate: Basic yearly rate (8% in the above example).

Continuous Compounding

 Considering the of ordinary compounding as the number 𝑚 (periods in a year)

goes to infinity.
 We know,

lim [1 + (𝑟/𝑚)] = 𝑒
→

[1 + (𝑟/𝑚)] = [1 + (𝑟/𝑚)] = {[1 + (𝑟/𝑚)] } → 𝑒

Hence, continuous compounding leads to exponential growth.

PRESENT VALUE

 Present Value (PV) is the current value of a future sum of money or stream of cash
flows given a specified rate of return.
 Discounting. The process of evaluating future obligations as an equivalent present
value.
 Discount Factor. The factor by which the future value must be discounted. If
compounding is done annually, 𝑘 year discount factor, 𝑑 = 1/(1 + 𝑟)
 If interval compounding (𝑚 period) then 𝑘 year discount factor,

𝑑 =
[ ( / )]

PRESENT AND FUTURE VALUES OF STREAMS

 Ideal Bank. An ideal bank applies the same rate of interest to both deposits and
loans, and it has no service charges or transaction fee. If an ideal bank has an
interest rate that is independent of the length of time for which it applies, and that
interest is compounded according to normal rules, it is said to be a constant ideal
bank.
 Future value of a stream. Given a cash flow stream (𝑥 , 𝑥 , … … , 𝑥 ) and interest
rate 𝑟 for each period, the future value of the stream is
 𝑭𝑽 = 𝒙𝟎 (𝟏 + 𝒓)𝒏 + 𝒙𝟏 (𝟏 + 𝒓)𝒏 𝟏 + ⋯ + 𝒙𝒏
 Present value of a stream. Given a cash flow stream (𝑥 , 𝑥 , … … , 𝑥 ) and interest
rate 𝑟 for each period, the present value of the stream is
𝒙𝒕 𝒙𝟐 𝒙𝒏
𝑷𝑽 = 𝒙𝟎 + + 𝟐
+ …+
𝟏 + 𝒓 (𝟏 + 𝒓) (𝟏 + 𝒓)𝒏
 The present value and future value are related by:
𝑭𝑽
𝑷𝑽 =
(𝟏 + 𝒓)𝒏
 Frequent compounding. Suppose interest is compounded at 𝑚 equally spaced
periods per year and cash flows occurs initially and at the end of each period for a
total of 𝑛 periods, forming a stream (𝑥 , 𝑥 , … … , 𝑥 ), then :
𝒏
𝒙𝒌
𝑷𝑽 = 𝒌
𝒓
𝒌 𝟎 𝟏+
𝒎
 Continuous compounding. Suppose that the nominal interest rate r is
compounded continuously and cash flow occur at times (𝑡 , 𝑡, … … , 𝑡 ),.. In that
case,
𝒏
𝒓𝒕𝒌
𝑷𝑽 = 𝒙(𝒕𝒌 )𝒆
𝒌 𝟎
where, cash flow at time 𝑡 is 𝑥(𝑡 ).
 Main theorem on present value. The cash flow streams 𝑥 = (𝑥 , 𝑥 , … … , 𝑥 )
and 𝑦 = (𝑦 , 𝑦 , … … , 𝑦 ) are equivalent for a constant ideal bank with interest rate
𝑟 if and only if the present value of two streams, evaluated at the bank’s interest
rate, are equal.

INTERNAL RATE OF RETURN

 Given a cash flow stream (𝑥 , 𝑥 , … … , 𝑥 ) associated with an investment [initial

deposit or payment (a negative flow) and the final redemption (a positive flow),
the present value will be :
𝑥
𝑃𝑉 =
(1 + 𝑟)
 Assuming that the cashflows correspond to constant ideal bank at interest rate 𝑟,
then by main theorem on present value, PV would be zero.
 IRR is the value of interest that renders the above PV equal to zero
 Let (𝑥 , 𝑥 , … … , 𝑥 ) be a cash flow stream then the IRR of this stream is a number 𝑟
satisfying the equation:
𝒙𝒕 𝒙𝟐 𝒙𝒏
𝟎 = 𝒙𝟎 + + 𝟐
+ …+
𝟏 + 𝒓 (𝟏 + 𝒓) (𝟏 + 𝒓)𝒏
 Let c = , i.e. r = (1/c) – 1, where c satisfies the polynomial equation:
0 = 𝑥 + 𝑥 𝑐 + 𝑥 𝑐 + ⋯+ 𝑥 𝑐
 Main theorem of IRR. Suppose the cash flow stream (𝑥 , 𝑥 , … … , 𝑥 ) has (𝑥 <
0 𝑎𝑛𝑑 𝑥 ≥ 0 for all 𝑘, 𝑘 = 1,2 … . , 𝑛. With at least one term being strictly positive
(>0). Then there is a unique positive root to the equation.
0 = 𝑥 + 𝑥 𝑐 + 𝑥 𝑐 + ⋯+ 𝑥 𝑐
 Furthermore, if ∑ 𝑥 > 0, then the corresponding IRR, 𝑟 = (1/𝑐) – 1, is positive.

EVALUATION CRITERIA

 The two most important methods are those based on Present Value and IRR.

Net Present Value (NPV)

 PV criteria evaluates alternatives by simply ranking them according to their PV’s –

the higher the PV, the more desirable the alternative.
 NPV is the difference between the terms present worth of benefits and present
worth of costs.
 The NPV criterion is quite compelling. It has a special advantage that the PV of
different investments can be added together to obtain a meaningful composite.
This is because of the PV of a sum of cashflow streams is equal to the sum of the
PV’s of the corresponding cashflows.
 Internal Rate of Return
 IRR can be used to rank alternative cashflow streams – the higher the IRR, the
more desirable the investment.
 However, a potential project is not worth considering unless its IRR> nominal rate.
 If IRR > nominal rate, the investment is considered better than what is available
externally in the financial market.

IRR vs NPV- which is the more appropriate criteria?

 Both NPV and IRR have attractive features and limitations.

 NPV
 NPV is the simplest to compute. It does not have ambiguity of several
possible roots like that in IRR equation.
 It can be broken into component pieces, unlike IRR.
 IRR has the advantage that it depends only on the properties of the cashflow
stream and not on the prevailing interest rate.
 In a situation where the proceeds of the investment can be repeatedly invested in
the same type of project but scaled in size, it makes sense to select the project with
the largest IRR – in order to get the greatest growth of capital.
 On the other hand, suppose that the investment is a one-time opportunity and
cannot be repeated. Then the NPV is the appropriate criterion, since it compares
the investment with what could be obtained through normal channels.
 Theorists (not practitioners) believe that the best criterion is that based on NPV as
if used intelligently it will provide consistency and rationality.
 There are many other factors that influence a good PV analysis-
 One significant issue is the selection of the interest rate to be used in
calculation, because there are several risk-free rates of interest in the
financial market.
 Also, generally the rates for borrowing are typically slightly higher than
those for lending.
 PV by itself doesn’t reveal much about the rate of return. Two alternative
investments might each have a NPV of $100, but one might require an
investment of $100 whereas the other require $1,000,000.
 NPV is not the whole story. It forms a solid starting point, but one must
supplement its use with additional structure.

CHAPTER 3

Fixed-Income Securities

THE MARKET FOR FUTURE CASH

 Securities. If there is a very developed market for an instrument so that it can be
traded freely, then that instrument is termed as security.
 Fixed Income Securities are financial instruments that are traded in well-
developed market and promise fixed income to the holder over a period of time.
They represent the ownership of definite cashflow stream.
 Saving Deposits. These are interest bearing bank deposit offered by commercial
banks, savings and loan institutions, and credit unions.
Examples. Demand Deposits, Time deposit account, certificate of deposit (CD).
 Money Market. The term money market relates to the market for short term (1
year or less) loans by corporations and financial institutions. For Example:
Commercial Paper, banker’s acceptance, Eurodollar deposits, Eurodollar CD’s etc.
 Government Securities. The Government obtains loans by issuing various types
of fixed-income securities. These securities are considered to be of the highest
credit quality since they are backed by the government itself. Common G-secs are:
 U.S. Treasury Bills. issued in denominations of $10,000 or more; fixed
terms to maturity – 13,26,52 weeks.
 U.S. Treasury Notes. sold in small denominations like $1000; maturities of
1-10 yrs.; owner receive a coupon payment every 6 months until maturity.
 U.S. Treasury Bonds. similar to treasury notes; maturity of more than 10
yrs.; some treasury bonds are callable.
  U.S. Treasury Strips. bonds issued in stripped form; each coupon and
principal are issued separately; generates a single cashflow with no
intermediate coupon payments; termed as zero-coupon bonds.
 Other Bonds. Bonds are issued by agencies of the federal govt, by state and local
governments and by corporations.
 Municipal Bonds. are issued by agencies of state and local governments.
There are two main types: general obligation bonds (backed by governing
body) and revenue bonds (backed by revenue or the agency responsible for
the project). The interest income associated with municipal bonds is
exempted from federal income tax, state tax and local tax.
 Corporate Bonds. are issued by corporations for the purpose of raising
capital for operations and new ventures. Some corporate bonds are traded
on exchange but most are traded over the counter. Features are: callable
bonds, sinking funds, debt subordination.
 Mortgages. A mortgage looks like the opposite of a bond. A future homeowner
usually will sell a home mortgage to generate immediate cash to pay for a home,
obligating himself to make periodic payments to the mortgage holder. Mortgage is
not usually thought of as securities, since they are written as contracts between
two parties, for e.g., homeowner and a bank. Mortgage are typically bundled into
large packages and traded among financial institutions. These mortgage-backed-
securities are quite liquid.
 Annuities. An annuity is a contract that pays the holder money periodically,
according to the predetermined schedule or formula over a period of time.

VALUE FORMULA

 Perpetual Annuities/Perpetuity. Perpetuity which pays a fixed sum periodically

forever. The PV of perpetuity can be easily derived. Suppose an amount ‘A’ is paid
at the end of each period, starting from the end of the 1stperiod and suppose that
the pre-period interest rate is ‘𝑟’.
A
𝐏=
(1 + r)
𝑨 𝑨 𝑨 𝑨 𝑷
𝑷= 𝒌
= + 𝒌
= +
(𝟏 + 𝒓) 𝟏+𝒓 (𝟏 + 𝒓) 𝟏+𝒓 𝟏+𝒓
𝒌 𝟏 𝒌 𝟐
 This is an infinite G.P. with common difference ,
𝐴
𝑃=
𝑟
 The present value 𝑃 that pays an amount ‘𝐴’ every period, beginning one period
from the present, is 𝑃 = .
  Finite-Life Streams. Suppose that the stream consists of 𝑛 periodic payments of
amount 𝐴, starting at the end of the current period and ending at period 𝑛.
 Then PV of finite stream will be:
𝐴 𝐴 𝐴
𝑃= + + ⋯+
1 + 𝑟 (1 + 𝑟) (1 + 𝑟)
 This is a finite G.P. with common difference ,

𝐴 (1 − ( ) ) 𝐴 1
𝑃= = = 1−
(1 + 𝑟) 1− 𝑟 (1 + 𝑟)
 Annuity Formula. Consider an annuity that begins payment one period from the
present paying an amount ‘𝐴’ each period for a total of 𝑛 periods. The PV i.e. P, 1
period annuity amount 𝐴 and the no. of periods are related by: 𝑃 = 1−( )
 Rewriting this equation as 𝐴 = 𝑓 (𝑃)
𝑟(1 + 𝑟) 𝑃
𝐴=
(1 + 𝑟) − 1
 Amortization: We need to express A as a function of P. This determines a periodic
payment that is equivalent to an initial payment of P. This process of substituting
periodic payments for current obligation is termed as amortization.
E.g.: Amount borrowed = $1000, r = 12%; you have to pay equal monthly payments
of such magnitude as to repay this loan over 5 years.
( )
 Using, 𝐴 = ( )
. ( . ) ×
 𝐴= = $22.2/month
( . )
Annual Percentage Rate
APR: An interest rate, or a nominal interest rate, refers only to the interest charged
on a loan, and it does not take any other expenses into account. In contrast, APR is
the combination of the nominal interest rate and any other costs or fees involved
in procuring the loan. As a result, an APR tends to be higher than a loan's nominal
interest rate.
It is that rate when, if applied to the loan amount without fees and expense,
would result in monthly payment exactly as before.
For example, if you were considering a mortgage for $200,000 with a 6% interest
rate, your annual interest expense would amount to $12,000, or a monthly
payment of $1,000. But say your home purchase also requires closing costs,
mortgage insurance and loan origination fees in the amount of $5,000. In order to
determine your mortgage loan's APR, these fees are added to the original loan
amount to create a new loan amount of $205,000. The 6% interest rate is then used
to calculate a new annual payment of $12,300. Divide the annual payment of
$12,300 by the original loan amount of $200,000 to get an APR of 6.15%.

For 30 years, At APR of 7.883%, loan amount is $203,150.

 . .
( ) ( ) ×
Monthly Payment, 𝐴 = = . = $1474
( ) ( )

At nominal interest 7.625% and the above monthly payment, the total amount of
loan $208,267 (using, 𝑃 = 1−( )
.
Total fee and expense are thus $208267- $203150 = $5117.
Since 1 point is the mortgage fee i.e. $2032(1% of $203150)
hence rest $(5117-2032) =$3085 is the other expense.

Running Amortization. Part payment of loan. It is a schedule showing paying off

the loan by making regular principal reductions.
E.g. 1: Same Example $1000|12% p.a. |A=22.22
Previous Payment Current Principal New
Balance Received Interest Amount Balance
(A) (B) (C) (D) = (B) (A)-(D)
–(C)
Jan 1 - - - 1000
Feb 1 1000 22.20 10 12.20 987.80
Mar 1 987.80 22.20 9.88 12.32 975.48
Apr 1 975.48 22.20 9.76 12.44 963.04
And the schedule continues until the balance becomes 0.

E.g. 2: A business decides to take a $5000 loan from the local bank for a period of 5
years at 9% interest rate.

 What will be the instalment for the above case?

 Will the schedule differ in case of fixed principal amount is deducted each
year rather than fixed instalment?
Fixed Instalment Payment
Year Beginning Total Interest Principal Ending
Balance Payment Paid Paid Balance

1 $5,000.00 $1,285.46 $450.00 $835.46 $4,164.54

2 4,164.54 1,285.46 374.81 910.65 3,253.88
3 3,253.88 1,285.46 292.85 992.61 2,261.27
4 2,261.27 1,285.46 203.51 1,081.95 1,179.32
5 1,179.32 1,285.46 106.14 1,179.32 .00
Totals $6.427.30 $1,427.31 $5,000.00
 Fixed Principal Payment
Year Beginning Total Interest Principal Ending
Balance ($) Payment ($) Paid ($) Paid ($) Balance
($)
1 5,000 1,450 450 1,000 4,000
2 4,000 1,360 360 1,000 3,000
3 3,000 1,270 270 1,000 2,000
4 2,000 1,180 180 1,000 1,000
5 1,000 1,090 90 1,000 0
Totals 6,350 1,350 5,000

Here in place of constant or fixed instalment, a fixed proportion of

principal is written off each year.

Annual worth. Annual Worth uses constant level cash flow for comparison. Defined
as the equivalent uniform annual worth of all estimated receipts (income) and
disbursements (costs) during the life cycle of a project.

Example. A machine of $100,000 today is expected to generate additional revenues

of $25,000 for next 10 years at the end of each year. If discount rate is 16%, is the
investment profitable?
. ( . ) ×
Equivalent Cost of machine is = 20690
( . )

Hence Annual Worth is $25000-$20690 = $4310. A positive worth signifies

profitability.

BOND DETAILS

 Bond is an obligation by the bond issuer to pay money to the bond holder
according to the rules specified at the time bond is issued.
accrued interest
AI = × coupon amount

 If a bond is purchased mid-way i.e. if coupon is to be received in 6 months and

you get your first coupon after 3 months, you receive an interest of extra three
months which you must pay to the previous owner who loses. This is accrued
interest.

Suppose we purchase a treasury bond of May 8 that matures on Aug 15, in some
distance year. Coupon Rate- 9%, coupon payments are made on Feb 15 and Aug 15.

Feb 15-May 8 (183 days)

May 8-Aug 15 (99 days)
  AI = × 4.5 = 2.04%
Hence $20.50 will be added back for a bond of FV=$1000.
 Bid Price (a price at which dealer is willing to pay for the bond) Ask Price (a price at
which the dealer is willing to sell a bond).

YEILD

Rate at which present value of stream of payments and final redemption value are equal
to current price. It is the IRR of the bond. The general formula for price of bond:

𝑃= + 1− 
[ (/ )] 

Where λ is the yield to maturity (YTM), F is the Face Value, C is the coupon payment and
n is coupon periods remaining to maturity.

Yield of different bonds adjust to each other. You will not want to buy a bond with 6%
yield when all others pay 8%.
Yield and Price are closely linked.
The price-yield curve shows the relation. A 10% bond pays 10% coupon over 30-year
period.

Yield and Price have inverse relation. A bond with YTM=0 means that face value and
coupon payments are not discounted or future streams are not discounted. Then for this
10% bond, price is equal to $300 of coupon (for 30 years) and $100 for face value i.e. $400.
Also, for YTM=10% this bond has Price equal to $100. Such bonds for which YTM =
Coupon rate are called Par Bonds.
 Higher the maturity, steeper is the yield curve implying greater sensitivity of prices to
yield.
Such changes in yield which result in price change pose a interest rate risk. Although this
does not affect the stream of payments but affects the price at the time of sale of such
bond.

Current Yield: Annual return on bond i.e. (Annual coupon/Bond Price) x 100

DURATION

 Bond prices with longer maturity are more sensitive to changes to interest rates. But
maturity is incomplete measure.
 Duration helps in understanding the sensitivity of interest. It is the weighted average of
the times the payments are made with weight being present value of individual cash
flows.
PV(t )t + PV(t )t + PV(t )t + ⋯ + PV(t )t
𝐷=
𝑃𝑉

where PV (𝑡 ) are the present values of cash flow occurring at time 𝑡 . D is itself in time
and will be 𝑡 ≤ 𝐷 ≤ 𝑡 .
 A zero-coupon bond making a single final payment at maturity will have duration
equal to maturity date.
 Duration measures the average maturity of the bond’s promised cash flows. It shows
the weighted average life of the bond taking into account the size and timing of cash
flows. Duration is a weighted average of the maturities of the cash payments.
 It shows the length of time that lapses before the average rupee/dollar present value
from the bond is received. Since the above formula is vague in describing the rate of
discount, Yield can be used to define the present value based weights.
 All else being equal, the longer the term to maturity of a bond, the longer its
duration.
 𝒏 𝒏
𝑪𝑷𝒕 𝑪𝑷𝒕
𝑫𝑼𝑹 = 𝒕 /
(𝟏 + 𝒊)𝒕 (𝟏 + 𝒊)𝒕
𝒕 𝟏 𝒕 𝟏
𝐷𝑈𝑅 = duration
𝑡 = years until cash payment is made
𝐶𝑃 = cash payment
𝑖 = interest rate
𝑛 = years to maturity of the security

Example:
Face Value: $100
Coupon Rate: 15% annually
Years to Maturity: 6
Redemption Value: $100
Yield to Maturity: 18%
Current Market Price: $89.50
Bond Duration at 18% YTM
Year Cash Present Proportion Proportion of the
Flow Value of bond’s bonds value x
@ 18% value times
1 15 12.71 0.142 0.1420
2 15 10.77 0.120 0.2407
3 15 9.13 0.102 0.3060
4 15 7.74 0.086 0.3458
5 15 6.56 0.073 0.3663
6 115 42.60 0.476 2.8556
89.51 4.2564

 All else being equal, when interest rates rise, the duration of a coupon bond falls.
Example: Face Value: $100
Coupon Rate: 15% annually
Years to Maturity: 6
Redemption Value: $100
YTM:20%
Bond Duration at 20%
Year Cash Present Proportion Proportion of the
Flow Value of bond’s bonds value x
@ 20% value times
1 15 12.50 0.150 0.1499
2 15 10.42 0.125 0.2499
3 15 8.68 0.104 0.3124
4 15 7.23 0.087 0.3471
5 15 6.03 0.072 0.3615
 6 115 38.51 0.462 2.7717
83.37 4.1924

 All else being equal, the longer the term to maturity of a bond, the longer its
duration.
Example: Face Value: $100
Coupon Rate: 15% annually
Redemption Value: $100
Yield to Maturity: 18%
Years to Maturity: 7
Bond Duration at 18% YTM with 7 years maturity
Year Cash Present Proportion Proportion of the
Flow Value of bond’s bonds value x
@ 20% value times
1 15 12.71 0.144 0.144
2 15 10.77 0.122 0.243
3 15 9.13 0.103 0.309
4 15 7.74 0.087 0.349
5 15 6.56 0.074 0.370
6 15 5.56 0.063 0.376
7 115 36.10 0.408 2.853
88.57 4.645

 Also, all else being equal, the higher the coupon rate on the bond, the shorter the
bond’s duration.
 Macaulay Duration.
Macaulay duration D is defined as
∑ c /[1 + /m)]
D=
PV
Where  is the yield to maturity and
𝑐
𝑃𝑉 =
[1 + (/𝑚)]

Where 𝑘/𝑚 is total time.

Macaulay duration formula The Macaulay duration for a bond with a coupon rate 𝑐
per period, yield 𝑦 per period, 𝑚 periods per year, and exactly 𝑛 periods remaining
is.
1+𝑦 1 + 𝑦 + 𝑛(𝑐 − 𝑦)
𝐷= −
𝑚𝑦 𝑚𝑐[(1 + 𝑦) − 1] + 𝑚𝑦
 More Properties on Duration:
The table shows various bonds with different coupons with different maturities but
constant yield of 5%
Duration of a Bond Yielding 5%
As Function of Maturity and Coupon Rate
Coupon rate
Years to 1% 2% 5% 10%
maturity
1 997 995 988 977
2 1,984 1,969 1,928 1,868
5 4,875 4,763 4,485 4,156
10 9,416 8,950 7,989 7,107
25 20,164 17,715 14,536 12,754
50 26,666 22,284 18,765 17,384
100 22,527 21,200 20,363 20,067
∞ 20,500 20,500 20,500 20,500
Duration does not increase appreciably with maturity. In fact, with fixed yield,
duration increases only to a finite limit as maturity is increased.

Since duration is always less than maturity, when maturity tends to infinity, the duration
does not tend to infinity.
Secondly, Duration does not rapidly vary with coupon rate as constant yield cancels out
that variation.
Duration and Sensitivity
Sensitivity of price changes to yield is captured by Duration.
𝑐
𝑃𝑉 =
[1 + (/𝑚)]

Derivate the above formula w.r.t. λ

𝑑𝑃𝑉 −(𝑘/𝑚)𝑐 𝑘/𝑚
= =− 𝑃𝑉
𝑑 [1 + (/𝑚)] 1 + (/𝑚)
Now since Price P is sum of PVk. Thus, using this relation, we get:

𝑑𝑃 𝑑𝑃𝑉 (𝑘/𝑚)𝑃𝑉 1
= =− =− 𝐷𝑃 = −𝐷 𝑃
𝑑 𝑑 1 + (/𝑚) 1 + (/𝑚)

DM is defined as Modified Duration.

𝐷 = 𝐷/[1 + (/𝑚)]
 It tends to equal to normal Duration D for large value of m or small value of λ.
= −𝐷

Defining the rate of change of price:
This equation can be modified to view the effect of small change in yield on price as:
∆𝑃 ≈ −𝐷 𝑃 ∆
The greater the duration of a security, the greater the percentage change in the market
value of the security for a given change in interest rates. Therefore, the greater the
duration of a security, the greater its interest-rate risk.

Duration of Portfolio
For several bonds’ different maturities for a single fixed income security. Assuming the
yield of all bonds to be same, the duration of portfolio is weighted sum of durations of
individual bonds.
∑ 𝑡 𝑃𝑉
𝐷 =
𝑃
∑ 𝑡 𝑃𝑉
𝐷 =
𝑃
Hence,

𝑃 𝐷 +𝑃 𝐷 = 𝑡 (𝑃𝑉 + 𝑃𝑉 )

Which gives, upon division by 𝑃 = 𝑃 + 𝑃 ,

𝐷= +

IMMUNISATION
 Immunization of a bond is a process of creating a bond portfolio through which the
interest rate risk and investment risk can either be eliminated or minimized.
Immunization cannot be achieved through single bond and creating a bond portfolio is
necessary for immunization.
 As soon as the duration of the portfolio match with the target period, then it is almost
immune to interest rate fluctuations.
 Example: Target period=3 years, $10,00,000 at present
Particulars A B
F.V. 1000 1000
M.P. 986.5 1035
Duration 5 2
Amount of money to be invested in each of
the bond:
3 = 𝐷𝑝
3 = 𝑊1𝐷𝐴 + 𝑊2𝐷𝐵
And, 𝑊1 + 𝑊2 = 1
 𝑊1 = 1 − 𝑊2
 3 = (1 − 𝑊2)5 + 𝑊2 × 2
 𝑊2 = 2/3 𝑎𝑛𝑑 𝑊1 = 1/3.
Also, 𝑊𝑖 =
 =
, ,
 𝑃 = 3,33,333.333
 Similarly, 𝑃 = 6,66,666.66.
Example 2: $1 million obligation is to be paid in 10 years. Since zero-coupon
bonds are unavailable, you decide to invest in three corporate bonds as shown:
Bond Choices
Rate Maturity Price Yield
Bond 1 6% 30 yr 69.40 9.00%
Bond 2 11% 10 yr 113.01 9.00%
Bond 3 9% 20 yr 100.00 9.00%
Three bonds are considered for the X corporation’s immunized portfolio
To match the obligation, we need to construct a portfolio using these three bonds. In
order to do so, we need the duration of these three bonds.
Since the duration of Obligation is 10 years, we need to match the duration of cash
flows with the duration of obligation. So first we find the duration.
For the given prices in decimal and not 32nd of a point, durations calculated at 9% of
yield are as:
D1 =11.44
D2 = 6.54
D3 = 9.61
Now using the combinations of two bonds, the durations must match. D2 = 6.54 and
D3 = 9.61 cannot be used since they fall short of the obligation of 10 years. So let’s
combine one such bond with a longer duration bond. So we use D1 and D2.
Next, we need to find present value of Obligation which comes out to be $414642.8.
To immunise a portfolio, we need to solve two equations:
𝑉 + 𝑉 = 𝑃𝑉
𝐷 𝑉 + 𝐷 𝑉 = 10𝑃𝑉

i.e., the value of money invested today in the portfolio of two bonds must equal the
present value of obligation. And, the duration of portfolio must equal the duration of
obligation.
And, the duration of portfolio must equal the duration of obligation.
𝑉 = $292,788.73 and 𝑉 = $121,854.27

This is the total value of investment in each of bonds 𝐷 and 𝐷 . In order to find the
number of bonds purchased for each type, we must divide the value of investment
with respective price of the bonds.
Hence 𝑄 = 4241 and 𝑄 = 1078.
The table shows the equality of present value of portfolio and obligation at 8% and
10% yield. The value of portfolio remains equal to that of obligation at each yield.
Hence, such a portfolio is considered to be immunised as immediate changes in yield
result in new portfolio which matches the new obligation at this new yield.
Once the yield changes, one must rebalance or re-immunise at new rate.
 Immunization Result
9.0
Bond 1
Price 69.04
Shares 4,241.00
Value 292,798.64
Bond 2
Price 113.01
Shares 1,078.00
Value 121,824.78
Surplus -19.44

 Immunisation does suffer from shortcomings. Firstly, all yields are not equal. And it is
hard to find a combination of long-term and short-term bonds with identical yields.
Also, given this, if yield changes, it is not necessary for yields on all such bonds to
change with equal effect which would make rebalancing difficult.
 CHAPTER 4

The Term Structure of Interest Rates

Yield Curve

 Yield curve displays yield as a function of time to maturity.

 Yields trace out an essentially smooth curve which rises gradually as the time to
maturity increases.
 Long bonds tend to offer higher yields than short bonds of the same quality. If long
bonds happen to have lower yields than short bonds, the result is said to be an inverted
yield curve.

The Term Structure

 The interest rate charged depends upon length of the time the funds are held is a basis of
term structure theory.
 Spot Rate: The Spot rate ‘𝑆 ’ is the rate of interest expressed in yearly terms charged for
money held from the present time(t = 0) until time t. Both the interest and the original
principal are paid at time t.
𝑺𝟏 < 𝑺𝟐 < 𝑺𝟑 ......

SPOT RATE CURVE

 a) Yearly. Under the yearly compounding convention, the spot rate 𝑆 is defined as
(𝟏 + 𝑺𝒕 )𝒕 is the factor by which a deposit held for 𝑡 years will grow.
b) 𝒎 periods per year. Under the 𝑚 periods per year compounding convention, the
spot rate 𝑆 is defined as (𝟏 + 𝑺𝒕 /𝒎)𝒎𝒕 is the corresponding factor.
c) Continuous. Under a continuous compounding convention, the spot rate 𝑆 is
defined so that 𝑒 . is the corresponding growth factor. This formula applies
directly to value of 𝑡.

Spot rate can be measured by recording the yield of zero-coupon bonds. Since a
zero-coupon bond promises to pay a fixed amount in the future, the ratio of
payment amount to the current price defines the spot rate for the maturity date of
the bond.

 Discount factor and present value: These are the factors by which future cashflows must
be multiplied to obtain an equivalent present value.
(a) Yearly for yearly compounding
1
𝑑 =
(1 + 𝑠 )
(b) 𝒎 periods per year for compounding 𝑚 periods per year,

1
𝑑 =
(1 + 𝑠 /𝑚)
(c) Continuous for continuous compounding,
d =𝑒

The discount factor transforms future cashflows directly into an equivalent present
value. Hence, given any cash flow stream,
𝑃𝑉 = 𝑥 + 𝑑 𝑥 + 𝑑 𝑥 + ⋯ + 𝑑 𝑥

 Determining the spot rate: Consider a 2-year bond. Suppose that bond has price ‘𝑝’,
makes coupon payments of amount ‘𝑐’ at the end of both years and has face value of ‘𝐹’,
then the price is equal to the discounted value of cashflow stream.
𝑪 𝑪+𝑭
𝑷= +
(𝟏 + 𝑺𝟏 ) (𝟏 + 𝑺𝟐 )𝟐

 Spot rate can also be determined by a subtraction process. Two bonds of different
coupon rates but identical maturity can be used to construct an equivalent zero-coupon
bond.
 Forward Rates

 Forward rates are interest rates for money to be borrowed between two dates in future
but under terms agreed upon today.

Let’s take a 2-year situation. Suppose S1 and S2 are known.

2-Alternatives:
o A - We leave $(1 + 𝑆 ) in a 2-year account.
o B - We place $1 in a one-year account and simultaneously make arrangements that
the proceeds, $(1 + 𝑆 ), will be lent for 1 year starting a year from now at a pre
agreed rate say 𝑓.
The final amount of money we will receive at the end of 2 years under:
o alternative A will be $(1 + 𝑆 )
o alternative B will be $(1 + 𝑆 )(1 + 𝑓).
 The concept of arbitrage states that due to market forces the above two will be equal, i.e.
(1 + 𝑆 ) = (1 + 𝑆 )(1 + 𝑓).
(1 + 𝑆 )
𝑓= −1
1+𝑆

 The forward rate between times 𝑡 and 𝑡 with 𝑡 < 𝑡 is denoted by 𝑓 , .It is a rate of
interest charged for borrowing money at time 𝑡 which is to be repaid with interest at
time 𝑡 . Generally, if j > i, then:
(a) Yearly: For yearly compounding, the forward rates satisfy, for j > i,
(1 + 𝑠 ) = (1 + 𝑠 ) (1 + 𝑓 . )
Hence,
/( )
(1 + 𝑠 )
𝑓. = −1
(1 + 𝑠 )

(b) 𝑚 periods per year. For m period per- year compounding, the forward rates
satisfy, for j > i, expressed in periods,
(1 + 𝑠 /𝑚) = (1 + 𝑠 /𝑚) (1 + 𝑓 . /𝑚)
Hence,
/( )
( / )
𝑓. = 𝑚 −𝑚
( / )

(c) Continuous For continuous compounding, the forward rates 𝑓 , are defined for
all 𝑡 and 𝑡 with 𝑡 > 𝑡 and satisfy
𝑒 =𝑒 𝑒 ( )

Hence,
 𝑠 𝑡 −𝑠 𝑡
𝑓 . =
𝑡 −𝑡

Term Structure Explanations

 Expectations Theory. The first expectation is that spot rates are determined by
expectations of what rates will be in the future. The 2-year rate is greater than the 1-year
rate, this is so because the market believes that the 1-year rate will most likely go up next
year. This majority belief that the interest rate will rise translates into a market
expectation. This argument is made more concrete by expressing the expectations in
terms of forward rates. This more precise formulation is the expectations hypothesis.
Weakness. According to this expectation, the primary weakness is that the market
expects rates to increase whenever the spot rate curve slopes upward; and this is
practically all the time. Thus, the expectations cannot be right even on average, since
rates don’t go up as often as expectations would imply.
 Liquidity Preference. The Liquidity Preference explanation asserts that investors usually
prefer short-term fixed income securities over long term securities. Investors do not like
to tie up capital in long term securities, since those funds may be needed before the
maturity date. Investors prefer their funds to be liquid rather than tied up.
 Market segmentations. The market segmentation explanation argues that the market for
fixed-income securities is segmented by maturity dates. This argument assumes that
investors have a good idea of the maturity date that they desire, based on their projected
need for future funds or their risk preference. The argument concludes that the group of
investors competing for long-term bonds is different from the group competing short-
term bonds. This viewpoint suggests that all points on the spot rate curve are mutually
independent.
 CHAPTER 6

Mean-Variance Portfolio Theory

ASSET RETURN
 An investment that can be bought and sold is frequently called an asset.
 Suppose you purchase an asset at time zero, for X0 and 1 year later you sell that asset
for X1, then the total return (𝑅) will be:
Total return =
 𝑅=
 Rate of return = =
 𝑅 = 1+𝑟
 𝑋 = (𝟏 + 𝒓)𝑋
 Short sales. The process of selling an asset that we do not own.
Borrow the asset---Sell at 𝑋 (receive) --- At later date, repay the borrower at 𝑋 (pay)
 Profit = 𝑋 - 𝑋
Hence, short selling is profitable if the asset price declines.
Here Potential for loss is unlimited because 𝑋 can increase arbitrarily.
 Portfolio Return.
o Portfolio is a master asset with n different assets.
o Suppose amount invested (in ith asset) is 𝑋 ; 𝑖=1,2,3,…..,𝑛.
Note: If short selling is allowed, 𝑋 can be negative.
o The amount invested can be represented as a fraction of total investment, i.e. 𝑋 =
𝑤 𝑋 ; where ∑ 𝑤 = 1
o Total return of ith asset, 𝑅 =
 𝑋=𝑅𝑋
 𝑋 =𝑅𝑤𝑋
 𝑋 =∑ 𝑅 𝑤 𝑋
∑

R= = =∑𝒏𝒊 𝟏 𝑤 𝑅


Similarly, r = ∑𝒏𝒊 𝟏 𝑤 𝑅 .
RANDOM RETURNS
 Expected value (or mean): E(𝑟 ) = 𝑟̅
 Variance: 𝐸[(𝑟 − 𝑟̅ 2)] = 𝜎
 Mean-standard deviation diagram: Random rate of return of asset can be represented on
a 2-D diagram called Mean-standard deviation diagram. Here standard deviation is used
as the horizontal axis because that gives both axes comparable units.
PORTFOLIO MEAN AND VARIANCE
 Mean Return of a portfolio. 𝐸(𝑟) = 𝑤 𝐸(𝑟 ) + 𝑤 𝐸(𝑟 ) + ⋯ + 𝑤 𝐸(𝑟 )
 Variance of a Portfolio Return. 𝜎 = 𝐸[(𝑟 − 𝑟̅ ) ]
= 𝐸[(∑ 𝑤 𝑟 − ∑ 𝑤 𝑟̅ ) ]
= 𝐸 (∑ 𝑤 (𝑟 − 𝑟̅ )) ∑ 𝑤 (𝑟 − 𝑟̅ )
= 𝐸 ∑ . 𝑤 𝑤 (𝑟 − 𝑟̅ )(𝑟 − 𝑟̅ )
=∑. 𝑤𝑤𝜎

If 𝑟 = 𝑤 𝑟 + 𝑤 𝑟 , then 𝑣(𝑟) = 𝑤 𝑣 (𝑟 ) + 𝑤 𝑣 (𝑟 ) + 2𝑤 𝑤 𝑐𝑜𝑣(𝑟 , 𝑟 )

 Diversification: “Don’t put all your eggs in one basket”

The variance of the return of a portfolio can be reduce by including additional assets
in the portfolio. This process is called diversification.
2-situations.
Situation A: n-assets; mean rate of return of each asset is m; cov(i,j)=0 i.e. assets are
not correlated; 𝑤 = .
So, 𝑟̅ = ∑ 𝑟 𝑤 =𝑟 𝑤 + 𝑟 𝑤 + 𝑟 𝑤 … … … + 𝑟 𝑤
 𝑟̅ = m. +m. +m. + ………+ m. = m
 𝒓=𝐦
Now, Var = ∑ . 𝑤 𝑤 𝜎 and cov(i,j)=0,
∴, var = 𝑤 𝜎11 + 𝑤 𝜎22 + ⋯ … + 𝑤 𝜎33.
 var = σ2+ σ2+……+ σ2.
𝛔𝟐
 var =
𝒏
 As n ⬆var ⬇, i.e. as diversification ⬆var ⬇.
Situation B: n-assets; mean rate of return of each asset is m; cov(i,j)=0.3σ2 i.e. assets
are correlated; wi= .
So, 𝑟̅ = ∑ 𝑟 𝑤 =𝑟 𝑤 + 𝑟 𝑤 + 𝑟 𝑤 … … … + 𝑟 𝑤
 𝑟̅ = m. +m. +m. + ………+ m. = m
 𝒓=𝐦
 Var = ∑ . 𝑤𝑤𝜎 =∑, . .σi,j = .∑ , σ, = .[∑ σ, + ∑ σ, ]

n cases n(n-1) cases

 Var = .[𝑛𝜎 + 𝑛(𝑛 − 1) 𝜎 ] = .[𝑛𝜎 + 𝑛(𝑛 − 1) 0.3𝜎 ]

.
 Var = + 0.3σ
Here, var cannot be zero even if n tends to infinity.
 100% Diversification not possible.
As n ⬆var ⬇, i.e. as diversification ⬆var ⬇.

 Diagram of a Portfolio:
Suppose 2 assets on a mean variance diagram. These 2 assets can be combined
according to weights to form a portfolio.
Let 𝑤 = 1 − 𝛼, 𝑤 = 𝛼 ; 0 ≤ 𝛼 ≤ 1 (If weights are negative then shorting is allowed).
As α varies, then new portfolio trace out a curve that include asset 1 and 2.

Portfolio diagram lemma. The curve in an 𝑟̅ − 𝜎 diagram defined by non negative

mixtures of two assets 1 and 2 lies within the triangular region defined by the two
original assets and the point on the vertical axis of height 𝐴 = (𝑟̅ 𝜎 )(𝑟̅ 𝜎 )/(𝜎 + 𝜎 )
Proof. The rate of return of the portfolio defined by  is 𝑟() = (1 −  )r + r . The
mean value of this return is
 𝒓(  ) = (1 −  )r̅ + r̅
This says that the mean value is between the original means, in direct proportion to
the proportions of the assets. In a 50-50 mix, for example, the new mean will be
midway between the original means.
Let us compute the standard deviation of the portfolio. We have, from the general
formula of the previous section,
𝜎(𝛼) = (1 − 𝛼) 𝜎 + 2𝛼(1 − 𝛼)𝜎 +𝛼 𝜎

Using the definition of the correlation coefficient 𝜌 = 𝜎 /(𝜎 𝜎 ), this equation can be
written
𝜎(𝛼) = (1 − 𝛼) 𝜎 + 2𝜌𝛼 (1 − 𝛼)𝜎 𝜎 + 𝛼 𝜎
This is quite a messy expression. However, we can determine its bounds. We know
that 𝜌 can range over −1 ≤ 𝜌 ≤ 1. Using 𝜌 = 1 we find the upper bound
𝜎(𝛼)∗ = (1 − 𝛼) 𝜎 + 2𝛼 (1 − 𝛼)𝜎 𝜎 + 𝛼 𝜎
= [(1 − 𝛼)𝜎 + 𝛼𝜎 ]
= (1 − 𝛼)𝜎 + 𝛼𝜎

Using 𝜌 = −1 we likewise obtain the lower bound

𝜎(𝛼)∗ = (1 − 𝛼) 𝜎 − 2𝛼 (1 − 𝛼)𝜎 𝜎 + 𝛼 𝜎
= [(1 − 𝛼)𝜎 − 𝛼𝜎 ]
= (1 − 𝛼)𝜎 − 𝛼𝜎

∴The two linear expression together with the linear expression of mean trace out a
kinked line.

THE FEASIBLE SET- Set of points that correspond to the portfolio.

It has 2 imp properties:
1. If there are at least 3 assets (not perfectly correlated and with different means), then the
feasible set will be a 2-D solid region.
2. The feasible region is convex to the left.

 The minimum-variance set and the efficient frontier

o The left boundary of a feasible set is called a minimum variance set.
o There is a special point on this set having minimum variance. It is termed as the
minimum variance point (MVP).
o An investor who agrees with the view point is said to be risk averse, since he
or she seeks to minimise risk. An investor who would select a point other than
the one of minimum standard deviation is said to be risk preferring.
o Nonsatiation: Everything else being equal investors always want more money;
hence they want the highest possible expected return for a given standard
deviation.
o These arguments imply that only the upper part of the minimum-variance set
will be of interest to investors who are risk averse and satisfy nonsatiation. This
upper portion of minimum-variance set is termed as the efficient frontier. This
provides the best mean-variance combinations for most investors.

THE MARKOWITZ MODEL

Consider an n asset scenario each with r as mean rate of return and σ as the individual
standard deviations with weights of n assets defined as 𝑤 . (Short selling weights are
allowed). For some value of portfolio mean, we find the portfolio of minimum variance
as:

Minimize ∑ . 𝑤𝑤𝜎
 Subject to ∑ 𝑤 𝑟̅ = 𝑟̅
∑ 𝑤 =1

The problem is for single period investment and relates to trade-off between expected
rate of return and variance of these returns. To solve, we construct a Lagrange Multiplier
equation.

1
𝐿= 𝑤𝑤𝜎 − 𝑤 𝑟̅ − 𝑟̅ − 𝜇 𝑤 −1
2
.

Differentiate the equation with respect to weights 𝑤 and equate to zero. This exercise can
be done for a two-variable case.
𝑑𝐿 1
= (2𝜎 𝑤 + 𝜎 𝑤 + 𝜎 𝑤 ) − 𝑟̅ − 𝜇
𝑑𝑤 2
𝑑𝐿 1
= (𝜎 𝑤 + 𝜎 𝑤 + 2𝜎 𝑤 ) − 𝑟̅ − 𝜇
𝑑𝑤 2

using the fact that 𝜎 = 𝜎 and setting these derivatives to zero, we obtain

𝜎 𝑤 + 𝜎 𝑤 − 𝑟̅ − 𝜇 = 0
𝜎 𝑤 + 𝜎 𝑤 − 𝑟̅ − 𝜇 = 0

So, we can say for Efficient Set

∑ 𝜎 𝑤 − 𝑟̅ − 𝜇 = 0 for 𝑖 = 1,2, … 𝑛

∑ 𝑤 𝑟̅ = 𝑟̅
∑ 𝑤 =1

𝑛 equations for covariance and two equations for constraints i.e. 𝑛 + 2 equations.

TWO FUND THEOREM:

The minimum variance set satisfies the system of 𝑛 + 2 linear equations.

Suppose there are two known solutions:

 ,𝜇 ,𝑤 = 𝑤 ,𝑤 ,…..,𝑤
𝑎𝑛𝑑

 ,𝜇 ,𝑤 = 𝑤 ,𝑤 ,…..,𝑤
each with expected returns as 𝑟 𝑎𝑛𝑑 𝑟 . Combinations can be formed by giving weights to
these two solutions i.e. α and (1 − 𝛼). Substituting them into the equations for efficient set,
the expected value of returns (𝜶)𝒓𝟏 + (𝟏 − 𝜶)𝒓𝟐 .
 The portfolio thus formed (𝜶)𝒘𝟏 + (𝟏 − 𝜶)𝒘𝟐 has legitimate weights whose sum is equal
to one.The result derived shows an important result, that suppose 𝑤 and 𝑤 are two
different portfolios in the minimum variance set, then as 𝜶 varies over −∞ < 𝜶 < ∞ the
portfolios defined by 𝜶. 𝑤 + (1 − 𝜶). 𝑤 cover the entire minimum variance set.

Alternatively, two efficient funds (portfolios) can be established so that any efficient
portfolio can be duplicated in terms of mean and variance, as a combination of these two.
In other words, all investors are seeking efficient portfolios need only to invest in
combinations of these two funds.

According to what has been discussed so far in two fund theorem, two mutual funds can
provide complete investment opportunity to everyone. This does implicitly assume that
everyone is concerned just about mean and covariances of returns and have single period
outlook.

INCLUSION OF RISK-FREE ASSET:

The previous analysis considers n assets all of which were risky i.e. some positive level of
𝜎. A risk-free asset has a deterministic return and hence zero risk. Hence a risk-free asset is
a pure interest-bearing instrument and corresponds to a lending or borrowing through
inclusion in the portfolio.

Inclusion of a risk-free assets creates mathematical degeneracy that simplifies the shape of
efficient frontier. Suppose the return on risk-free asset is 𝑟 and consider another asset
which is risk with returns as 𝑟 having mean return as 𝑟 and variance as 𝜎 .
It is obvious that covariance between the two will be zero.
Forming a portfolio using weights 𝛼 for risk-free asset and (1 − 𝛼) for risky asset. The
mean return of portfolio will be 𝛼 𝑟 + (1 − 𝛼)𝑟̅ and the standard deviation will be
(1 − 𝛼)𝜎. For a moment consider 𝜎 = 0 we see:

Mean = 𝛼𝑟 + (1 − 𝛼)𝑟̅

Standard Dev. = 𝛼𝜎 + (1 − 𝛼)𝜎

The equation show that both means and standard deviation of portfolio vary linearly with
𝛼 thus the points representing portfolio trace out to be straight line.

Taking it further, now suppose there are 𝑛 assets (with mean rate of returns as 𝑟̅ and
known covariances as 𝜎 ) are used to form a portfolio along with a risk-free asset with
rate of return 𝑟 .
 The left diagram shows the inclusion of risk-free asset converts the curved feasible set into
a triangular one. The triangle is infinite in case of borrowing of risk-free asset (or short
selling) and gets restricted when the risk-free asset is only lent.

ONE FUND THEOREM

Since the inclusion of risk-free asset forms a new feasible region, the line segment
extended from the risk-free point is tangent to the original feasible set. The point F
corresponds to tangency of the line with the overall efficient set. Any Efficient point i.e.
points on the line can be obtained/expressed as a combination of this asset and the risk-
free asset.

There is a single fund F of risky assets that any efficient portfolio can be constructed as a
combination of F and risk-free assets.
 CHAPTER 7

The Capital Asset Pricing Model

 Two major decisions for Financial Investment, firstly to decide the best possible action
or choosing a portfolio that gives optimal solution. Other decision pertains to, correct,
arbitrage-free equilibrium price of the asset.
 The focus of previous chapter was on finding such an optional portfolio.
This chapter focuses on the pricing strategy of such portfolios.

FINDING EQUILIBRIUM
 Assuming every investor is mean-variance optimiser, and every investor follows the
probabilistic method for assigning same mean values to returns on assets. Taken the
fact that transactions costs don't exist and there is also available an asset with unique
risk-free rate.
 One-fund theorem projected that every such investor will prefer the single risk fund
in addition to risk free asset. This fund will be same for all as the mean, variance and
covariances will be same.
 The mix will of Risky and Risk-free asset will vary depending upon the taste and
preference (Risk Averse vs Risk Takers) of individuals.
 Those in avoidance of risk will prefer a greater percentage of risk-free asset in the
portfolio where as aggressive investors will put in larger percentage of investment
towards risky assets.
 Lately, everyone ends up with a portfolio of risk-free asset and a single risky one fund
(which is the only fund).
 CAPM answers as the what will be that one fund. If everyone buys just one fund, and
their purchases add up to the market, then that one fund must be the Market Portfolio.
(It must contain shares of every stock in proportion to that stocks representation in the
entire market).
 Assets weight in the portfolio would then be equal to the proportion of capital devoted
to that asset. The weight of the asset would be equal to proportion of assets total capital
value/market capital value. This is termed as Capitalisation Weights.
Market Capitalization Weights
Security Shares Relative Price Capitalization Weight in
outstanding share in market
market
Jazz, Inc 10,000 1/8 $6.00 $60,000 3/20
Classical, 30,000 3/8 $4.00 $120,000 3/10
Inc
Rock Inc. 40,000 1/2 $5.50 $220,000 11/20
Total 80,000 1 $400,000 1
 Taking this example of three companies which form the market. The market weights
will be proportional to the capitalisation. The efficient fund thus created will be the
market. If price of the asset changes, the share proportion does not change, but the
capitalisation weights do change.
We also need not solve for optimal portfolio in this case as market itself the that optimal
portfolio.
We need not solve for equilibrium conditions as others will be solving the problem.

 To see why, consider that returns depend upon current and final price of the asset.
Other investors solve the mean variance problem and place orders to acquire the
portfolio. when the orders don't match, the prices must change, hence demand will
raise price of asset and vice versa. These price changes will affect estimates of returns
and the optimal portfolio will be re-calculated and the process continues until demand
equals supply: Equilibrium Argument.
 In ideal world of mean-variance, everyone will have same estimate and buys same
portfolio i.e. the Market. Price adjustment drive market to efficiency and it is the efforts
of few to adjust portfolios to price changes so that others follow. Only few people need
to devote time.
 The CAPM asks what would happen if all investors shared an identical investable
universe and used the same input list to draw their efficient frontiers. Obviously, their
efficient frontiers would be identical.
 Facing the same risk-free rate, they would then draw an identical tangent line and
naturally all would arrive at the same risky portfolio, P.
 Because the market portfolio is the aggregation of all of these identical risky
portfolios, it too will have the same weights. Therefore, if all investors choose the
same risky portfolio, it must be the market portfolio, that is, the value weighted
portfolio of all assets in the investable universe. CML.
  When we sum over, or aggregate, the portfolios of all individual investors, lending and
borrowing will cancel out (because each lender has a corresponding borrower), and the
value of the aggregate risky portfolio will equal the entire wealth of the economy.
 This is the market portfolio, M. The proportion of each stock in this portfolio equals the
market value of the stock (price per share times number of shares outstanding) divided
by the sum of the market value of all stocks. 4 This implies that if the weight of GE
stock, for example, in each common risky portfolio is 1%, then GE also will constitute
1% of the market portfolio.

THE CAPITAL MARKET LINE

 Shows the efficient set with single fund of risky asset being: Market Portfolio being
labelled as M. This is also called the price line as prices adjust such that efficient assets
fall on this line.
 The equation of capital market line 𝑟 and 𝜎 are the expected value and standard
deviation of the market rate of return of an arbitrary efficient asset.

 The slope of capital market line is 𝐾 = (𝑟̅ − 𝑟 )/𝜎 is this value called Price of Risk:
denoting how much the expected return on the portfolio should if std. deviation of that
rate increases by 1 unit.
THE PRICING MODEL

The CML relates to the expected rate of return of an efficient portfolio to its standard
deviation, but it does not relate the expected rate of return of an individual asset to its
individual risk. We can derive that using CAPM:

𝑟̅ − 𝑟 = 𝛽 (𝑟̅ − 𝑟 );

Where, 𝛽 =
Proof: For any 𝛼 consider the portfolio consisting of a portion 𝛼 invested in asset i and a
portion 1 − 𝛼 invested in the market portfolio M. (We allow 𝛼 < 0, which corresponds to
borrowing at the risk-free rate.) The expected rate of return of this portfolio is:
𝑟̅ = 𝛼𝑟̅ + (1 − 𝛼)𝑟̅
and the standard deviation of the rate of return is
𝜎 = [𝛼 𝜎 + 2𝛼(1 − 𝛼)𝜎 + (1 − 𝛼) 𝜎 ] /
For varying the value of weights alpha, we trace a curve. For 𝛼 = 0, the point corresponds
to market portfolio M. This curve will be tangent to the CML at this point and cannot cross
the CML (or else it would violate the definition of CML). We need to solve for tangency
condition, whereby slope of CML must equal the slope of curve.
First, we have
̅
= 𝑟̅ − 𝑟̅
( ) ( )
=
Thus,
| =
we then use the relation
̅ ̅ /
=
/
to obtain
̅ ( ̅ ̅ )
| =
This slope must equal the slope of the capital market line. Hence,
( ̅ ̅ ) ̅
=
We now just solve for 𝑟̅ , obtaining the final result
̅
𝑟̅ = 𝑟 + 𝜎 = 𝑟 + 𝛽 (𝑟̅ − 𝑟 )
This is clearly equivalent to the stated formula.

Observations:
 The value 𝛽 of is referred to as the beta of an asset which characterises its risk.
(𝒓𝒊 – 𝒓𝒇 ) is termed as Expected Excess Rate of Return on ith asset. The rate of
return expected to exceed the risk- free rate.
(𝒓𝑴 – 𝒓𝒇 ) is termed as Expected Excess Rate of Return on Market Portfolio.
 Hence Expected Excess Rate of Return on ith asset is proportional (by beta) to the
Expected Excess Rate of Return on Market Portfolio. Beta can also be called the
normalised version of covariance of asset with market.
 We consider a case where asset is completely uncorrelated with market i.e. β = 0
In this case, 𝒓𝒊 = 𝒓𝒇 which means that no matter how risky the asset is (𝝈𝒊 is large),
the expected rate of return will be equal to that of risk-free asset. Risk on the
asset (which is uncorrelated to the market) can be diversified.
  We could purchase small amounts of assets and resulting variance would be small
and the composite return would approach 𝒓𝒇
 Another case is where β is negative. In such cases 𝒓𝒊 < 𝒓𝒇 (even though asset 𝝈𝒊 is
large), such assets will tend to reduce overall portfolio risk when combined. Some
investors are willing to accept lower risk.
 The overall risk of portfolio is still in terms of 𝝈 but for concern of individual
assets we refer to their β’s.
 The CAPM changes our concept of risk of an asset from that of σ to that of β

Beta of A Portfolio (In Terms of Beta of Individual Assets)

For n asset portfolio where each asset holds weight 𝑤 , 𝑤 , . . , 𝑤 such that return on
portfolio
𝑟 = ∑𝑟 𝑤
For 𝐶𝑜𝑣(𝑟, 𝑟 ) = ∑ 𝑤 . 𝐶𝑜𝑣(𝑟, 𝑟 )
Thus, 𝛽 = ∑ 𝑤 𝛽
Portfolio beta is the weighted sum of the betas of individual assets in the portfolio.

The Security Market Line

Liner relation of CAPM is called SML.

The two graphs are plotted in terms of covariance and beta and represent the risk-
reward structure of an asset under CAPM conditions.

Systematic Risk

CAPM provides an insight that β is the most important measure of risk. To look more
closely, let us consider the following equation:
𝑟̅ = 𝑟 + 𝛽 𝑟̅ − 𝑟 + 𝑒 ;
 for rate of return of 𝑖 asset where 𝐸(𝑒 ) = 0 and cov(𝑒 , 𝜎 ) = 0

We further have:
𝝈𝟐𝒊 = 𝜷𝟐𝑰 𝝈𝟐𝑴 + 𝒗𝒂𝒓(𝒆𝒊 )
The variance of ith asset 𝜎 has two parts:
o 𝜷𝟐𝑰 𝝈𝟐𝑴 is called the systematic risk associated with the whole market. It cannot be
reduced and diversified because every asset with non-zero beta contents this risk.
o 𝒗𝒂𝒓(𝒆𝒊 )shows the extent of non-systematic risk/ specific risk which is diversifiable.
Beta is thus a measure systematic risk and in most important since it directly combines
with the systematic risk of other assets.

For any asset on CML, standard deviation is 𝛽𝜎 which

has only systematic risk and return equal to:
𝒓𝒊 = 𝒓𝒇 + 𝜷𝒊 𝒓𝑴 − 𝒓𝒇
Any similar assets with non-systematic risk
will be horizontally on the plane and not on CML.

Investment Implication
 CAPM solves the Markowitz problem using the argument that market portfolio is
that one fund (and only fund) of risky assets that one needs to hold, supplemented
by risk-free asset.
 Investor should thus purchase market portfolio. Hence investor must possess little
of every stock with proportions related to capitalisation.
 To avoid assembling the portfolio, one simple method is to invest in mutual funds
or index funds (since they tend to duplicate the portfolio of major stock market
index and are thought to represent the market) and alter the portfolio with market
conditions.
 CAPM assumes everyone has identical information about the (uncertain) returns of
all assets.

CAPM as Pricing Formula

The model is indeed a pricing formula although it does not contain the price specifically.
To define prices out of the rate of returns, we resort to the following method:
𝑸 𝑷
Define rate of return, 𝒓 = ,
𝑷
Where, 𝑃 = purchase price
𝑄 = price at which an asset is sold (random).
We substitute the rate of return derived above in the definition of return in the CAPM
equation and solve for P.
= 𝑟 + 𝛽 𝑟̅ − 𝑟
Solving for P, we obtain
𝑃= ̅

This shows price of an asset with payoffs Q and risk of beta.

Linearity of Prices/ Certainty Equivalent

The pricing formula is linear i.e. the price of sum of two assets is the sum of their prices.
It looks simple but is not linear firstly:
𝑃 = ̅
,𝑃 = ̅

It does not seem obvious that

𝑄 +𝑄
𝑃 +𝑃 =
1+𝑟 +𝛽 𝑟̅ − 𝑟
Where, 𝛽 is the beta of new asset.
For this we convert the formula to a linear form called the Certainty Equivalent Form.
𝑸
𝒓= -1
𝑷
the value of beta then is:
???????????????check???????????????????? Type these equations?????????

The term in the bracket is treated as certain and then discounted at risk free rate to obtain
the price and the certainty equivalent ensures the formula of price is linearly related to Q.
The reason for linearity lies in no arbitrage argument. That if price of new asset is not the
sum of prices of individual asset, then there is possibility of making arbitrage profits.
Year: 2015
Q1 (a) What is duration and how is it calculated?
5
Duration helps in understanding the sensitivity of interest. It is the
weighted average of the times the payments are made with weight
being present value of individual cash flows.
𝑃𝑉(𝑡 )𝑡 + 𝑃𝑉(𝑡 )𝑡 + 𝑃𝑉(𝑡 )𝑡 + ⋯ + 𝑃𝑉(𝑡 )𝑡
𝐷=
𝑃𝑉

where 𝑃𝑉 (𝑡 ) are the present values of cash flow occurring at time 𝑡 . 𝐷

is itself in time and will be 𝑡 ≦ 𝐷 ≦ 𝑡 .
Duration measures the average maturity of the bond’s promised cash
flows. It shows the weighted average life of the bond taking into
account the size and timing of cash flows. Duration is a weighted
average of the maturities of the cash payments.
(b) What are the important properties of duration?
1. Duration of a coupon paying bond is always less than its maturity.
For a non-coupon paying bond, the duration is the same as its maturity.
2. Bonds with longer maturities have longer durations. This is because
the coupon payments will be spread over longer periods and will be
more affected by inflations.
3. The bond with higher coupon rates have lower duration, and vice
versa.
4. Since duration is always less than maturity, when maturity tends to
infinity, the duration does not tend to infinity.
5. The duration of a bond increases immediately on the day a coupon is
paid. However, throughout the life of the bond, the duration is
continually decreasing as time to the bond’s maturity decreases.
(c) Find the duration 𝑫 and the modified duration 𝐷 of a perpetual
annuity that pays an amount 𝑨 at the beginning of each year, with the
first such payment being 1 year from now. Assume a constant interest
rate r compounded yearly.
 5

𝑆 = + + + ⋯ … … … … … … … ….
( ) ( )

𝑆 = + + + ⋯ … … … … … … … ….
( ) ( ) ( )
- - - - - …………………

(1 − )S1 = + + + ⋯ … … … … … … … ….
( ) ( )

( )𝑆 =

( )
𝑆 =
Price = PV = + + + ⋯ … … … … … … … ….
( ) ( )

∑ . ( )
Duration = = =

𝐷 = = =

Q2 (a) The spot rate curve usually slopes gradually upwards as

maturity increases. Give reasons for this typical shape.
(Same as Q2 (b) (i) 2018 question paper)
(b) Two stocks are available. The corresponding expected rate of
return are 𝒓𝟏 and 𝒓𝟐 the corresponding variances and covariances are
by 𝝈𝟐𝟏 ,𝝈𝟐𝟐 and 𝝈𝟏𝟐 . What is invested in each of the two stocks to
minimise the total variance of the rate of return of the resulting
portfolio? What is the mean rate of return of this portfolio?
5

Two stocks: 1 and 2

  Expected Rate of return = 𝑥̅ and 𝑥̅
 Variance = 𝝈𝟐𝟏 and 𝝈𝟐𝟐
 Covariance =𝝈𝟏𝟐
Let weights be 𝑤 = (1 − 𝛼)and 𝑤 = 𝛼
Then, the rate of return of the portfolio will be:
𝑟̅ = (1 − 𝛼)𝑥̅ + 𝛼𝑥̅
And variance of the portfolio will be:
𝜎 = (1 − 𝛼) 𝜎 + 2. 𝛼. (1 − 𝛼)𝜎 + 𝛼 𝜎
To minimize variance:
( )
𝑃𝑢𝑡, =0
( )
 = −2(1 − 𝛼 )𝜎 + 2(1 − 2𝛼 )𝜎 + 2𝛼𝜎
 0 = −2𝜎 + 2𝛼𝜎 + 2𝜎 − 4𝛼𝜎 + 2𝛼𝜎
 2𝜎 − 2𝜎 = 2𝛼𝜎 − 4𝛼𝜎 + 2𝛼𝜎
 𝜎 − 𝜎 = 𝛼𝜎 − 2𝛼𝜎 + 𝛼𝜎
 =𝛼

 (1 − 𝛼 ) =

Now, Mean rate of return:

𝑟̅ = (1 − 𝛼 )𝑟̅ + 𝛼𝑟̅
 𝑟̅ = 𝑟̅ + 𝑟̅
̅ ̅ ( ̅ ̅ )
 𝑟̅ =

(c) (Two correlated assets) the correlation P between asset A and B is

0.1 and the other data are given in the table ahead:

Asset 𝒓 𝝈
A 10.0% 15%
B 18.0% 30%
(i) Find the proportions of weights (𝜶 𝒂𝒏𝒅 (𝟏 − 𝜶) of asset A and B to
define the portfolio having minimum standard deviation.
(ii) What is the value of minimum standard deviation?
(iii) What is the expected return of this portfolio?

2+2+2

(a) Using part (b) we know that :

(i) 1−𝛼 =

𝜎12
And ρ =
𝜎1 𝜎2
 0.1 =
. × .

 𝜎 = 0.0045
( . ) ( . )
 1−𝛼 = ( . ) ( .
( . ) )
.
 1−𝛼 =
.

 1 − 𝛼 = 0.826

 𝛼 = 0.174
 weight of asset A = 82.6%

And weight of asset B = 17.4%

(ii) Minimum variance

𝜎 =
(0.826) (0.15) + (0.174) (0.3) 2(0.826)(0.174)(0.0045)
𝜎 = 0.0194
(iii) Expected return
𝑟̅ = (1 − 𝛼)𝑟̅ + 𝛼𝑟̅
𝑟̅ = (0.826)(0.10) + (0.174)(0.18)
𝑟̅ = 0.11392
𝑟̅ = 11.392%
 Q3 (a) Indicate on a diagram the feasible set of at least 3
assets not perfectly correlated and with different means.

7.5

If there are at least 3 assets (not perfectly correlated and with different
means), then the feasible set will be a 2-D solid region.

(b) Use the Markowitz Model to find the solution when there is a risk
free and risky asset which are available.

7.5

A risk-free asset has a deterministic return and hence zero risk. Hence a
risk-free asset is a pure interest-bearing instrument and corresponds to
a lending or borrowing through inclusion in the portfolio.

Inclusion of a risk-free assets creates mathematical degeneracy that

simplifies the shape of efficient frontier. Suppose the return on risk-free
asset is 𝑟 and consideranother asset which is risk with returns as r
having mean return as 𝑟̅ and variance as σ2.
It is obvious that covariance between the two will be zero.
Forming a portfolio using weights 𝛼 for risk-free asset and (1 − 𝛼) for
risky asset. The mean return of portfolio will be 𝛼 𝑟 + (1 − 𝛼)𝑟̅ and
the standard deviation will be (1 − 𝛼)𝜎. For a moment consider 𝜎 = 0
we see:
Mean = 𝛼𝑟 + (1 − 𝛼)𝑟̅

Standard deviation = 𝛼𝜎 + (1 − 𝛼)𝜎

The equation show that both means and standard deviation of portfolio
vary linearly with α thus the points representing portfolio trace out to be
straight line.

Taking it further, now suppose there are 𝑛 assets (with mean rate of
returns as 𝑟̅ and known covariance as 𝜎 ) are used to form a portfolio
along with a risk-free asset with rate of return 𝑟 .

The left diagram shows the inclusion of risk-free asset converts the
curved feasible set into a triangular one. The triangle is infinite in case
of borrowing of risk-free asset (or short selling) and gets restricted when
the risk-free asset is only lent.

(The solution part is not in syllabus).

 Year: 2016
1. (a) Which is a better criterion between NPV and IRR in the selection of
a project?
5
Both NPV and IRR have attractive features and limitations.

 NPV
o NPV is the simplest to compute. It does not have ambiguity of several
possible roots like that in IRR equation.
o It can be broken into component pieces, unlike IRR.
 IRR has the advantage that it depends only on the properties of the cash
flow stream and not on the prevailing interest rate.
 In a situation where the proceeds of the investment can be repeatedly
invested in the same type of project but scaled in size, it makes sense to
select the project with the largest IRR – in order to get the greatest
growth of capital.
On the other hand, suppose that the investment is a one-time
opportunity and cannot be repeated. Then the NPV is the appropriate
criterion, since it compares the investment with what could be obtained
through normal channels.
 Theorists (not practitioners) believe that the best criterion is that based
on NPV as if used intelligently it will provide consistency and
rationality.
 There are many other factors that influence a good PV analysis-
o One significant issue is the selection of the interest rate to be used in
calculation, because there are several risk-free rates of interest in the
financial market.
o Also, generally the rates for borrowing are typically slightly higher
than those for lending.
o PV by itself doesn’t reveal much about the rate of return. Two
alternative investments might each have a NPV of $100, but one might
require an investment of $100 whereas the other requires $1,000,000.
o NPV is not the whole story. It forms a solid starting point, but one
must supplement its use with additional structure.
 (b) Suppose ONGC launches a pension scheme for its retired
employees for that the company has to meet an obligation of making
perpetual payment of rupees 200cr per annum. In order to construct
the pension fund, the fund manager uses secured government bonds.
The YTM of all bonds is 15%. If the duration of 10-year maturity
bonds with coupon rates of 10% (paid annually) is 8 years and the
duration of 20-year maturity bonds with coupon rates of 8% (paid
annually) is 12 years, what will be the amount of each of these coupon
bearing bonds in terms of market value that the fund manager wants
to hold to immunize the obligation?
5
YTM=15%
A: 10-year bond—10% coupon rate—8years duration
B: 20-year bond—8% coupon rate—12years duration

𝑃𝑉 = = 173.913 𝐶𝑟.
( . )
.
Duration of a portfolio: 𝐷 = = = 7.67 years
( )( . )
𝐷 = 𝑤 𝐷 +𝑤 𝐷
7.67 = 𝑤 (8) + 𝑤 (12)
7.67 = 𝑤 (8) + (1 − 𝑤 )(12)
𝑤 = 1.0825 > 1
So, he should buy 10-year bonds for ₹ 173.913 Cr, although this would
not fully immunize his portfolio but this is the best he can do.

(c) Consider two 5-year bonds, one has a 9% coupon and selling price
₹101, the other has 8% coupon and sells for ₹95. How will you
construct a 5-year zero-coupon bond from prices of above coupon
bearing bonds?
5
 9% coupon – ₹101  x
 8% coupon – ₹ 95  y

To make net coupon value zero: 9𝑥 + 8𝑦 = 0

To make face value 100: 𝑥 + 𝑦 = 1
 Solving the equations,
𝑥 = −8
𝑦= 9

 Price = −8 × 101 + 9 × 95 = 47.

Q2 (a) Consider a portfolio that consist of two risky assets 1 and 2

with weights 𝜶 and (𝟏 − 𝝈). The mean return of the assets 1 and 2 are
𝑟 and 𝑟 respectively. How will you construct the portfolio diagram
when the weight 𝜶 varies from 0 to 1 and short selling or borrowing is
not allowed? How much will u invest in both the assets 1 and 2 in
order to make the portfolio risk free? What will be shape of portfolio
diagram if you short sell the asset 2 and use the proceeds to invest in
asset 1?

10

Suppose 2 assets on a mean variance diagram. These 2 assets can be

combined according to weights to form a portfolio.
Let 𝑤 = 1 − 𝛼, 𝑤 = 𝛼 ; 0 ≤ 𝛼 ≤ 1 (If weights are negative then
shorting is allowed).
As 𝛼 varies, then new portfolio traces out a curve that include asset 1
and 2.

Portfolio diagram lemma. The curve in an 𝑟̅ − 𝜎 diagram defined by non-

negative mixtures of two assets 1 and 2 lies within the triangular region
defined by the two original assets and the point on the vertical axis of
height 𝐴 = (𝑟̅ 𝜎 ) + (𝑟̅ 𝜎 )/(𝜎 + 𝜎 )
Proof. The rate of return of the portfolio defined by  is 𝑟() =
(1 −  )r + r . The mean value of this return is
𝒓(  ) = (1 −  )r̅ + r̅
This says that the mean value is between the original means, in direct
proportion to the proportions of the assets. In a 50-50 mix, for example,
the new mean will be midway between the original means.
Let us compute the standard deviation of the portfolio. We have, from the
general formula of the previous section,
𝜎(𝛼) = (1 − 𝛼) 𝜎 + 2𝛼(1 − 𝛼)𝜎 + 𝛼 𝜎

Using the definition of the correlation coefficient 𝜌 = 𝜎 /(𝜎 𝜎 ), this

equation can be written
𝜎(𝛼) = (1 − 𝛼) 𝜎 + 2𝜌𝛼(1 − 𝛼)𝜎 𝜎 + 𝛼 𝜎
This is quite a messy expression. However, we can determine its bounds.
We know that 𝜌 can range over −1 ≤ 𝜌 ≤ 1. Using 𝜌 = 1 we find the
upper bound
𝜎(𝛼)∗ = (1 − 𝛼) 𝜎 + 2𝛼(1 − 𝛼)𝜎 𝜎 + 𝛼 𝜎
= [(1 − 𝛼)𝜎 + 𝛼𝜎 ]
= (1 − 𝛼)𝜎 + 𝛼𝜎
Using 𝜌 = −1 we likewise obtain the lower bound

𝜎(𝛼)∗ = (1 − 𝛼) 𝜎 − 2𝛼(1 − 𝛼)𝜎 𝜎 + 𝛼 𝜎

= [(1 − 𝛼)𝜎 − 𝛼𝜎 ]
= |(1 − 𝛼)𝜎 − 𝛼𝜎 |
∴ The two linear expression together with the linear expression of mean
trace out a kinked line.

(b) Briefly explain two-fund theorem.

The minimum variance set satisfies the system of 𝑛 + 2 linear equations.

Suppose there are two known solutions:

 ,𝜇 ,𝑤 = 𝑤 ,𝑤 ,…..,𝑤
𝑎𝑛𝑑
 ,𝜇 ,𝑤 = 𝑤 ,𝑤 ,…..,𝑤
each with expected returns as 𝑟 𝑎𝑛𝑑 𝑟 . Combinations can be formed by
giving weights to thesetwo solutions i.e. α and (1-α). Substituting them
into the equations for efficient set, the expected value of returns
(𝜶)𝒓𝟏 + (𝟏 − 𝜶)𝒓𝟐 .
The portfolio thus formed (𝜶)𝒘𝟏 + (𝟏 − 𝜶)𝒘𝟐 has legitimateweights
whose sum is equal to one.The result derived shows an important
result, that suppose w1and w2 are two different portfolios in the
minimum variance set, then as 𝜶 varies over −∞ < 𝜶 < ∞ the
portfolios defined by 𝜶. 𝒘𝟏 + (1 − 𝜶).w2 cover the entire minimum
variance set.

Alternatively, two efficient funds (portfolios) can be established so that

any efficient portfolio can be duplicated in terms of mean and variance,
as a combination of these two. In other words, all investors are seeking
efficient portfolios need only to invest in combinations of these two
funds.

According to what has been discussed so far in two fund theorem, two
mutual funds can provide complete investment opportunity to
everyone. This does implicitly assume that everyone is concerned just
about mean and covariance of returns and have single period outlook.

Q3 (a) Suppose that Tata Motors is going to launch a new hatchback.

The current price of a share of Tata Motor is ₹850. If the new car
successfully hit the market, the share can be expected to yield ₹1000
but due to uncertainty in the market performance, the estimated
standard deviation of the return is 𝝈 = 𝟑𝟎%. Currently the risk-free
interest rate is 9%. Suppose NIFTY 50 mirrors the market portfolio
and its expected return is 18% and the standard deviation of this rate
of market portfolio is 10%. Will the investment in Tata Motors be
profitable?
5

The current price of Tata Motors is ₹850, the actual expected rate of
return is:

𝑟̅ = 1000/850 – 1 = 0.1764 = 17.64%

And Standard deviation of the return is 30%. The risk-free rate is 9%
which has zero level of standard deviation of return. The market
expected return is 18% and the standard deviation is 10%.

Total Portfolio Risk-free Market

Portfolio
Expected 17.65% 9% 18%
return
Standard 30% 0% 10%
deviation
The expected return of the market portfolio is greater than Tata
portfolio, also its variance (standard deviation) of the return is lower
than the Tata’s. So, the market portfolio will be preferred over the Tata
Motor’s portfolio.

Now, the choice between the market portfolio and the risk-free portfolio
depends on the investor’s preference of risk. If he cares more about the
expected return, he would go for market portfolio.

But in either of the cases the investment in the Tata Motor’s will not be
profitable.

(b) What is security market line? Why should the return of any asset
fall on the SML under the equilibrium conditions assumed by
CAPM?

Liner relation of CAPM is called SML.

The two graphs are plotted in terms of covariance and beta and
represent the risk-reward structure of an asset under CAPM conditions.
It shows different level of systematic, or market, risk of various
marketable securities plotted against the expected return of the entire
market at a given point in time.

(c) Suppose if the mean return of assets is 𝒓A = 25%, mean return of

market portfolio 𝒓m=20%, risk-free rate 𝒓f=10% and σm = 30%what is
the beta of an efficient portfolio? What is asset A’s correlation with
the market portfolio? Calculate the systematic risk of asset A.
5
𝑟̅ = 25%
𝑟̅ = 20%
𝑟̅ = 10%
𝜎 = 30%

Due to presence of risk-less asset, an efficient portfolio lies on the CML

𝒓𝒎 𝒓𝒇
𝑟̅ = 𝑟̅ + 𝜎
𝒎
. .
0.25 = 0.1 + 𝜎
.
 𝜎 = 0.45

SML: 𝑟̅ = 𝒓𝒇 + 𝛽 [𝒓𝒎 − 𝒓𝒇 ]
0.25 = 0.1 + 𝛽 [0.2 − 0.1]
𝛽 = 1.5  Systematic risk

Correlation:
𝜎(𝑟̅ ,𝑟̅ ) = 𝛽 = 1.5 [ 0.3/0.45] = 1
 Year 2017

Q1 (a) Define NPV and IRR. Which of these two criteria is the most
appropriate for investment evaluation? Explain your answer with the
help of an example.
7
Definitions:
 NPV is the difference between the terms present worth of benefits and
present worth of costs.
 IRR is the value of interest that renders the above NPV equal to zero.

NPV v/s IRR:

(Same as: 2014 Q1 (a))

Example:

(Same as: Nov-2018: Q3 (a) (ii))

(b) A major lottery advertises that it pays the winner $10 million.
However, this prize money is paid at the rate of $500,000 each year
(with the first -payment being immediate) for a total of 20 payments
what is present value of this prize at 10% interest?
4
𝒙 𝒙 𝒙
𝑷𝑽 = 𝒙𝟎 + 𝟏 + 𝟐 𝟐 + … + 𝒏 𝒏
𝟏 𝒓 (𝟏 𝒓) (𝟏 𝒓)

 𝑃𝑉 = 500000 + (
+ + ⋯………….+
. ) ( . ) ( . )

[ ]
 𝑃𝑉 = 500000 .
[ ]
.
 𝑃𝑉 = $4,682,460.045
(c) What do you understand if: -
(i) 𝜷 = 𝟎
(ii) β = Negative
4

We know,
 ̅
𝑟̅ = 𝑟 + 𝜎 = 𝑟 + 𝛽 (𝑟̅ − 𝑟 )

 When asset is completely uncorrelated with market then 𝛽 = 0.

In this case,𝑟̅ = 𝑟 which means that no matter how risky the asset is (𝜎
is large), the expected rate of return will be equal to that of risk-free
asset. Risk on the asset (which is uncorrelated to the market) can be
diversified.
 When 𝛽 is negative, i.e.𝑟̅ < 𝑟 (even though asset 𝜎 is large), such assets
will tend to reduce overall portfolio risk when combined. Some
investors are willing to accept lower risk.

Q2 (a) Define Immunization. What problems does it solve and what are
the shortcomings of this procedure?
5

 Immunization of a bond is a process of creating a bond portfolio through

which the interest rate risk and investment risk can either be eliminated
or minimized. Immunization cannot be achieved through single bond
and creating a bond portfolio is necessary for immunization. As soon as
the duration of the portfolio match with the target period, then it is
almost immune to interest rate fluctuations.

 Example: Target pd=3 years, $10,00,000 at present

Particulars A B
F.V. 1000 1000
M.P. 986.5 1035
Duration 5 2
Amount of money to be invested in each of the bond:
3=𝐷
3=𝑊 𝐷 +𝑊 𝐷
And, 𝑊 + 𝑊 = 1
𝑊 = 1 − 𝑊
 3 = (1 − 𝑊 )5 + 𝑊 × 2
 𝑊 = 2/3 𝑎𝑛𝑑 𝑊 = 1/3.
 Also, 𝑊 =
 =
, ,
 𝑃 = 3,33,333.333
 Similarly, 𝑃 = 6,66,666.66.

 Immunisation does suffer from short comings. Firstly, all yields are not
equal. And it is hard to find a combination of long-term and short-term
bonds with identical yields. Also, given this, if yield changes, it is not
necessary for yields on all such bonds to change with equal effect which
would make rebalancing difficult.
(b) A debt of $25000 is to be amortized over 7 years at 7% interest
compounded annually. What value of monthly payments will achieve
this?
4
𝐴 1
𝑃𝑉 = [1 − ]
𝑟 (1 + 𝑟)
𝐴 1
25000 = . 1−
.
1+
 𝐴 = $377.32

(c) Define the terms:

I. Risk averse
II. Risk preferring
III. Non satiation
6
 The left boundary of a feasible set is called a minimum variance set.
 There is a special point on this set having minimum variance. It is
termed as the minimum variance point (MVP).
 An investor who agrees with the view point is said to be risk averse,
since he or she seeks to minimise risk.
 An investor who would select a point other than the one of minimum
standard deviation is said to be risk preferring.
 Non satiation: Everything else being equal investors always want more
money; hence they want the highest possible expected return for a
given standard deviation.

Conclusion: These arguments imply that only the upper part of the
minimum-variance set will be of interest to investors who are risk
averse and satisfy non satiation. This upper portion of minimum-
variance set is termed as the efficient frontier. This provides the best
mean-variance combinations for most investors.

Q3 (a) Differentiate between systematic and non-systematic risk which

of them can be reduced by diversification?
6

CAPM provides an insight that β is the most important measure of risk.

To look more closely, let us consider the following equation:
𝑟̅ = 𝑟 + 𝛽 𝑟̅ − 𝑟 + 𝑒 ;
o for rate of return of 𝑖 asset where 𝐸(𝑒 ) = 0 and cov(𝑒 , 𝜎 ) = 0
o
o We further have:
o 𝝈𝟐𝒊 = 𝜷𝟐𝑰 𝝈𝟐𝑴 + 𝒗𝒂𝒓(𝒆𝒊 )
o The variance of ith asset 𝜎 has two parts:
 o 𝜷𝟐𝑰 𝝈𝟐𝑴 is called the systematic risk associated with the whole market. It
cannot be reduced and diversified because every asset with non-zero
beta contents this risk.
o 𝒗𝒂𝒓(𝒆𝒊 )shows the extent of non-systematic risk/ specific risk which is
diversifiable.
Beta is thus a measure systematic risk and in most important since it
directly combines with the systematic risk of other assets

For any asset on CML, std dev is βσM which has only
systematic risk and return equal to

𝒓 = rf + βi(𝒓𝑴 – rf).
Any similar assets with non-systematic risk will lie
horizontally on the plane and not on CML.

(b) Assume that the expected rate of return on the market portfolio is
23% and risk-free rate is 7%. The standard deviation of the market is
32% assume that the market is efficient.
i. Write the equation for CML.
ii. If the expected return of 39% is desired what is the standard deviation
of this position?
iii. If you invest $300 in the risk-free assets and $700 in the market
portfolio how much money should you expect to have at the end of
the year?
9
𝒓𝒎 𝒓 𝒇
i. 𝑟̅ = 𝑟 + 𝜎
𝒎
𝟎.𝟐𝟑 𝟎.𝟎𝟕
𝑟̅ = 0.07 + 𝜎
𝟎.𝟑𝟐

𝑟̅ = 0.07 + 0.5𝜎
ii. 𝑟̅ = 0.39
 0.39=0.07+0.5σ
 σ = 0.64
iii. With a standard deviation of 0.32
Risk-free asset will generate 300(1 + 0.07) = $321
Market portfolio will generate 700(1 + 0.23) = $861
 Total amount expected = $321 + $861 = $1182
 Year 2018 (May)

Q1 (a) Some firms prefer the IRR rule to the NPV criterion. Explain
carefully four major drawbacks of depending solely on the IRR rule.
Why is it still widely used in finance?
8+2
 NPV is the difference between the terms present worth of benefits and
present worth of costs.
 IRR is the value of interest that renders the above NPV equal to zero.
Drawbacks of IRR:

1. This method assumed that the earnings are reinvested at the internal
rate of return for the remaining life of the project. If the average rate of
return earned by the firm is not close to the internal rate of return, the
profitability of the project is not justifiable.

2. It involves tedious calculations.

3. This method gives importance only to the profitability but not

consider the earliest recouping of capital expenditure. The reason is that
sometimes Internal Rate of Return method favours a project which
comparatively requires a longer period for recouping the capital
expenditure. Under the conditions of future is uncertainty, sometimes
the full capital expenditure cannot be recouped if Internal Rate of
Return followed.

4. The results of Net Present Value method and Internal Rate of Return
method may differ when the projects under evaluation differ in their
size, life and timings of cash inflows.

It is still widely used in finance because:

1. It considers the time value of money even though the annual cash
inflow is even and uneven.

2. The profitability of the project is considered over the entire economic

life of the project. In this way, a true profitability of the project is
evaluated.
 3. There is no need of the pre-determination of cost of capital. Hence, a
lot of financial firms prefer IRR method than NPV method.

4. The ranking of project proposals is very easy under Internal Rate of

Return since it indicates percentage return.

(b) Consider two mutually exclusive projects as under:

Project C0 C1 C2 C3
A -100 60 60 0
B -100 0 0 140

(i) Calculate the NPV of each project using discount rates 10% and 20%.
(ii) Plot these on a graph.

3+2

NPV of a Project A
𝑁𝑃𝑉 (10%) = −100 + + + = 4.1322
. ( . ) ( . )

𝑁𝑃𝑉 (20%) = −100 + + + = -8.3333.

. ( . ) ( . )
𝑁𝑃𝑉 of a Project B
𝑁𝑃𝑉 (10%) = −100 + = 5.1841
( . )

𝑁𝑃𝑉 (20%) = −100 + = −18.9815

( . )
Q2 (a) Derive an expression for modified duration. What is the
relationship between duration and price sensitivity?
6

Sensitivity of price changes to yield is captured by Duration.

𝑃𝑉 = ∑
[ (/ )]

Derivate the above formula w.r.t. 𝜆

( / ) /
= =− 𝑃𝑉
 [ (/ )] (/ )
Now since Price P is sum of 𝑃𝑉 . Thus, using this relation, we get:
( / )
=∑ = −∑ =− 𝐷𝑃 = −𝐷 𝑃
  (/ ) (/ )

𝑫𝑴 is defined as Modified Duration.

𝐷 = 𝐷/[1 + (/𝑚)]
It tends to equal to normal Duration D for large value of m or small
value of λ.
 = −𝐷


Defining the rate of change of price:

This equation can be modified to view the effect of small change in yield
on price as: ∆𝑃 ≈ −𝐷 𝑃 ∆
The greater the duration of a security, the greater the percentage
change in the market value of the security for a given change in
interest rates. Therefore, the greater the duration of a security, the
greater its interest-rate risk.

(b) Consider two 10-year bonds, one has a 10% coupon and selling
price Rs 98.72, the other has 8% coupon and sells for Rs 85.89 and both
have same face value normalized to 100. Construct a zero-coupon
bond and find its price.
2+2
A zero-coupon bond can be constructed by purchasing an amount N Aof
bond A and selling an amount NB so that:NAcA+NBcB= 0
Solution:
 10% coupon – ₹98.72  x
 8% coupon – ₹85.89  y

To make net coupon value zero:10𝑥 + 8𝑦 = 0

To make face value 100: 𝑥 + 𝑦 = 1
Solving the above equations,
 𝑥 = −4
𝑦= 5
 Price = −4 × 98.72 + 5 × 85.89 = 𝑅𝑠. 34.57

(c) State the two-fund theorem and its implications. What are the
assumptions under which the theorem hold?

3+2

The minimum variance set satisfies the system of n+2 linear equations.

Suppose there are two known solutions:

 ,𝜇 ,𝑤 = 𝑤 ,𝑤 ,…..,𝑤
𝑎𝑛𝑑
 ,𝜇 ,𝑤 = 𝑤 ,𝑤 ,…..,𝑤
each with expected returns as 𝑟 𝑎𝑛𝑑 𝑟 . Combinations can be formed by
giving weights to these two solutions i.e. α and (1-α). Substituting them
into the equations for efficient set, the expected value of returns
(α)𝒓𝟏 + (𝟏 − 𝜶)𝒓𝟐 .
The portfolio thus formed (𝜶)𝒘𝟏 + (𝟏 − 𝜶)𝒘𝟐 has legitimate weights
whose sum is equal to one.The result derived shows an important
result, that suppose w1and w2 are two different portfolios in the
minimum variance set, then as 𝜶 varies over −∞ < 𝜶 < ∞ the
portfolios defined by (𝜶)𝒘𝟏 + (𝟏 − 𝜶)𝒘𝟐 cover the entire minimum
variance set.

Alternatively, two efficient funds (portfolios) can be established so that

any efficient portfolio can be duplicated in terms of mean and variance,
as a combination of these two. In other words, all investors are seeking
efficient portfolios need only to invest in combinations of these two
funds.

According to what has been discussed so far in two fund theorem, two
mutual funds can provide complete investment opportunity to
everyone. This does implicitly assume that everyone is concerned just
about mean and covariance of returns and have single period outlook.
 Assumptions:

1. All agents are mean-variance optimizers.

2. All agents know the probability distribution of the n assets.
3. All agents have the same risk-free rate of borrowing and lending.
4. There are no transaction costs in the market.

Q3 (a) Using CAPM, show how the expected rate of return of an

individual asset relates to its individual risk.

The CAPM helps to calculate investment risk and what return an

investor should expect from the investment. The Capital Market Line
relates to the expected rate of return of an efficient portfolio to its
standard deviation, but it does not relate the expected rate of return of
an individual asset to its individual risk. We can derive that using
CAPM:
𝑟̅ − 𝑟 = 𝛽 (𝑟̅ − 𝑟 );
Where, 𝛽 =
??????????????????????check??????????????? Missing proof photo

For varying the value of weights alpha, we trace a curve. For 𝛼 = 0,the
point corresponds to market portfolio M.This curve will be tangent to
the CML at this point and cannot cross the CML (or else it would violate
the definition of CML). We need to solve for tangency condition,
whereby slope of CML must equal the slope of curve.
First, we have
̅
= 𝑟̅ − 𝑟̅
( ) ( )
=
 Thus,
| =
we then use the relation
̅ ̅ /
=
/
to obtain
̅ ( ̅ ̅ )
| =
This slope must equal the slope of the capital market line. Hence,
( ̅ ̅ ) ̅
=
We now just solve for 𝑟̅ , obtaining the final result
̅
𝑟̅ = 𝑟 + 𝜎 = 𝑟 + 𝛽 (𝑟̅ − 𝑟 )
This is clearly equivalent to the stated formula.

(b) Consider a portfolio of 250 shares of firm A worth $30/share and

1500 shares of firm B worth 20/share. You expect a return of 4% for
stock A and a return of 9% for stock B.

(i) What is the total value of the portfolio? What are the portfolio
weights and what is the expected return?
(ii) Suppose firm A’s share price falls to $24 and firm B’s price goes
up to $22. What is the new value of the portfolio? What return did it
earn? After the price change what are the new portfolio weights?
10
(i) Total value = 250(30) + 1500(22) = $37500
Portfolio weights:
( )
𝑊 = = 0.2

( )
𝑊 = = 0.8

Expected Return:
𝑟̅ = (0.2)(0.04) + (0.8)(0.09) = 0.08
 𝑟̅ = 8%

(ii) New Total value = 250(24) + 1500(22) = $39000

Portfolio weights:
( )
𝑊= = 0.1538

( )
𝑊 = = 0.8462

Expected Return:
𝑟̅ = (0.1538)(0.04) + (0.8462)(0.09) = 0.0823
𝑟̅ = 8.23%

Year: 2018-Nov-Dec

1. (a) Explain the annual worth method with the help of an example.
How does if differ from NPV analysis.
5,2.5
Annual Worth uses constant level cash flow for comparison. It is the
equivalent net amount that is generated by the project if all amounts are
converted to a fixed n-year annuity starting from the first year.
Example: A machine of $100,000 today is expected to generate
additional revenues of $25,000 for next 10 years at the end of each year.
If discount rate is 16%, is the investment profitable?
. ( . ) ×
Equivalent Cost of machine is = 20690
( . )
Hence Annual Worth is $25000-$20690 = $4310.A positive worth
signifies profitability.
Difference b/w NPV and Annual worth analysis:
Suppose a project has an associated cash flow stream
(𝑥 , 𝑥 , … … , 𝑥 ) over n years. A Present Value analysis uses a constant
idea bank with interest rate 𝑟 to transform this stream into an
equivalent one of the form (𝑣, 0,0, … . ,0) where 𝑣 is the NPV of the
stream.
 Whereas, an annual worth analysis uses the same ideal bank to
transform the sequence to one of the form (0, 𝐴, 𝐴, 𝐴 … . . , 𝐴). The value 𝐴
is annual worth of the project (over 𝑛 years).

(b) (i) Rank the following bonds in terms of descending duration

(without calculating) and give reasons for the same.
5
Bond Coupon Time to Yield to
Maturity (in Maturity
years)
A 10% 30 10
B 0% 30 10
C 10% 30 7
D 10% 5 10

Duration is directly proportional to the time to maturity and inversely

proportional to the coupon payment and yield to maturity. The
duration of a zero-coupon bonds shall be highest when compared with
a coupon bond of same maturity.
 Therefore, the highest duration shall be for Bond B.
 Then in order comes Bond C because of higher coupon rate but lower
YTM than other bonds.
 After that the Bond having lowest Time to Maturity (other things same),
is Bond D. Therefore, it has lowest duration.
Hence the proper order shall be: B,C,A,D.
(ii) Calculate the duration D of a 10%, 30-year bond, if it is at par.
2.5
𝐷= 1−
( )
Hence,
.
𝐷= 1− = 9.938
( . )

(c) Differentiate between Macaulay duration and modified duration?

7.5
 Macaulay Duration:
 Macaulay duration D is defined as
∑ (𝑘/𝑚) /[1 + (/𝑚)]
𝐷=
𝑃𝑉
Where  is the yield to maturity and
𝑐
𝑃𝑉 =
[1 + (/𝑚)]
where k/m is total time.
Macaulay duration formula. The Macaulay duration for a bond with a
coupon rate 𝑐 per period, yield 𝑦
Per period, 𝑚 periods per year, and exactly 𝑛 periods remaining, is
( )
𝐷= − [( ) ]
(3.3)

 Modified Duration:
Sensitivity of price changes to yield is captured by Duration.
𝑃𝑉 = ∑
[ (/ )]

Derivate the above formula w.r.t. 𝜆

( / ) /
= =− 𝑃𝑉
[ (/ )] (/ )

Now since Price P is sum of 𝑃𝑉 . Thus, using this relation, we get:

𝑑𝑃 𝑑𝑃𝑉 (𝑘/𝑚)𝑃𝑉 1
= =− =− 𝐷𝑃 ≡ −𝐷 𝑃
𝑑𝜆 𝑑𝜆 1 + (𝜆/𝑚) 1 + (𝜆/𝑚)
DM is defined as Modified Duration.
𝐷 = 𝐷/[1 + (𝜆/𝑚)]
It tends to equal to normal Duration D for large value of m or small
value of λ.
= −𝐷
Defining the rate of change of price:
This equation can be modified to view the effect of small change in yield
on price as: ∆𝑃 ≈ −𝐷 𝑃 ∆𝜆
The greater the duration of a security, the greater the percentage
change in the market value of the security for a given change in
 interest rates. Therefore, the greater the duration of a security, the
greater its interest-rate risk.

2. (a) Consider a portfolio consisting of n assets all having a rate of

return with mean 𝝁, variance 𝝈𝟐 and covariance 𝜽𝝈𝟐 no matter how
large n is made. Elucidate with a diagram.
7.5
𝑛-assets; mean rate of return of each asset is m; 𝒄𝒐𝒗(𝒊, 𝒋) = 𝜽𝝈𝟐 i.e. assets
are correlated; 𝑤 = .
So, 𝑟̅ = ∑ 𝑟 𝑤 = 𝑟 𝑤 + 𝑟 𝑤 + 𝑟 𝑤 … … … + 𝑟 𝑤
 𝑟̅ = 𝑚. + 𝑚. + 𝑚. + … … … + 𝑚. = 𝑚
 r ̅= 𝐦
Var = ∑ . 𝑤𝑤𝜎 = ∑, . .σi,j = .∑ , σ, = .[∑ σ, +
∑ σ, ]

n cases n(n-1) cases

 𝑉𝑎𝑟 = . [𝑛𝜎 + 𝑛(𝑛 − 1) 𝜎 ] = . [𝑛𝜎 + 𝑛(𝑛 − 1)𝜃𝜎 ]

.
 𝑉𝑎𝑟 = + Ө𝜎
Here, var cannot be zero even if n tends to infinity.
As n tends to infinity, variance = 𝜽𝝈𝟐
 100% Diversification not possible.
 As n ⬆var ⬇, i.e. as diversification ⬆var ⬇.
 (b) (i) Give 3 explanations for the shape of the spot rate curve.
3
The spot rate curve is analogous to the yield curve.
1. Expectation Hypothesis: An upward sloping curve is explained by
expected future spot rates being higher than the current spot rates.
2. Liquidity preference Hypothesis: An upward sloping yield curve can be
consistent even with expectations of falling spot rates if liquidity
premium is high enough.
3. Segmentation Hypothesis: An upward sloping yield curve is evidence of
supply pressure in the long-term market and demand pressure in the
short- term market.

(ii) Distinguish between implied forward rates and market forward

rates. Specify the forward rates under different compounding
traditions.
4.5

Forward rates are interest rates for money to be borrowed between two
dates in future but under terms agreed upon today.

Market Forward rate: In market there could be more than one rate for
any particular forward period. For example, the forward rate for
borrowing may differ that from that for lending. Thus, when discussing
market rates one must be specific.

implied forward rates: In theoretical discussions the definition of

forward rates is based on an underlying set of spot rates. These
calculated forward rates are termed implied forward rates. They are
defined to satisfy the following equation (𝑓𝑜𝑟 𝑖 < 𝑗):
(1 + 𝑠 ) = (1 + 𝑠 ) (1 + 𝑓 . )

Forward rates under different compounding traditions:

(a) Yearly:: For yearly compounding, the forward rates satisfy, for 𝑗 > 1.
1+𝑠 = (1 + 𝑠 ) 1 + 𝑓 .
 Hence,

1+𝑠
𝑓. = −1
(1 + 𝑠 )

(b) m periods per year: For m period per- year compounding, the
forward rates satisfy, for j > i, expressed in periods,
(1 + s /m) = (1 + s /m) (1 + f . /m)( )
Hence,
/( )
( / )
f. =m −m
( / )

(c) Continuous: For continuous compounding, the forward rates f , are

defined for all t and t with t > t and satisfy
e =e e ( )

Hence,
s t −s t
f . =
t −t

(c) Describe the Markowitz Problem. How does it lead to the two-
fund theorem?
2.5,5
THE MARKOWITZ PROBLEM:

Consider an n asset scenario each with 𝑟 as mean rate of return and 𝜎

the individual standard deviations with weights of n assets defined as
𝑤 .(Short selling weights are allowed).For some value of portfolio mean,
we find the portfolio of minimum variance as:

Minimize ∑ . 𝑤𝑤𝜎

Subject to ∑ 𝑤 𝑟̅ = 𝑟̅
∑ 𝑤 =1
 The problem is for single period investment and relates to trade-off
between expected rate of return and variance ofthese returns.To solve,
we construct a Lagrange Multiplier equation.

1
𝐿= 𝑤𝑤𝜎 − 𝑤 𝑟̅ − 𝑟̅ − 𝜇 𝑤 −1
2
.

Differentiate the equation with respect to weights 𝑤 and equate to zero.

This exercise can be done for a two-variable case
𝑑𝐿 1
= (2𝜎 𝑤 + 𝜎 𝑤 + 𝜎 𝑤 ) − 𝑟̅ − 𝜇
𝑑𝑤 2
𝑑𝐿 1
= (𝜎 𝑤 + 𝜎 𝑤 + 2𝜎 𝑤 ) − 𝑟̅ − 𝜇
𝑑𝑤 2
using the fact that 𝜎 = 𝜎 and setting these derivatives to zero, we obtain
𝜎 𝑤 + 𝜎 𝑤 − 𝑟̅ − 𝜇 = 0
𝜎 𝑤 + 𝜎 𝑤 − 𝑟̅ − 𝜇 = 0
So, we can say for Efficient Set
∑ 𝜎 𝑤 − 𝑟̅ − 𝜇 = 0 for 𝑖 = 1,2, … 𝑛
∑ 𝑤 𝑟̅ = 𝑟̅
∑ 𝑤 =1
𝑛 equations for covariance and two equations for constraints i.e.
n+2equations. This leads to two fund theorem.

TWO FUND THEOREM:

The minimum variance set satisfies the system of n+2 linear equations.

Suppose there are two known solutions:

 ,𝜇 ,𝑤 = 𝑤 ,𝑤 ,…..,𝑤
𝑎𝑛𝑑
 ,𝜇 ,𝑤 = 𝑤 ,𝑤 ,…..,𝑤
 each with expected returns as 𝑟 𝑎𝑛𝑑 𝑟 . Combinations can be formed by
giving weights to thesetwo solutions i.e. α and (1-α). Substituting them
into the equations for efficient set, theexpected value of returns (𝜶)𝒓𝟏 +
(𝟏 − 𝜶)𝒓𝟐 .
The portfolio thus formed (𝜶)𝒘𝟏 + (𝟏 − 𝜶)𝒘𝟐 has legitimateweights
whose sum is equal to one.The result derived shows an important
result, that suppose w1and w2 are two different portfolios in the
minimum variance set, then as 𝜶 varies over −∞ < 𝜶 < ∞ the
portfolios defined by 𝜶. 𝒘𝟏 + (1 − 𝜶). 𝒘𝟐 cover the entire minimum
variance set.

Alternatively, two efficient funds (portfolios) can be established so that

any efficient portfolio can be duplicated in terms of mean and variance,
as a combination of these two. In other words, all investors are seeking
efficient portfolios need only to invest in combinations of these two
funds.

According to what has been discussed so far in two fund theorem, two
mutual funds can provide complete investment opportunity to
everyone. This does implicitly assume that everyone is concerned just
about mean and covariance of returns and have single period outlook.

Q3 (a) (i) Consider the cash flow sequence (-2,2,4), find the internal
rate of return of this cashflow stream.
2.5
𝒙𝒕 𝒙𝟐 𝒙𝒏
𝟎 = 𝒙𝟎 + + + …+
𝟏 𝒓 (𝟏 𝒓)𝟐 (𝟏 𝒓)𝒏

0 = 𝑥 + 𝑥 𝑐 + 𝑥 𝑐 + ⋯+ 𝑥 𝑐

Where, 𝑟 = (1/𝑐) – 1
0 = −2 + 2𝑐 + 4𝑐2
0 = −2 + 4𝑐 − 2𝑐 + 4𝑐2
0 = −2(1 − 2𝑐) − 2𝑐 (1 − 2𝑐)
0 = (−2 − 2𝑐) (1 − 2𝑐)
𝑐 = −1 (𝑅𝑒𝑗𝑒𝑐𝑡) 𝑜𝑟 𝑐 = 0.5
𝑟 = 1/0.5 – 1 = 1
  𝐼𝑅𝑅 = 100%
(ii) Using two methods of evaluating investment decisions, the IRR
and the NPV methods, evaluate the two flows associated with
harvesting trees to be sold for lumber: (A) (-1,2) cut early (B) (-1,0,3)
cut later. Assume that the prevailing interest rate is 10%.
5
NPV criterion:
(a) NPV = −1 + 2/1.1 = 82
(b) NPV =−1 + 3/(1.1) = 1.48
Hence according to the net present value criterion, it is best to cut
later.
IRR criterion:
(a) −1 + 2𝑐 = 0
(b) −1 + 3𝑐 = 0
As usual, 𝑐 = 1/(1 + 𝑟). These have the following solutions:
(b) 𝑐 = = ; 𝑟 = 1.0
√
(c) 𝑐 = = ; 𝑟 = √3 − 1 ≈ 7

In other words, for (a), cut early, the internal rate of return is
100%, whereas for (b) it is about 70%. Hence under the internal
rate of return criterion, the best alternative is (a). Note this is
opposite to the conclusion obtained from the net present value
criterion.

(b) Show how the variance of a portfolio return can be calculated

easily from the covariances of the pairs of asset returns and the asset
weights used in the portfolio.
7.5
Variance of a Portfolio Return:
 𝜎 = 𝐸[(𝑟 − 𝑟̅ ) ] = 𝐸[(∑ 𝑤 𝑟 − ∑ 𝑤 𝑟̅ ) ]
= 𝐸 (∑ 𝑤 (𝑟 − 𝑟̅ )) ∑ 𝑤 (𝑟 − 𝑟̅ )
= 𝐸 ∑ . 𝑤 𝑤 (𝑟 − 𝑟̅ )(𝑟 − 𝑟̅ )
 =∑. 𝑤𝑤𝜎

If 𝑟 = 𝑤 𝑟 + 𝑤 𝑟 , then 𝑣(𝑟) = 𝑤 𝑣(𝑟 ) + 𝑤 𝑣(𝑟 ) + 2𝑤 𝑤 𝑐𝑜𝑣(𝑟 , 𝑟 )

(c) (i) Derive general formulas for mean return of a portfolio and
variance of portfolio return for a portfolio comprising of n assets with
rates of return 𝑟 , 𝑟 ,………….., 𝑟 , expected values of rates of return
𝑬(𝑟 ) = 𝑟 , 𝑬(𝑟 ) = 𝑟 , … … … . . , 𝑬(𝑟 ) = 𝑟 , variance of the return of the
asset 𝒊, 𝝈𝒊 𝟐 , and covariance of return of asset i with asset j,
𝝈𝒊𝒋 respectively.
1.5+3
Mean of a portfolio:
𝑟 = 𝑤 𝑟 +𝑤 𝑟 +⋯+𝑤 𝑟
we may take the expected values of both sides, and using linearity
(property 2 of the expected value in Section 6.2), we obtain
𝐸 (𝑟) = 𝑤 𝐸 (𝑟 ) + 𝑤 𝐸 (𝑟 ) + ⋯ + 𝑤 𝐸(𝑟 )

 Variance of a Portfolio Return: 𝜎 = 𝐸[(𝑟 − 𝑟̅ ) ] = 𝐸 [(∑ 𝑤𝑟 −

∑ 𝑤 𝑟̅ ) ]
= 𝐸 (∑ 𝑤 (𝑟 −
𝑟̅ )) ∑ 𝑤 (𝑟 − 𝑟̅ )
= 𝐸 ∑. 𝑤 𝑤 (𝑟 −
𝑟̅ )(𝑟 − 𝑟̅ )
=∑. 𝑤𝑤𝜎

If 𝑟 = 𝑤 𝑟 + 𝑤 𝑟 , then 𝑣(𝑟) = 𝑤 𝑣(𝑟 ) + 𝑤 𝑣(𝑟 ) + 2𝑤 𝑤 𝑐𝑜𝑣(𝑟 , 𝑟 )

(ii) Consider a portfolio comprising of two assets 1 and 2 with weights

𝒘𝟏 = ¼ and𝒘𝟐 = ¾ respectively. Given, (𝒓𝟏 ) = 𝟎. 𝟏𝟐 𝒂𝒏𝒅 (𝒓𝟐 ) =
𝟎. 𝟏𝟓, 𝝈𝟏 = 𝟎. 𝟐𝟎, 𝝈𝟏𝟏 = 𝟎. 𝟏𝟖, 𝝈𝟏𝟐 = 𝟎. 𝟎𝟏 , calculate the mean and
variance of this portfolio.

𝑟̅ = (1/4)(0.12) + (3/4)(0.15) = 0.1425

𝜎 = (1/4) 2
(0.20)2 +(3/4)2 (0.15)2 + (1/4)(3/4)(0.01) + (3/4)(1/
4)(0.01) = 0.24475
 Q4 (a) ‘The CAPM changes our concept of risk of an asset from that of
𝝈 to that of 𝜷.’ Show this by deriving the relationship between the
expected rate of return of an individual asset with its individual risk.
7.5
The Capital Market Line relates to the expected rate of return of an
efficient portfolio to its standard deviation, but it does not relate the
expected rate of return of an individual asset to its individual risk. We
can derive that using CAPM:
𝑟̅ − 𝑟 = 𝛽 (𝑟̅ − 𝑟 );
Where, 𝛽 =

??????????????????check???????????????????

******MISSING*****

For varying the value of weights alpha, we trace a curve. For α=0, the
point corresponds to market portfolio M. This curve will be tangent to
the CML at this point and cannot cross the CML (or else it would violate
the definition of CML). We need to solve for tangency condition,
whereby slope of CML must equal the slope of curve.
First we have
̅
= 𝑟̅ − 𝑟̅
( ) ( )
=
Thus,
| =
we then use the relation
̅ ̅ /
=
/
to obtain
̅ ( ̅ ̅ )
| =
This slope must equal the slope of the capital market line. Hence,
( ̅ ̅ ) ̅
=
We now just solve for 𝑟̅ , obtaining the final result
̅
𝑟̅ = 𝑟 + 𝜎 = 𝑟 + 𝛽 (𝑟̅ − 𝑟 )
This is clearly equivalent to the stated formula.
 The value 𝛽 of is referred to as the beta of an asset which characterises
its risk.

(𝒓𝒊 – 𝒓𝒇 ) is termed as Expected Excess Rate of Return on 𝒊𝒕𝒉 asset. The

rate of return expected to exceed the risk- free rate.

(𝑟 − 𝑟 )is termed as Expected Excess Rate of Return on Market

Portfolio.

 Hence Expected Excess Rate of Return on 𝒊𝒕𝒉 asset is proportional (by

beta) to the Expected Excess Rate of Return on Market Portfolio. Beta
can also be called the normalised version of covariance of asset with
market.
 We consider a case where asset is completely uncorrelated with market
i.e. 𝜷 = 𝟎
In this case, (𝒓𝒊 – 𝒓𝒇 ) which means that no matter how risky the asset is
(𝜎 is large), the expected rate of return will be equal to that of risk-
free asset. Risk on the asset (which is uncorrelated to the market) can
be diversified.
 We could purchase small amounts of assets and resulting variance
would be small and the composite return would approach 𝑟 .
 Another case is where 𝛽 is negative. In such cases 𝒓𝒊 < 𝒓𝒇 (even though
asset 𝜎 is large), such assets will tend to reduce overall portfolio risk
when combined. Some investors are willing to accept lower risk.
 The overall risk of portfolio is still in terms of σ but for concern of
individual assets we refer to their 𝛽’𝑠.
 The CAPM changes our concept of risk of an asset from that of 𝝈 to
that of 𝜷.

(b) How is CAPM expressed as a pricing model? Derive the Certainty

Equivalent form of the CAPM.
7.5
The model is indeed a pricing model although it does not contain the
price specifically.
To define prices out of the rate of returns, we resort to the following
method:
𝑸 𝑷
Define rate of return, 𝒓 = ,
𝑷
Where, P = purchase price
Q = price at which an asset is sold(random).

We substitute the rate of return derived above in the definition of return

in the CAPM equation and solve for P.
= 𝑟 + 𝛽 𝑟̅ − 𝑟
Solving for P, we obtain
𝑃= ̅

This shows price of an asset with payoffs Q and risk of beta.

Linearity of Prices/ Certainty Equivalent
The pricing formula is linear i.e. the price of sum of two assets is the
sum of their prices.
It is not linear firstly:
𝑃 = ,𝑃 =
̅ ̅

𝑃 +𝑃 =
̅

Where, 𝛽 is the beta of new asset.

For this we convert the formula to a linear form called the Certainty
Equivalent Form.
 r =
𝑸
( )-
𝑷
1
the value of beta then is:
𝑐𝑜𝑣 − 1 ,𝑟
𝛽=
𝜎
This becomes
𝑐𝑜𝑣(𝑄, 𝑟 )
𝛽=
𝑃𝜎
Substituting this into the pricing formula and dividing by P yields
𝑄
1=
𝑃 1 + 𝑟 + 𝑐𝑜𝑣(𝑄, 𝑟 ) 𝑟̅ − 𝑟 /𝜎
Finally, solving for P we obtain the following formula:
Certainty equivalent pricing formula. The price P of an asset with
payoff Q is
1 𝑐𝑜𝑣(𝑄, 𝑟 ) 𝑟̅ − 𝑟
𝑃= 𝑄−
1+𝑟 𝜎

The term in the bracket is treated as certain and then discounted at risk
free rate to obtain the price and the certainty equivalent ensures the
formula of price is linearly related to Q.
The reason for linearity lies in no arbitrage argument. That if price of
new asset is not the sum of prices of individual asset, then there is
possibility of making arbitrage profits.

(c) Distinguish between capital market line and security market line.
7.5

THE CAPITAL MARKET LINE

 Shows the efficient set with single fund of risky asset being: Market
Portfolio being labelled as M. This is also called the price line as prices
adjust such that efficient assets fall on this line.
  The equation of capital market line 𝑟̅ and 𝜎 are the expected value and
standard deviation of the market rate of return of an arbitrary efficient
asset.

 The slope of capital market line is K = (𝑟̅ – 𝑟 )/𝜎 is this value called
Price of Risk: denoting how much the expected return on the portfolio
should if std. deviation of that rate increases by 1 unit.
THE SECURITY MARKET LINE
Linear relation of CAPM is called SML.
The two graphs are plotted in terms of covariance and beta and
represent the risk-reward structure of an asset under CAPM conditions.

Q5 (a) (i)Consider a world in which there are only two risky assets, A
and B, and a risk-free asset F. The two risky assets are in equal supply
 in the market; that is, 𝑴 = ½ (𝑨 + 𝑩). (The following information is
known):

𝒓𝒇 = 𝟎. 𝟏𝟎, 𝝈𝟐 𝑨 = 𝟎. 𝟎𝟒, 𝝈𝑨𝑩 = 𝟎. 𝟎𝟏, 𝝈𝟐 𝑩 = 𝟎. 𝟎𝟐, and 𝒓𝑴 = 𝟎. 𝟏𝟖.

Find 𝝈𝟐 𝑴 , 𝜷𝑨 and 𝜷𝑩 .

2+1.5+1.5
𝑀 = ½ (𝐴 + 𝐵)
𝒓𝒇 = 0.10, 𝝈𝟐 𝑨 = 0.04, 𝝈𝑨𝑩 = 0.01, 𝝈𝟐 𝑨 = 0.02, and𝒓𝑴 = 0.18.

 𝝈𝟐 𝑴 = (½) [𝝈𝟐 𝑨 + 2𝝈𝑨𝑩 +𝝈𝟐 𝑩 ] = (½) [ 0.04 + 2(0.01) + 0.02] = 0.02

 𝝈𝟐 𝑨𝑴 = (½) [𝝈𝟐 𝑨 + 𝝈𝑨𝑩 ] = (½)[ 0.04 + 0.01] = 0.025
𝝈𝟐 𝑨𝑴
𝜷𝑨 = = 0.025/0.02 = 1.25
𝝈𝟐 𝑴

 𝝈𝟐 𝑩𝑴 = (½) [𝝈𝟐 𝑩 + 𝝈𝑨𝑩 ] = (½)[ 0.02 + 0.01] = 0.015

𝝈𝟐 𝑩𝑴
𝜷𝑩 = = 0.015/0.02 = 0.75
𝝈𝟐 𝑴

(ii) The security market line expresses the risk reward structure of
assets according to CAPM. Comment.

2.5

The security market line expresses the risk reward structure of assets
according to the CAPM, and emphasizes that the risk of an asset is a
function of its covariance with the market or, equivalently, a function of
its beta.

The two graphs are plotted in terms of covariance and beta and
represent the risk-reward structure of an asset under CAPM conditions.
(b) (i) A negative value of beta implies that 𝒓 > 𝒓𝒇 . Do you agree?
Give reasons for your answer.
2.5
Negative value of beta implies that 𝒓 > 𝒓𝒇 , that is , even though the
asset may have very high risk (as measured by its 𝜎), its expected rate of
return should be even less than the risk-free rate.

Reason: Such an asset reduces the overall portfolio risk when it is

combined with the market . Investors are therefore willing to accept the
lower expected value for this risk-reducing potential. Such assets
provide a form of insurance. They do well when everything else does
poorly.

(ii) (A) Consider a risky venture with a per unit share price of ₹875
which is expected to increase to ₹1000 after a year. The standard
deviation of the return of the venture is 𝝈 = 𝟒𝟎%. Currently the risk-
free rate is 10%. The expected rate of return on the market portfolio is
17%, with a standard deviation of 12%. Find the expected rate of
return of this venture and the expected rate of return predicted by
Capital Market Line. Compare the two and comment.

2.5+1

Expected rate of return predicted by Capital Market Line:

̅ 𝒓
𝑟̅ = 𝒓 +
( . . )
𝑟̅ = 0.10 + × 0.4 = 0.33 = 33%
.

However, the actual expected rate of return is only 𝑟̅ = 1000/875 – 1 =

0.14 = 14%. Therefore the point representing the oil venture lies well
below the capita market line.
 This doesn’t mean that the venture is necessarily a poor one, but it
certainly does not constitute an efficient portfolio.

(B) Given that the beta of the risky venture is 𝜷 = 𝟎. 𝟔, find the value
of the share of the risky venture based on CAPM.

1.5
$𝟏𝟎𝟎𝟎
𝑷= = $𝟖𝟕𝟔
𝟏.𝟏𝟎 𝟔(𝟏𝟕 𝟏𝟎)

(c) (i) Explain the property of non-satiation with respect to the

efficient frontier.

2.5

o Non satiation: Everything else being equal investors always want more
money; hence they want the highest possible expected return for a
given standard deviation.
o These arguments imply that only the upper part of the minimum-
variance set will be of interest to investors who are risk averse and
satisfy non satiation. This upper portion of minimum-variance set is
termed as the efficient frontier. This provides the best mean-variance
combinations for most investors.

(ii) Calculate the beta coefficients for the securities from the
following information:
𝝈𝒊 𝝈𝒊𝑴

Security A 0.5 0.6

Security B 0.6 -0.2

Market Portfolio 0.2 1

𝝈𝒊 is the standard deviation of the rate of return on asset 𝒊; 𝝈𝒊𝑴 is the

correlation coefficient between the return on asset I and the return on
market portfolio.

5
 We know,
𝝆𝒊𝑴 = 𝝈𝒊𝑴 /𝝈𝒊 𝝈𝑴
For Security A:
ρ𝑨𝑴 = 𝝈𝑨𝑴 / 𝝈𝑨 𝝈𝑴
 0.6 = 𝝈𝑨𝑴 / 0.5 × 0.2
 𝝈𝑨𝑴 = 0.06

For Security B:
ρ𝑩𝑴 = 𝝈𝑩𝑴 / 𝝈𝑩 𝝈𝑴
 −0.2 = 𝝈𝑩𝑴 / 0.6 × 0.2
 𝝈𝑩𝑴 = −0.024

And, 𝜷𝒊 = 𝝈𝒊𝑴 /𝜎

𝜷𝑨 = 0.06 /(0.2) = 1.5
 𝜷𝑩 = −0.024/(0.2) = −0.6