FMP-II

Interest Rates

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© EduPristine For FMP-II (2016)
Agenda

This reading also covers the following readings from Valuation and Risk Models
 Spot, forward and Par Rates
 Returns, Spreads and Yields

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 Interest Rates

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Types of interest rates

 Interest rate is the amount of money a borrower promises to pay to the lender over and above the
principal amount
• Treasury Rates: This is the rate an investor receives when he invests in Treasury bills and Treasury bonds.
Treasury bills are short term while Treasury bonds are longer term (> 1 year)
• Corporate bond rates: These are rates on long term bonds issued by a corporate
• LIBOR: This is the London Interbank Offer Rate (LIBOR) and the rate at which banks make a large wholesale
deposit or loan with/to another bank
 1 month, 3 months, 6 months and 12 month LIBORs
 Opportunity cost for AA rated banks
 Not entirely risk free
• Repo rates and Reverse Repo: Repo rate is the rate at which banks borrow money from the central bank.
Reverse Repo rate is the rate at which the central bank borrows money from banks
• A Repurchase agreement (also known as a repo or Sale and Repurchase Agreement) allows a borrower to use
a financial security as collateral for a cash loan at a fixed rate of interest
• A repo is equivalent to a cash transaction combined with a forward contract

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Calculation of interest rates

 There are many ways to calculate interest rates – annual, semi annual, quarterly, continuously
compounding and so on
 Each rate can be expressed in the form of another rate. For example an interest rate of 10%
compounded semi-annually would fetch (1 + 10% / 2) * (1 + 10% / 2) = 1.1025 (remember
6months rate is 10% / 2) on $1 after one year. This is equivalent of 10.25% annual rate
 Amount compounded annually would be given by:
• A = P (1+ r)t
 A  terminal amount
 P  principal amount
 r  annual rate of interest
 t  number of years for which the principal is invested
 If amount compounded n times a year then:
• A = P ( 1+ r/n )nt
 When n  ∞ then we call it continuous compounding:
• A = Pert (this formula is derived using limits and continuity)

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Question

 If the interest rate is 10% per annum compounded continuously, then what is the effective annual
interest rate?

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Solution

 At continuous compounding, $1 after an year will become 1.ert = e0.1x1 = 1.10517

Had it been just annual compounding, then the interest rate required for $1 to rise up to $1.10517
would have been 1.10517 – 1 = 10.517% which is the effective annual rate.

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Question

 If the interest rate is 10% per annum compounded semi-annually then what is the equivalent
continuously compounded interest rate.

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Solution

 A = 1(1+10%/2)2 = 1e(rx1)
=> 1.1025 = er
=> r = 0.09758 = 9.758%

 Alternatively, following formulae can be used to calculate the interest rates:

• Rc = Continuous compounding interest rate
• Rm = Periodic compounding interest rate with ‘m’ periods per year
• Rc = m.ln(1+ Rm/m)

• Rm = m[e^(Rc/m) – 1]

• Try solving the above problems using these formulae

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Introduction – Bonds

 A bond is a debt security usually issued by a company or the government to raise funds
 Example: A company ABC issues bonds of worth $100. An investor ‘X’ buys the bond by paying
$100 to the company ABC. ABC promises to repay the money back to X after 5 years and also pay
5% of the $100 principle every year, semi-annually
 In the above example:
• Face Value: $100
• Coupon rate: 5%
• Time to maturity: 5 years
C+P

C C C C

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Bond pricing

 Bonds are either zero coupon bonds (having no interest payments) or coupon bonds
(with periodic interest payments)
 The price of a bond is the present value of all the coupon payment and the final principal payment
received at the end of its life
 1 
T
1 
B   Ce  Pe
 rt  rT
 (1  YTM) 
n
1
t 1 B  I    F
• B  the bond price  YTM  (1  YTM) n
• C  coupon payment  
• r  zero interest rate at time t
• P  bond principal
• T  time to maturity
 The yield of a bond is the discount rate (applied to all future cash flows) at which the present
value of the bond is equal to its market price
• Yield to Maturity = Investor’s Required Rate of Return
 The par yield is the coupon rate at which the present value of the cash flows equal to the par
value (principal value) of the bond
 If we are looking at a semi-annual 5 year coupon bond with a par value of $100 then the coupon
payment would be solved using the following equation:
5
100   (C / 2)e  rt  100e 5 r
t 1

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Question

 A 10 year bond has a yield of 12% with a 7% coupon payment annually

• What is the bond price?
• What will happen if the yield of the bond increases by 1%
 1 
1 
 (1  YTM) n  1
 Use : B  I    F
 YTM  (1  YTM) n
Or  

Years 1 2 3 4 5 6 7 8 9 10
Yield 12%
Coupon payments 7 7 7 7 7 7 7 7 7 7
Principal payment 100
PV factor 0.892857 0.797193878 0.711780248 0.635518078 0.567427 0.506631 0.452349 0.403883 0.36061 0.321973
Total PVs 6.25 5.580357143 4.982461735 4.448626549 3.971988 3.546418 3.166445 2.827183 2.52427 34.45114
Bond price 71.74888

Years 1 2 3 4 5 6 7 8 9 10
Yield 13%
Coupon payments 7 7 7 7 7 7 7 7 7 7
Principal payment 100
PV factor 0.884956 0.783146683 0.693050162 0.613318728 0.54276 0.480319 0.425061 0.37616 0.332885 0.294588
Total PVs 6.19469 5.482026784 4.851351136 4.293231094 3.79932 3.36223 2.975425 2.633119 2.330194 31.52095
Bond price 67.44254

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Why do Bonds Have Different Yields?
 Default risk – The higher the default risk, the higher the required YTM
 Liquidity – The less liquid the bond, the higher the required YTM
 Call features – Increase required YTM
• A bond that can be redeemed by the issuer prior to its maturity
 Extendible feature – Reduce required YTM
• An extendible bond gives its holder the right to "extend" its initial maturity at a specific date
or dates
 Retractable feature – Reduce required YTM
• A bond that features an option for the holder to force the issuer to redeem the bond before maturity at par
value

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Treasury zero rates

 In the case of treasury rates there are some key facts to know:
• Treasury bills are issued at a discount from face value and are paid at their par (face amount) at maturity.
The purchase price is expressed as a price per hundred dollars
• Bills are sold at a discount. The discount rate is determined at auction
• Bills pay interest only at maturity. The interest is equal to the face value minus the purchase price
• Bills are sold in increments of $100. The minimum purchase is $100
 Boot Strap Method to determine zero rates
• Consider the bond prices of Treasury bonds given below in column 4. Calculate the continuously
compounded zero rates for 6 months, 12 months, 18 months and 24months

Continuously
Bond Principal Time to Maturity Annual Coupon Bond Price Compounded 0-
rate
100 0.5 10 99.5 10.76
100 1.0 10 98.4 11.43
100 1.5 10 96.5 12.31
100 2.0 10 94.3 13.01

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Forward rate agreements (FRAs)
 In general:
R 2 T2  R 1T1
Ft1,t2 
T2  T1

 A forward rate agreement (FRA) is an over the counter agreement where the forward interest rate,
Ft1,t2 ,is fixed for a certain principal between times T1 and T2

 The payer of the fixed interest rate is also known as the borrower or the buyer. The buyer hedges against
the risk of rising interest rates, while the seller hedges against the risk of falling interest rates

 Payment to the long at settlement = Notional Principal X (Rate at settlement – FRA Rate) (days/360)
----------------------------------------------------------
1 + (Rate at settlement) (days / 360)

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Question

 An FRA settles in 30 days

• Has $1mn notional
• Is Based on 90-day LIBOR
• Forward rate of 5%, Actual 90-day LIBOR at settlement is 6%
 What is the value of FRA at settlement?

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Solution

 Value at the end of agreement = (6% – 5%) * (90/360)* $1mn = $2,500

 Value at settlement: 2,500 / (1 + (90/360)*6%) = $2,463

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Duration

 Duration it is the measure of how long on an average the holder of the bond has to wait before he
receives his payments on the bond
 A coupon paying bond’s duration would be lower than n as the holder gets some of his payments
in the form of coupons before n years
 Macaulay’s duration: is the weighted average of the times when the payments are made. And the
weights are a ratio of the coupon paid at time t to the present bond price
n
n*Mt *C
t  (1  y)
(1  y) n

Macaluay Duration  t 1

Current bond price

 Where:
• t = Respective time period
• C = Periodic coupon payment
• y = Periodic yield
• n = Total no of periods
• M = Maturity value

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Duration (Cont.…)

 Macaulay duration is also used to measure how sensitive a bond or a bond portfolio's price is to
changes in
interest rates
 A bond’s interest rate risk is affected by:
• Yield to maturity
• Term to maturity
• Size of coupon
 From Macaulay’s equation we get a key relationship:
B
  DY
B
 In the case of a continuously compounded yield the duration used is modified duration given as:
Macaulay Duration
D* 
r
1
n
 Consider a bond trading at 96.54 with duration of 4.5 years. In this case:
• ΔB = - 96.54* 4.5 Δy => ΔB = -434.43 Δy
• If there is 10 basis points increase ( + Δy) in the yield then the bond price would change by:
• ΔB = -434.43 * ( 0.001)
• ΔB = -0.43443
• Hence, B = 96.54 – 0.43443 = 96.10

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Convexity

 Duration is a good measure when the changes in yield are small

 However if the yield changes are high then we use the measure of convexity along with duration
 Convexity is a measure of the curvature of the price / yield relationship
1 d 2B
C
B dy 2
 Note that this is the second partial derivative of the bond valuation equation w.r.t. the yield
 Hence, convexity is the rate of change of duration with respect to the change in yield

Bond price ($)

P* Actual bond price

Tangent

Y* Yield

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… Convexity

 The convexity of the price / YTM graph reveals two important insights:
• The price rise due to a fall in YTM is greater than the price decline due to a rise in YTM, given an identical
change in the YTM
• For a given change in YTM, bond prices will change more when interest rates are low than when they are high
 To make the convexity of a semi-annual bond comparable to that of an annual bond, we can divide
the convexity by 4
 In general, to convert convexity to an annual figure, divide by m2, where m is the number of
payments per year

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Calculating Bond Price Changes

 We can approximate the change in a bond’s price for a given change in yield by using duration and
convexity:


VB    D Mod  i  VB   0.5  C  VB   i
2


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Question

 If yields rise by 1% per period, then by what price will the bond fall by? Assume C = 16.65.

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Solution

VB   359 
.  0.01  964.54  0.5  16.75  964.54  0.01
2
  34.63  0.81  3382
.

Note:

The 1st term “-D x ∆y x P” , can either be a positive or a negative amount, indicating an increase or
decrease in price respectively. Depending on the sign of ∆y.

The 2nd term “0.5 x C x (∆y)2 x P”, will always be a positive amount.

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Theories of the Term Structure

(1  ilt ) n  (1  ist )(1  ist )...(1  ist

year 1 year 2 yearn
Three theories are used to explain the shape of the )
term structure
 Expectations theory
• The long rate is the geometric mean of expected
future short interest rates.
(1  ilt ) n  rpn  (1  ist )(1  ist )...(1  ist
year 1 year 2 yearn
 Liquidity preference theory )
• Investors must be paid a “liquidity premium” to hold Where rpn is the risk premium associated with an n
less liquid, long-term debt. year bond
 Market segmentation theory
• Investors decide in advance whether they want to
invest in short term or the long term.
• Distinct markets exist for securities of short term
bonds and long term bonds.
• Supply demand conditions decide the prices.

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Term structures
 The term structure of interest rates is graphed as though each coupon payment of a non callable
fixed-income security were a zero-coupon bond that “matures” on the coupon payment date
 The yield curve describes the yield differential among treasury issues of differing maturities
 The Yield Curve is the graph created by putting term to maturity on the X axis, YTM on the Y axis
and then plotting the yield at each maturity
• Upward sloping: This is the most persistent shape historically when short-term interest rates and inflation
are low
• Downward sloping (Declining): This occurs at peaks in the short-term interest rate cycle, when inflation is
expected to decrease in the future
• Flat: This shape is evident during periods of interest rates transitions
• Humped: This occurs when rates are transitioning or perhaps market participants are attracted in large
numbers to particular maturity segment of the market, thereby creating the hump

Rising Declining Flat Humped

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Yield Curves for Different Risk Classes Risk Premiums (Yield Spreads)

16

14

12

Yield Spread
10
Percent

0
1 mth 3 mths 6 mths 1 yr 2yrs 5 yrs 7 yrs 10 yrs 30 yrs

Term Left to Maturity

BBB Corporates Government Bonds

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Questions

 Discuss the yield curve below and the economic impacts it conveys:

Yield

Maturity

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Questions

 What is the 1.5 year par yield of a $100 bond when the zero rates (Continuous Compounded) on
6months, 12 months and 18 months are 4%, 4.5% and 5%?

 Sol: Let it be C
100 = C*e-0.5*.04 + C*e-1*.045 + (100 + C)*e-1.5*.05
100 = C*(0.9801987 + 0.955997 + 0.927743) + 92.7743
C = 7.225651/2.863939
C = 2.523
Coupon = 2.523*2 = 5.05%

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Summary of Bond Features
 Bond price is inversely related to yield
 When yield > coupon, bond trades at discount; and when yield < coupon, bond trades at a
premium.
 Zero coupon bonds – Bonds that do not pay any coupon and are issued at a price below its face
value.
 Treasury zero rates – the yield obtained from treasury zero coupon bonds
 Treasury yield curve (term structure) – the curve representing treasury zero rates vis-à-vis the
maturity of the treasury ZCB. Generally it is upward sloping.
 Forward rate – rates implied by zero rates for a period of time in the future.
 Duration – Linear measure of interest rate risk of a bond.
 Convexity – rate of change of duration w.r.t. to interest rate changes.

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 Prices, Discounts Factors and Arbitrage
(This Reading is From Valuation & Risk Models)

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Day count conventions

 Day count defines the way in which interest is accrued over time. Day count conventions normally
used in US are:
• Actual / actual  treasury bonds
• 30 / 360  corporate bonds
• Actual/360  money market instruments
 The interest earned between two dates

(Number of days between dates)*(Interest earned in reference period)

=
(Number of days in reference period)

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Examples
 Actual / 360
• The interest price of a 91-day T-bill is 9%. Find the dollar amount of interest paid over the 91 day period and
the corresponding rate of interest
• Dollar interest is $100*0.09*91/360 = $2.275
• Rate of interest = 2.275/(100-2.275) = 2.328 %
 Actual / Actual
• A treasury bond with face value $100 pays a semi-annual coupon of 12%. Coupon payment dates are Mar 1
and Sept 1. Find the interest earned between Mar 1 & July 3
• Reference period Mar 1 to Sept 1 is 184 days
• Desired period is Mar 1 and July 3, is 124 days
• Interest earned is 124/184*6 = $4.043
 30 / 360
• A corporate bond with face value $100 pays a semi-annual coupon of 12%. Coupon payment dates are Mar 1
and Sept 1. Find the interest earned between Mar 1 & July 3
• Reference period Mar 1 to Sept 1 is 6 months with each month @30 days =180 days
• Desired period is Mar 1 and July 3, is 4*30 + 2 = 122 days
• Interest earned is 122/180*6 = $4.0666

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Treasury bonds

 The prices for treasury bonds are quoted in dollars and 1/32nd of a dollar
• $82–27 is equivalent to $82.84375

 Cash price / dirty price is the price at which the investor buys a bond from the market
• Cash price = Quoted price + accrued interest
 Accrued interest is the interest which the nearest coupon that is due generates

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Treasury bond futures
 Treasury bond futures are the most commonly traded futures contract on the Chicago board of
trade (CBOT)
 When the Treasury bond futures contract expires, any government bond with maturity more than
15 years on the first day of the delivery month and is not callable for the next 15 years from that
day can be delivered

Conversion price
 When a bond is delivered the party with the short position, the amount transacted is:
• Quoted futures price + accrued interest
• Where, Quoted futures price = settlement price * conversion factor
 The conversion factor is equal to the quoted price the bond would pay per dollar or principal on
the first day of the delivery month on the assumption that the interest rate for all maturities
equals 6% per annum (with semi annual compounding)

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Question

 The last coupon payment of $10 was paid on a treasury bond on June 19, 2009. The next coupon
is due on December 19, 2009 and we are currently on September 1, 2009. If the quoted price is
$82–27 then the cash price would be?

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Solution
 The number of days for which the December coupon has accrued interest is the time period
between June 19 to September 1 (74 days). The actual time period between June 19 and
December 19 is 183 days. Hence the interest the December coupon accrues is:
• $10 * (74/183) = 4.04
 Hence, cash price = 82.84375 + 4.04= $86.887

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Cheapest to deliver bond

 At any given time during the delivery month there are many bonds that can be delivered in the
CBOT futures contract
 The party with the short position can chose to deliver the cheapest bond when it comes to
delivery, hence he would chose the cheapest to deliver bond
 Net pay out for delivery (he has to buy a bond and deliver it):
• Quoted bond price – (settlement price * conversion factor)
• Consider an example in the table below where the short position holder has 3 options for delivery.
His cheapest to deliver bond is Bond 2
Cheapest to Deliver Bond (All figures in $)
Settlement Future Price: 94.23
Bond Quoted Bond Price Conversion Factor
1 99.6 1.033
2 135.67 1.432
3 122.45 1.257

Cost of Delivering ($)

Bond 1 99.6 - (94.23*1.033) = 2.26041
Bond 2 135.67 - (94.23*1.432) = 0.73264
Bond 3 122.45 - (94.23*1.257) = 4.00289

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 One Factor Risk Metrics and Hedges
(This Reading is From Valuation and Risk Models)

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DV01

 DV01: The price value of basis point (PVBP) or dollar value of basis point (DV01) change is the
absolute change in the bond price from one basis point change in yield.
• DV01 =  price at YTM0 – price at YTM1 
 YTM0 = the initial YTM
 YTM1 = the YTM 1 basis point above or below YTM0 (YTM1 = YTM0 ± 0.0001)

DVBP  [ D *P0 ]  0.0001

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DV01 – Application to hedging
 Hedge ratio is calculated using DV01 with the help of following relation

HR = DV01 (per $100 of initial position)

DV01 (per $100 of hedging instrument)

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Duration based hedging strategies

 Considering a situation where an asset that is interest rate dependant is hedged using an interest
rate futures contract
 In such cases the number of contracts to hedge is given by the equation below:

PDP
N* 
FC DF

• FC Contract price for interest rate futures

• DF Duration of asset underlying futures at maturity
• P Value of portfolio being hedged
• DP Duration of portfolio at hedge maturity

 When hedges are constructed using interest rates it is important to note that interest rate and
futures prices move in opposite directions . So if one is expecting to lose money when the interest
rate falls, one should long futures contracts so that they can hedge their losses by gains in futures
prices

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Question

 An investor has invested $10m in government bonds and is expecting the interest rates to rise in
the next 6 months so he decides to hedge himself by interest rate futures. It is currently June and
he decides to use the December T-bond futures contract for the hedge. If the current futures price
is 97.2345 and the duration of the portfolio of government bonds at the end of 6 months is 7.1
years. The duration of the cheapest to deliver T-bond in December is given as 9.121 years. What
position should the investor take in the futures contract? How many futures contract should long /
short for the hedge if each contract is for the delivery of $100,000 face value?

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Solution

 The number of contracts that should be shorted is:

10,000,000 7.1
  80
(97.2345100,000 / 100) 9.121

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Portfolio Duration and Convexity

 Portfolio duration is the weighted sum of durations of individual securities.

 Weight of each security is the value of the security divided by the value of the portfolio.
 Portfolio convexity is calculated in the same way portfolio duration. It is the weighted sum of
durations of individual securities.

Negative Convexity
 Callable bonds exhibit negative convexity when yields fall below certain level.
 At lower yield, there is incentive for the issuer to call the bond.
 Price curve of the bond bends away from the normal curve thereby exhibiting negative convexity.

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Barbell and Bullet Strategy

 In barbell strategy, investor uses the bonds of short and long maturities and does not invest in the
bonds of intermediate maturity.
 In bullet strategy, investor uses the bonds concentrated in intermediate maturity range.
 In volatile rate environment, barbell strategy is preferred over bullet strategy.

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 Multi Factor Risk Metrics and Hedges
(This Reading is From Valuation and Risk Models)

© EduPristine For FMP-II (2016)

Weakness of Single Factor Approach

 Single factor approach assumes that all the future rate changes are driven by single factor.
 The same change in interest rate is assumed for the entire yield curve.
 In practice, change in short term interest rate might be different from the change in long term
interest rate.
 Same hedging instrument cannot be used for hedging the change in short term interest rate and
long term interest rate.

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Key Rate Exposures

 Key Rates are the rates selected at key point on the yield curve. These are usually 2, 5, 10 and 30
year rates.
 Key rate exposures hedge risk by using rates from a small number of available liquid bonds.
 Partial ‘01 is used to measure the risk of the bond or swap portfolio in terms of liquid money
markets and swap instruments.
 Forward ’01 is used to measure the risk of the bond or swap portfolio in terms of shifts in the
forward rates.

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Key Rate Shift

 Key Rate Shift technique is a approach to nonparallel shift in the yield curve.
 This technique allows to determine changes in all the rates due to the changes in key rates.
 Choice has to be made as to which key rates shifts and how the key rate movement relate to prior
or subsequent maturity key rates.

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Key Rate ‘01 and Key Rate Durations

 Key Rate ‘01 measures the dollar change in the value of the bond for every basis point shift in the
key rate
• Key Rate ‘01 = (-1/10,000) * (Change in Bond Value/0.01%)

 Key rate duration provides the approximate percentage change in the value of the bond
• Key Rate Duration = (-1/BV) * (Change in Bond Value/Change in Key rate)

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Hedging with Key Rate Exposures

 Hedging positions can be created in response to key rate shifts.

 This can be done by equating individual key rate exposures next to key rate shifts to the overall key
rate exposure for that particular key rate change.
 The above step indicates either a long position or a short position in securities to protect against
the changes in interest rate surrounding key rate shifts.

© EduPristine For FMP-II (2016) 51

 Mechanics of Options Markets

© EduPristine For FMP-II (2016)

Agenda

 Introduction to Options
• What are Options
• Intrinsic Value of Options
• Returns to Option buyers and sellers
• Put Call Parity
• Bounds and Option Values
• Determinants of Option Values
• Some special cases
• Summary

© EduPristine For FMP-II (2016) 53

What are Options?

 Options are contracts that give its buyer the right to buy or sell a particular asset
• In future
• At a pre-decided price (i.e. exercise or strike price)
• Without any obligations
 The seller of the option collects a payment (Premium) from the buyer for providing the option
 Types of options:
• Call or Put Options
 Call Option: Gives option holder the right to buy the asset at an agreed price
 Put Options: Gives option holder the right to sell the asset at an agreed price
• European or American Options
 European options: Are those that can only be exercised on expiration
 American options: May be exercised on any trading day on or before expiration

© EduPristine For FMP-II (2016) 54

Question

 Early exercise of an option is more likely for:

A. European calls options on stocks paying large dividends.
B. American call options on stocks paying small dividends.
C. American put options deep in the money and close to maturity.
D. American put options on stocks paying large dividends.

© EduPristine For FMP-II (2016) 55

Solution

 C.
A. European options cannot be exercised early
B. Small dividends will not make much of a difference in the price fall in the stock
C. A deep in the money put option should always be exercised early because it is likely that the stock might
recover from the fall
D. Though this might be profitable if the stock prices significantly fall after the ex-dividend date but the third
option is likely to provide more profit

© EduPristine For FMP-II (2016) 56

Question

 Assuming the stock price and all other variables remain the same what will be the impact of an
increase in the risk-free interest rate on the price of an American put option?
A. No impact
B. Negative
C. Positive
D. Cannot be determined

© EduPristine For FMP-II (2016) 57

Solution

 B. (Negative)

© EduPristine For FMP-II (2016) 58

 Intrinsic Value of Options

© EduPristine For FMP-II (2016)

Intrinsic Value of Options

 Intrinsic value: is the maximum of zero and the value of the option if the option were exercised
immediately
• At the money:
 When the price of the underlying is the same as the strike price of the option, the option is termed at the money
and exercising it carries a nil pay-off
• In the money:
 When the price of the underlying is greater than the strike price carried by a call option, the call option is termed in
the money, as exercising it results in a positive pay off
 When the price of the underlying is less than the strike price carried by a put option, the put option is termed in
the money, as exercising it results in a positive pay off
• Out of the money:
 When the price of the underlying is less than the strike price carried by a call option, the call option is termed out
of the money, as exercising it will result in a nil pay off
 When the price of the underlying is greater than the strike price carried by a put option, the put option is termed
out of the money, as exercising it will result in a nil pay off

© EduPristine For FMP-II (2016) 60

Intrinsic Value of Options
 Illustration: Pay-offs from buying a Call
• Call option is written on the stock of XYZ Corporation with a strike price of 5. Consider options on a stock
whose price is expected to range from 0–10 at the time of expiration.
• If share price is less than 5, then the pay off to the option buyer is nil.
• If the price is more than 5, the pay-off moves upward linearly with the share price.
Stock Strike
Call Value Put Value
Price Price
Call Value
0 0 5 5
1 0 4 5
6
2 0 3 5

Call-Pay off
3 0 2 5 4
4 0 1 5
5 0 0 5
2
6 1 0 5
7 2 0 5
8 3 0 5 0
0 4 8 12
9 4 0 5
Stock Price
10 5 0 5

© EduPristine For FMP-II (2016) 61

Intrinsic Value of Options
 Illustration: Pay-offs from selling a Call
• Call option is written on the stock of XYZ Corporation with a strike price of 5. Consider options on a stock
whose price is expected to range from 0–10 at the time of expiration.
• If share price is less than 5, then the pay off to the option seller is nil.
• If the price is more than 5, the pay-off moves downward linearly with the share price.
Stock Strike
Call Value Put Value Short Call-Pay off
Price Price
0 4 8 12
0 0 -5 5 0
1 0 -4 5
2 0 -3 5

Short Call-Pay off

3 0 -2 5 -2
4 0 -1 5
5 0 0 5 -4
6 -1 0 5
7 -2 0 5
-6
8 -3 0 5
9 -4 0 5 Stock Price
10 -5 0 5

© EduPristine For FMP-II (2016) 62

Intrinsic Value of Options
 Illustration: Pay-offs from buying a Put
• Put option is written on the stock of XYZ Corporation with a strike price of 5. Consider options on a stock
whose price is expected to range from 0–10 at the time of expiration.
• If share price is more than 5, then the pay off to the option buyer is nil.
• If the price is less than 5, the pay-off moves linearly with the share price.
Stock Strike
Call Value Put Value
Price Price Put Value
0 0 5 5 6
1 0 4 5

Put-Pay off
2 0 3 5
3 0 2 5 4
4 0 1 5
5 0 0 5 2
6 1 0 5
7 2 0 5
0
8 3 0 5 0 4 8 12
9 4 0 5 Stock Price
10 5 0 5

© EduPristine For FMP-II (2016) 63

Intrinsic Value of Options
 Illustration: Pay-offs from selling a Put
• Put option is written on the stock of XYZ Corporation with a strike price of 5. Consider options on a stock
whose price is expected to range from 0–10 at the time of expiration.
• If share price is more than 5, then the pay off to the option seller is nil.
• If the price is less than 5, the pay-off moves downward linearly with the share price.
Stock Strike
Call Value Put Value Short Put-Pay off
Price Price
0 4 8 12
0 0 -5 5 0
1 0 -4 5
2 0 -3 5

Short Put-Pay off

3 0 -2 5 -2
4 0 -1 5
5 0 0 5 -4
6 -1 0 5
7 -2 0 5
-6
8 -3 0 5
9 -4 0 5 Stock Price
10 -5 0 5

© EduPristine For FMP-II (2016) 64

Payoffs from Options

Long call payoff = Max (ST – X,O) Long put payoff = Max (X – ST,O)

0 0
X ST X ST

X X
0 ST 0 ST

Short call Short put

© EduPristine For FMP-II (2016) 65

 Returns to Option Buyers and Sellers

© EduPristine For FMP-II (2016)

Returns to Option Sellers

 Returns to Option sellers:

• The price that the option writer gets for underwriting the contract is called premium.
• If the option is not exercised, the option writer makes profit from the premium.
• If the option is exercised, the option writer may make profit or loss depending on the spot price of the
underlying asset at the time.
 Example: A Call option writer gets premium of 1 for an option with strike price of 5
• He makes a profit if:
 The option is not exercised when spot price is less than 5. The profit is 1 (i.e. premium).
 The option is exercised and spot price is more than 5 but less than 6.
 The profit to the call writer is less than 1.
 If the spot price is 6, the writer has no profit and no loss.
 For all spot prices more than 6, the call writer makes losses, which increase linearly with increase in spot prices.

© EduPristine For FMP-II (2016) 67

Returns to Option Sellers

Short put payoff with premium

0 2 4 6 8 10 12
2
1

Short put payoff

Stock Option Short-Call Put Strike 0
Price Premium Value Value Price -1
-2
0 1 1 -4 5
-3
1 1 1 -3 5 -4
2 1 1 -2 5 -5
Stock price
3 1 1 -1 5
4 1 1 0 5
5 1 1 1 5 Short call payoff with premium
6 1 0 1 5 0 2 4 6 8 10 12
2
7 1 -1 1 5 1

Short call payoff

8 1 -2 1 5 0
9 1 -3 1 5 -1
-2
10 1 -4 1 5
-3
-4
-5
Stock price

© EduPristine For FMP-II (2016) 68

Returns to Option Buyers

 Profit to Option buyers:

• The pay-off are distinct from the profit (or loss) to Long call payoff with premium
the option holder. 5
• To estimate the profit, the premium (price of 4

Long call payoff

option) is to be subtracted from the pay-off. 3
 Illustration: In continuation to above, further 2
consider options which carries a premium of 1. 1
0
Stock Option Call Put Strike -1
Price Premium Value Value Price -2
0 1 -1 4 5 0 2 4 6 8 10 12
Stock price
1 1 -1 3 5
2 1 -1 2 5 Long put payoff with premium
3 1 -1 1 5
5
4 1 -1 0 5 4

Long put payoff

5 1 -1 -1 5 3
6 1 0 -1 5 2
1
7 1 1 -1 5
0
8 1 2 -1 5 -1
9 1 3 -1 5 -2
10 1 4 -1 5 0 2 4 6 8 10 12
Stock price
© EduPristine For FMP-II (2016) 69
 Put Call Parity

© EduPristine For FMP-II (2016)

Put Call parity

 Consider the Pay-off of a trader who has the following position:

• A Call Option with a Strike Price of 5 and
• A Bond with a maturity value of 5.

Share Price Call Bond Value

Strike Price Bond + Call
at Expiration Pay-Off at Maturity
0–5 0 5 5 5
6 1 5 5 6
7 2 5 5 7
8 3 5 5 8
9 4 5 5 9
10 5 5 5 10

© EduPristine For FMP-II (2016) 71

Put Call parity

 Consider, now, the Pay-off of a trader who has:

• A Put Option with a Strike Price of 5 and
• An equivalent unit of the underlying asset.

Share Price Put Pay-Off (Exercise Stock Stock+

at Expiration Price 5) Pay-off Put

0 5 0 5

1 4 1 5

2 3 2 5

3 2 3 5

4 1 4 5

5–10 0 5–10 5–10

© EduPristine For FMP-II (2016) 72

Put Call parity

 The Pay-offs are exactly the same

12
10
Total payoff

8
6
4
2
0
0 2 4 6 8 10 12

Share price

© EduPristine For FMP-II (2016) 73

Question: Put Call parity

 According to Put Call parity for European options, purchasing a put option on ABC stock will be
equivalent to
A. Buying a call, buying ABC stock and buying a Zero Coupon bond
B. Buying a call, selling ABC stock and buying a Zero Coupon bond
C. Selling a call, selling ABC stock and buying a Zero Coupon bond
D. Buying a call, selling ABC stock and selling a Zero Coupon bond

© EduPristine For FMP-II (2016) 74

Solution: Put Call parity

 B: p + S0 = c + Ke-rT

© EduPristine For FMP-II (2016) 75

Put Call parity

 Put Call parity provides an equivalence relationship between the Put and Call options of a
common underlying and carrying the same strike price.
 It can be expressed as:
• Value of call + Present value of strike price = value of put + share price
 If value of put is not available, it can be derived as:
• Value of put = Value of call + present value of strike price - share price
 Put-call parity relationship, assumes that the options are not exercised before expiration day, i.e. it
follows European options.
 This holds true for American options only if they are not exercised early.
 In case of dividend-paying stocks, either the amount of dividend paid should be known in advance
or it is assumed that the strike price factors the future dividend payment.
 The mathematical representation of Put Call Parity is:
X d
Pr emium(C )  PV of strike price  PV of dividends
1  rt 1  rt
= Initial stock price (S) + Put premium (P)

Put Call Parity is valid only for European options, for American Options this relationship turns into an inequality

© EduPristine For FMP-II (2016) 76

Question: Put Call parity

 Consider a 1-year European call option with a strike price of $27.50 that is currently valued at
$4.10 on a $25 stock. The 1-year risk-free rate is 6%.What is the value of the corresponding put
option?
A. 4.1
B. 5
C. 6
D. 25

© EduPristine For FMP-II (2016) 77

Solution: Put Call parity

 p + S0 = c + D + Xe-rt

© EduPristine For FMP-II (2016) 78

 Bounds and Option Values

© EduPristine For FMP-II (2016)

Bounds and Option Values

 The value of an option changes over its life.

 Consider the earlier illustration of the call
• If the share price of is below 5 on the exercise date, the call will be worthless.
• If the stock price is above 5, the call will be worth 5 less than the value of the stock.
• Even before maturity of the option, its value can never remain below this lower-bound line.
• For options that still have some time to run, the heavy lower line is thus the lower-bound limit on the
market price of the option.
• The diagonal line in the plot is the upper bound limit to the option price, because the stock gives a higher
ultimate pay-off than the option.

4
Call payoff

0
0 1 2 3 4 5 6 7 8 9 10 11
Share price

© EduPristine For FMP-II (2016) 80

Bounds and Option Values
 The value of an option changes over its life.
 Consider the earlier illustration of the call
• If at the option’s expiration, stock price > exercise price, the option is worth the stock price minus the
exercise price.
• If the stock price < exercise price, the option is worthless. But the share owners still have a valuable financial
asset in the form of stock of ABC Corporation.
• The value of the option would lie between these two bounds throughout the option’s life.

4
Call payoff

0
0 1 2 3 4 5 6 7 8 9 10 11
Share price

© EduPristine For FMP-II (2016) 81

 Determinants of Option Values

© EduPristine For FMP-II (2016)

Bounds and Option Values

Option Minimum Value Maximum Value

European call (c) ct ≥ Max(0,St-(X/(1+RFR)t) St

American Call (C) Ct ≥ Max(0, St-(X/(1+RFR)t) St

European put (p) pt ≥Max(0,(X/(1+RFR)t)-St) X/(1+RFR)t

American put (P) Pt ≥ Max(0, (X-St)) X

Where t is the time to expiration

© EduPristine For FMP-II (2016) 83

Question
 If a European call option is written on a dividend paying stock, an increase in which of the
following will not automatically result in an increased option price?
A. The stock price
B. The risk-free rate
C. The time to expiration
D. The volatility of the stock price

© EduPristine For FMP-II (2016) 84

Solution
 C.
• The time to expiration

© EduPristine For FMP-II (2016) 85

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