MA634 Financial Risk Management

Time Value of Money, Interest Rates and Bonds

Time Value of Money

A stream of cash flows can be shown on a timeline, which visually maps out when each payment
occurs. Using a timeline is the first step in organizing and solving financial problems. For example,
consider the following timeline. It describes an outflow of $1000 (negative cash flow) at time 0 (today),
with two positive (inflow) cash flows of $400 and $600 at times 1 and 2 respectively. The period can
be any period, months or years depending on the context.

Date 0 1 2

Cash Flow −1000 400 600

Financial decisions often require comparing or combining cash flows that occur at different points in
time. Most important principle is “A dollar today and a dollar in one year are not equivalent. Having
money now is more valuable than having money in the future; if you have the money today you can
earn interest on it.” Therefore, one needs to convert the cash flows into the same units or move them
to the same point in time to compare or combine cash flows that occur at different points in time.
Further, to move a cash flow forward in time, you must compound it. That is, if the market interest
rate for the year is r, then we multiply by the interest rate factor, 1 + r, to move the cash flow from
the beginning to the end of the year. This process of moving a value or cash flow forward in time
is known as compounding. In general, to take a cash flow C forward n periods into the future, we
must compound it by the n intervening interest rate factors. If the interest rate r is constant, then

Future Value of a Cash Flow F Vn = C × (1 + r)n

Similarly, to move a cash flow backward in time, you must discount it. This process of moving a value
or cash flow backward in time—finding the equivalent value today of a future cash flow—is known as
discounting.

C
Present Value of a Cash Flow P V = C ÷ (1 + r)n =
(1 + r)n

Consider a stream of cash flows

0 1 2 N
···
C0 C1 C2 CN

This timeline provides the general formula for the present value of a cash flow stream:

N N
C1 C2 CN X X Cn
P V = C0 + + 2
+ · · · + N
= P V (C n ) =
(1 + r) (1 + r) (1 + r) (1 + r)n
n=0 n=0

1
Future Value of a Cash Flow Stream is given by

F VN = P V × (1 + r)N = C0 (1 + r)N + C1 (1 + r)N −1 + . . . + CN

Perpetuities and Annuities

• A perpetuity is a constant cash flow C paid every period, forever.

0 1 2 3
···
C C C

The present value of a perpetuity is

C
r

• An annuity is a constant cash flow C paid every period for N periods.1

0 1 2 N
···
C C C

The present value of an annuity is  

1 1
C× 1−
r (1 + r)N

The future value of an annuity at the end of the annuity is

1
(1 + r)N − 1


C×
r

• In a growing perpetuity or annuity, the cash flows grow at a constant rate g each period.

0 1 2 3 4
···
C C × (1 + g) C × (1 + g)2 C × (1 + g)3

The present value of a growing perpetuity is

C
r−g

The present value of a growing annuity is

0 1 2 N
···
C C(1 + g) C(1 + g)N −1

 N !
1 1+g
PV = C × 1−
r−g 1+r
1
We assume that it is paid in arrears, starting at the end of the first period. The other term is annuity due, that
means the first payment begins immediately.

2
Exercise: Verify the above formulae.

Internal Rate of Return

Sometimes, you may know the present value and cash flows of an investment, but not the interest rate
that links them. This rate, known as the internal rate of return (IRR), is the interest rate that makes
the net present value of the cash flows equal to zero.

Example
Anderson, Patel, and Rivera are considering investing in Olivia’s new software company. They agree
to provide her with $1,000,000 today to fund development. In return, Olivia promises to pay them
$125,000 at the end of each year for the next 30 years. Assuming Olivia makes all payments as
agreed, what is the internal rate of return (IRR) on Anderson, Patel, and Rivera’s investment?

Solution
Here’s the timeline (from Anderson, Patel, and Rivera’s perspective):

0 1 2 30
···
−$1,000,000 $125,000 $125,000 $125,000

The cash flows form a 30-year annuity. Setting the net present value equal to zero:
 
1 1
1,000,000 = 125,000 × 1−
r (1 + r)30

The IRR can be found by solving for r.

3
Interest Rates
Interest rates can be expressed in many forms. Although they are usually quoted as annual rates,
interest payments may be made more frequently, such as monthly or semiannually. When analyzing
cash flows, it’s essential to use a discount rate that aligns with the timing of those cash flows, ensuring
it reflects the actual return achievable over that specific period.

Key Concept: A 10% per annum interest rate may sound clear, but its meaning depends on how it
is measured and compounded.

Measuring Interest Rates

(i) Effective Annual Rate (EAR): Interest rates are often expressed as the effective annual rate
(EAR), which represents the total amount of interest actually earned over the course of a year.
For example, an EAR of 5% means $100 becomes $105 after one year:

100 × (1.05) = 105

Over 2 years, it becomes

100 × (1.05)2 = 110.25

We can find equivalent interest rates for periods longer or shorter than one year. This is done
by raising (1 + r) to the appropriate power. For shorter periods, use a fractional power. For
example: 5% EAR to 6-month rate:

(1 + r)1/2 = (1.05)1/2 ≈ 1.0247

This means about 2.47% every six months. One can verify this as, two 6-month periods at 2.47%
each:
(1.0247)2 ≈ 1.05

This matches the original 5% EAR. Thus, we have the following general formula for converting
rates,
Equivalent n-Period Rate = (1 + r)n − 1

• If n > 1: Gives the rate for more than one period.

• If n < 1: Gives the rate for a fraction of a period.

(ii) Annual Percentage Rates (APR)

APR is the annual rate of simple interest earned in one year. It does not include the effect of
compounding. Because of this, APR is usually less than the actual amount of interest you will
earn. To find the true annual return, we must convert APR to the Effective Annual Rate
(EAR).

Example: Monthly Compounding Suppose a bank offers a savings account with 6% APR
and monthly compounding. Then, the monthly interest rate = 6%/12 = 0.5% = 0.005. Over
one year, we have
$1 × (1.005)12 = 1.061678

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 This equals an EAR of 6.1678%. Thus, the EAR is higher than the quoted APR due to
compounding (earning interest on interest). The APR with k compounding periods/year
shows the rate earned each period:

APR
Rate per period =
k

Finally, if we are compounding for k periods/year, we have

 k
APR
1 + EAR = 1+
k

Compounding Frequency Matters:

• Annual compounding: $100 grows to $110.

• Semiannual compounding: $100 × 1.05 × 1.05 = $110.25.

• Quarterly compounding: $100 × 1.0254 = $110.38.

General Formula:
R mn
 
Future Value = A 1 +
m
• A = Initial Amount

• R = Annual Interest Rate

• m = Compounding periods per year

• n = Number of years

When m = 1, rate is called Equivalent Annual Interest Rate.

Observation: Higher compounding frequency → Higher future value.

Converting Between Two Compounding Frequencies:

 m1  m2
R1 R2
1+ = 1+
m1 m2
"  m1 #
R1 m2
R2 = m2 1+ −1
m1

Example:
R1 = 6%, m1 = 2, m2 = 4
" 2 #
0.06 4
R2 = 4 1 + − 1 = 0.0596 (5.96%)
2

Continuous Compounding
Concept:

• When compounding frequency m → ∞, we get continuous compounding.

5
• Future Value:
FV = AeRn

Example:
A = 100, R = 0.1, n = 1 =⇒ 100e0.1 = 110.52

Conversion Between Continuous and Periodic Rates:

Rm m
 
Rc
e = 1+
m

Formulas:  
Rm
Rc = m ln 1 +
m
 
Rm = m eRc /m − 1

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Zero Rates
The n-year zero-coupon interest rate is the rate earned on an investment starting today, lasting n
years, with no intermediate payments. It is also called: n-year spot rate, n-year zero rate. Suppose a
5-year zero rate with continuous compounding is quoted as 5% per annum. This means that $100, if
invested for 5 years, grows to

Future Value = 100 × e0.05×5 = 128.40

Most observed market rates are not pure zero rates. For example: 5-year risk-free bond with 6% coupon
does not directly reveal zero rate. We next discuss bond pricing and then we discuss bootstrap method
to determine zero rates from bond prices.

Bond Pricing
Most bonds make periodic coupon payments to their holders, while the principal—also referred to
as the par value or face value—is repaid at maturity. The theoretical price of a bond is determined
by calculating the present value of all future cash flows the bondholder will receive. While some
bond traders apply a single discount rate to all cash flows, a more precise method is to discount each
payment using its corresponding zero rate.

Example: 2-Year Bond with 6% Coupons (Semiannual) with following zero rates

Maturity (yrs) Zero Rate (%)

0.5 5.0
1.0 5.8
1.5 6.4
2.0 6.8

Price = 3e−0.05×0.5 + 3e−0.058×1.0 + 3e−0.064×1.5 + 103e−0.068×2.0 = 98.39

Bond Yield

• single discount rate that equates bond price to its market value.

• For a bond with price $98.39 and cash flows:

3e−y×0.5 + 3e−y×1.0 + 3e−y×1.5 + 103e−y×2.0 = 98.39

• Solve iteratively (trial and error or Newton-Raphson) to get:

y = 6.76% (continuously compounded)

Par Yield

• coupon rate that makes bond price = par value.

• Assume semiannual coupons (c/2 per period).

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That is,
c −0.05×0.5 c −0.058×1.0 c −0.064×1.5  c  −0.068×2.0
e + e + e + 100 + e = 100
2 2 2 2

Determining Zero Rates: Bootstrap Method

Consider the following example data on the prices of five bonds.

Principal ($) Maturity (yrs) Annual Coupon∗ ($) Price ($) Yield (%)∗∗
100 0.25 0 99.6 1.6064 (Q)
100 0.50 0 99.0 2.0202 (SA)
100 1.00 0 97.8 2.2495 (A)
100 1.50 4 102.5 2.2949 (SA)
100 2.00 5 105.0 2.4238 (SA)

Notes: a) *Coupon payments assumed semiannual. b) **Yield compounding: Q = Quarterly, SA =

Semiannual, A = Annual.

The 3-month bond has the effect of turning an investment of 99.6 into 100 in 3 months. The continu-
ously compounded 3-month rate R is therefore given by solving

100 = 99.6eR×0.25

It is 1.603% per annum. The 6-month continuously compounded rate is similarly given by solving

100 = 99.0eR×0.5

It is 2.010% per annum. Similarly, the 1-year rate with continuous compounding is given by solving

100 = 97.8eR×1.0

It is 2.225% per annum. The fourth bond lasts 1.5 years. Suppose the 1.5-year zero rate is denoted
by R. It follows that

2e−0.02010×0.5 + 2e−0.02225×1.0 + 102e−R×1.5 = 102.5

This reduces to
e−1.5R = 0.96631

or
ln(0.96631)
R=− = 0.02284
1.5

The 1.5-year zero rate is therefore 2.284%. This is the only zero rate that is consistent with the
6-month rate, 1-year rate, and the data in Table. Exercise Find 2-year zero rate.

A chart showing the zero rate as a function of maturity is known as the zero curve.

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Forward Rates and Calculations
Forward interest rates are the interest rates for future time periods, based on today’s zero rates. For
example, suppose zero rates are given in the Table below and are continuously compounded.

Year (n) Zero Rate (%)

1 3.0
2 4.0
3 4.6
4 5.0
5 5.3

A zero rate of 3% for 1 year means $100 invested today grows to

100e0.03×1 = 103.05

after 1 year.

A zero rate of 4% for 2 years means $100 invested today grows to

100e0.04×2 = 108.33

after 2 years.

The forward rate for year 2 is 5%. This is the rate between the end of year 1 and the end of year 2,
implied by the 1-year zero rate (3%) and the 2-year zero rate (4%). It’s the rate for year 2 that makes
the overall return for 2 years equal to 4% per year.

To check: invest $100 for 1 year at 3% and for the next year at 5%:

100e0.03×1 e0.05×1 = 108.33

This matches the result from investing for 2 years at 4%:

100e0.04×2 = 108.33

In general, if R1 and R2 are the zero rates for maturities, T1 and T2 , respectively, and RF is the
forward interest rate for the period of time between T1 and T2 , then

R2 T2 − R1 T1
RF =
T2 − T1

Alternatively, we can write the following.

T1
RF = R2 + (R2 − R1 )
T2 − T1

This shows that

• If zero curve is upward sloping (R2 > R1 ), then RF > R2 .

• If zero curve is downward sloping (R2 < R1 ), then RF < R2 .

9
Calculating forward rates for the above table, we have

Year n Forward rate

for nth year
(% per annum)
2 5.0
3 5.8
4 6.2
5 6.5

If a large financial institution can borrow or lend at the rates in the above Table, it can lock in the
forward rates.

For example, it can borrow $100 at 3% for 1 year and invest the money at 4% for 2 years. The result
is a cash outflow of
100e0.03×1 = 103.05

at the end of year 1 and an inflow of

100e0.04×2 = 108.33

at the end of year 2. Since

108.33 = 103.05e0.05 ,

a return equal to the forward rate (5%) is earned on 103.05 during the second year.

Alternatively, it can borrow $100 for 4 years at 5% and invest it for 3 years at 4.6%. The result is a
cash inflow of
100e0.046×3 = 114.80

at the end of the third year and a cash outflow of

100e0.05×4 = 122.14

at the end of the fourth year. Since

122.14 = 114.80e0.062 ,

money is being borrowed for the fourth year at the forward rate of 6.2%.

If a big investor believes future interest rates will be different from today’s forward rates, they can use
various trading strategies to try to profit. One common approach is using a contract called a forward
rate agreement.

Forward Rate Agreements (FRAs)

A forward rate agreement (FRA) is a forward contract obligating two parties to agree that a
certain interest rate will apply to a principal amount during a specified future time. Obviously,
forward rates play a crucial role in the valuation of FRAs. The T2 cash flow of an FRA that promises
the receipt or payment of RX is:

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 cash flow (if receiving RX ) = L × (RX − R) × (T2 − T1 )

cash flow (if paying RX ) = L × (R − RX ) × (T2 − T1 )

where:

• L = principal

• RX = annualized rate on L, expressed with compounding period T2 − T1

• R = annualized actual rate, expressed with compounding period T2 − T1

• Ti = time i, expressed in years

The value of an FRA if receiving or paying is:

value (if receiving RX ) = L × (RX − RForward ) × (T2 − T1 ) × e−R2 ×T2

value (if paying RX ) = L × (RForward − RX ) × (T2 − T1 ) × e−R2 ×T2

where:
RForward = forward rate between T1 and T2

Note that R2 is expressed as a continuously compounded rate.

Additionally, any interest should be due at the end of the FRA period (e.g., six months in the example
that follows); however, the common practice is for the FRA to be settled at the beginning of the FRA
period. Therefore, any payoffs must be discounted at the relevant discount rate (e.g., the floating rate
for three months in the example that follows).

EXAMPLE: Computing the Value of an FRA

Suppose the three-month and six-month LIBOR spot rates are 4% and 5%, respectively (continuously
compounded rates). An investor enters into an FRA in which she will receive 8% (assuming quarterly
compounding) on a principal of $5,000,000 between months 3 and 6. Calculate the value of the FRA.

Answer:
 
1
RForward = 0.05 + (0.05 − 0.04) × = 0.06 = 6%
2−1

 0.06 
RForward (with quarterly compounding) = 4 × e 4 − 1 = 4 × 0.006452 = 6.05%

value = $5,000,000 × (0.0800 − 0.0605) × (0.50 − 0.25) × e−(0.05)(0.5) = $23,773

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Interest Rate Risk
When companies borrow money, they have to pay interest. If interest rates go up, borrowing becomes
more expensive and profits may fall. Many companies also have fixed long-term obligations, like lease
payments or pension promises. If interest rates go down, the value of these future payments (in today’s
terms) goes up, which can hurt the company’s value. Thus, when interest rates are volatile, interest
rate risk is a concern for many firms.

Interest Rate Risk Measurement: Duration

Observe that the sensitivity of zero-coupon bonds to interest rates increases with their maturity. For
example, for a 10-year, zero-coupon bond, an increase of one percentage point in the yield to maturity
from 5% to 6% causes the bond price per $100 face value to fall from

100 100
= $61.39 to = $55.84
1.0510 1.0610

or a price change of (55.84 − 61.39)/61.39 = −9%. The price of a five-year bond drops only 4.6% for
the same yield change. The interest rate sensitivity of a single cash flow is roughly proportional to
its maturity. The farther away the cash flow is, the larger the effect of interest rate changes on its
present value.

Now consider a bond or portfolio with multiple cash flows. How will its value change if interest rates
rise? Because the interest rate sensitivity of a cash flow depends on its maturity, the interest rate
sensitivity of a security with multiple cash flows depends on their value-weighted maturity. Thus, we
formally define a security’s duration as follows:2
X P V (Ct )
Duration = × t,
t
B

where Ct is the cash flow on date t, P V (Ct ) is its present value (evaluated at the bond’s yield), and
X
B= P V (Ct ),
t

is the total present value of the cash flows, which is equal to the bond’s current price. Therefore, the
duration weights each maturity t by the percentage contribution of its cash flow to the total present
value, P V (Ct )/B. Note that the duration of a bond measures how long the holder must wait to
receive the present value of cash flows.

More specifically, suppose a bond provides cash flows ci at time ti (1 ≤ i ≤ n). The bond price B and
continuously compounded yield y are related by:
n
X
B= ci e−yti ,
i=1

The (Macaulay) duration of the bond, D, is defined as:

Pn −yti
i=1 ti ci e
D= ,
B
2
This measure is also called the Macaulay duration.

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Or equivalently:
n
ci e−yti
X  
D= ti .
B
i=1

Example: What is the duration of a 10-year, zero-coupon bond? What is the duration of a 10-year
bond with 10% annual coupons trading at par?

Solution: For a zero-coupon bond, there is only a single cash flow. Thus, the duration is equal to
the bond’s maturity of 10 years.

For the coupon bond, because the bond trades at par, its yield to maturity equals its 10% coupon
rate. Table below shows the calculation of the duration of the bond.

Computing the Duration of a Coupon Bond

t (years) Ct P V (Ct ) P V (Ct )/P [P V (Ct )/P ] × t

1 10 9.09 9.09% 0.09

2 10 8.26 8.26% 0.17
3 10 7.51 7.51% 0.23
4 10 6.83 6.83% 0.27
5 10 6.21 6.21% 0.31
6 10 5.64 5.64% 0.34
7 10 5.13 5.13% 0.36
8 10 4.67 4.67% 0.37
9 10 4.24 4.24% 0.38
10 110 42.41 42.41% 4.24

Duration =
Bond price = 100.00
6.76 yrs

Remark 0.1. Because the bond pays coupons prior to maturity, its duration is shorter than its 10-year
maturity. Moreover, the higher the coupon rate, the more weight is put on these earlier cash flows,
shortening the duration of the bond.

Duration and Interest Rate Sensitivity:

When a small change ∆y in yield occurs, we have (by Taylor’s approximation)

dB
∆B = ∆y
dy

This becomes:
n
X
∆B = −∆y ci ti e−yti
i=1

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Thus, the key duration relationship is:

∆B = −BD∆y

Or equivalently:
∆B
= −D∆y
B

If y is expressed with annual compounding, the relationship becomes.3 :

BD∆y
∆B = − .
1+y

More generally, if y is expressed with a compounding frequency of m times per year:

BD∆y
∆B = − y .
1+
m

Define:
D
D∗ = y ,
1+
m
which is the Modified Duration. Then:

∆B = −BD∗ ∆y.

Example 0.2. Let the bond price B is 94.213 and the duration D is 2.653. Calculate the change in
bond price (∆B) when the yield increases by 10 basis points (0.1%), i.e., ∆y = +0.001.

∆B = −94.213 × 2.653 × ∆y

or
∆B = −249.95 × ∆y

When the yield on the bond increases by 10 basis points (= 0.1%), ∆y = +0.001. The duration
relationship predicts that
∆B = −249.95 × 0.001 = −0.250

so that the bond price goes down to 94.213 − 0.250 = 93.963. How accurate is this? Valuing the bond
in terms of its yield in the usual way, we find that, when the bond yield increases by 10 basis points to
12.1%, the bond price is

5e−0.121×0.5 + 5e−0.121×1.0 + 5e−0.121×1.5 + 5e−0.121×2.0 + 5e−0.121×2.5 + 105e−0.121×3.0 = 93.963

which is (to three decimal places) the same as that predicted by the duration relationship.
3
 
∂B X ∂ Ct X Ct 1 X
= = − t=− t · P V (Ct )t
∂r t
∂r (1 + r/k)kt t
(1 + r/k)kt+1 1 + r/k t
The result follows by dividing by B to express the price change in percentage terms.

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What if we have semiannual compounding,

B = 94.213, D = 2.653, y = 12.3673% (semiannual compounding).

The modified duration is:

D 2.653
D∗ = y = = 2.499.
1+ 0.123673
m 1+
2
When yield increases by 10 basis points (∆y = +0.001), the new price is 93.978,

∆B = −94.213 × 2.499 × 0.001 = −0.235.

If r, the APR used to discount a stream of cash flows, increases to r + ε, where ε is a small change,
then the present value of the cash flows changes by approximately:

ε
Percent Change in Value ≈ −Duration × ,
1 + r/k

where k is the number of compounding periods per year of the APR.

Example: Suppose the yield of a 10-year bond with 10% annual coupons increases from 10% to
10.25%. Use duration to estimate the percentage price change. How does it compare to the actual
price change?

Solution: In an earlier example, we found that the duration of the bond is 6.76 years. Thus, the
percentage price change is given by

0.25%
%Price Change ≈ −6.76 × = −1.54%
1.10

Indeed, calculating the bond’s price with a 10.25% yield to maturity, we get:
 
1 1 100
10 × 1− + = $98.48
0.1025 (1.1025)10 (1.1025)10

which represents a 1.52% price drop.

As we see, we can use duration to measure the interest rate sensitivity of a security or a portfolio. We
now consider ways firms use duration to hedge the risk.

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Duration-Based Hedging
A firm’s market capitalization equals the market value of its assets minus the market value of its
liabilities. When interest rates change, they can alter these values, which in turn affects the firm’s
equity. The firm’s sensitivity to interest rate movements can be gauged by calculating the duration
of its balance sheet. By restructuring the balance sheet to shorten this duration, the firm can hedge
against interest rate risk.

Savings and Loans: An Example. Think about a typical savings and loan (S&L) bank. It takes in
short-term deposits from customers, like money in checking accounts, savings accounts, and certificates
of deposit. It then uses that money to make long-term loans, such as for cars and houses. The problem
is that these loans usually last much longer than the deposits. When the timeframes of a bank’s assets
(loans) and liabilities (deposits) are very different, it’s called a duration mismatch. This mismatch
can cause trouble if interest rates change a lot.

For example, consider the table below that shows the market-value balance sheet for XYZ S&L. It
lists the market value and duration of each asset and liability.

Market-Value Balance Sheet for XYZ Savings and Loan

Assets Value Duration Liabilities Value Duration

($m) (yrs) ($m) (yrs)

Cash Reserves 10 0 Checking and Savings 120 0

Auto Loans 120 2 Certificates of Deposit 90 1
Mortgages 170 8 Long-Term Financing 75 12
Total Assets 300 Total Liabilities 285
Owner’s Equity 15
Total Liabilities and Equity 300

Duration of a Portfolio A portfolio of securities with market values A and B and durations DA
and DB , respectively, has the following duration:

A B
DA+B = DA + DB ,
A+B A+B

That is, duration of a portfolio is the value-weighted average of the durations of each investment in
the portfolio. Therefore, the duration of XYZ’s assets and liabilities is

10 120 170
DA = ×0+ ×2+ × 8 = 5.33 years
300 300 300

Similarly, the duration of Acorn’s liabilities is:

120 90 75
DL = ×0+ ×1+ × 12 = 3.47 years
285 285 285

XYZ’s assets and liabilities don’t match well. Because its assets have a much longer duration, a rise
in interest rates will make them lose value faster than its liabilities. This means XYZ’s equity could
shrink a lot if interest rates go up. Let’s calculate the duration of XYZ’s equity (note that it could be

16
considered a portfolio that is long the assets and short the liabilities):

Equity = Assets − Liabilities

Duration of equity is given by

A L
DE = DA−L = DA − DL
A−L A−L

300 285
= × 5.33 − × 3.47 = 40.67 years
15 15
Therefore, if interest rates rise by 1%, the value of XYZ’s equity will fall by about 40% (because the
value of assets decreasing by approximately 5.33% × 300 = $16 million, while the value of liabilities
decrease by only 3.47% × 285 = $9.9 million).

Exercise: Consider a portfolio A that consists of:

• A 1-year zero-coupon bond with a face value of $2,000

• A 10-year zero-coupon bond with a face value of $6,000

Consider another Portfolio B consisting of a 5.95-year zero-coupon bond with a face value of $5,000.
Assume that the current yield on all bonds is 10% per annum Show that both portfolios have the
same duration.

To keep its equity safe from changes in interest rates, XYZ needs an equity duration of zero. This is
called a duration-neutral or immunized portfolio, meaning small changes in interest rates won’t affect
the equity’s value.

To reduce its risk from interest rate fluctuations, XYZ would like to reduce the duration of its equity
from 40.7 to 0. To achieve equity duration neutrality, XYZ must either shorten the duration of its
assets or lengthen the duration of its liabilities. One way to reduce the duration of the asset is by
selling a portion of its mortgages in exchange for cash.

We compute the amount to sell from the following formula: let P be the value of the original portfolio
and S be the amount of assets sold, and let DP and DS be their respective durations. Let DB be the
duration of the new assets bought. Then, new portfolio duration DP∗ is

P S S
DP∗ = DP + DB − DS
P P P

Solving for S leads to S = (DP − DP∗ )P/(DS − DB ).

Change in Portfolio Duration × Portfolio Value

Amount to Exchange =
Change in Asset Duration
Because the duration of the mortgages will change from 8 to 0 if the XYZ sells the mortgages. XYZ
must sell
15
(40.7 − 0) × = $76.3 million worth of mortgages.
8−0
If it does so, the duration of its assets will decline to

10 + 76.3 120 170 − 76.3

Increased cash balance: ×0+ ×2+ × 8 = 3.30 years.
300 300 300

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Thus, its equity duration will fall to

300 285
× 3.30 − × 3.47 = 0, as desired.
15 15

Remark 0.3. Duration matching is a helpful way to manage interest rate risk, but it has some
drawbacks.

• The duration of a portfolio depends on the current interest rate. As interest rates change, the market
values of the securities and cash flows in the portfolio change as well, which in turn affects the weights
used when computing the duration as the value-weighted average maturity. Hence, maintaining a
duration-neutral portfolio will require constant adjustment as interest rates change.

• It only works for parallel shifts in the yield curve. That is, it protects you if all interest rates move
up or down by the same amount. But if short-term rates rise and long-term rates stay the same, it
won’t fully protect you. That’s called a change in the slope of the yield curve, and you’d need extra
strategies to handle that.

• It doesn’t protect against credit risk changes. If two bonds have the same maturity but different credit
risk, duration matching won’t help if their yields move differently. For example, in 2008, Treasury
rates fell a lot, but corporate bond yields with the same maturity actually went up.

Convexity and Bond Price Changes

The duration relationship applies only to small changes in yields. Consider the figure below. At the
origin, the slopes of the two curves are the same. This means both bond portfolios change in value
by the same percentage when yields move a little. But when yields change a lot, the two portfolios
act differently. Portfolio X’s curve bends more than Portfolio Y’s, and this bending is measured by
something called convexity.

Figure 1: Two bond portfolios with the same duration

A measure of convexity is Pn 2 −yti

1 d2 B i=1 ci ti e
C= =
B dy 2 B
From Taylor series expansions, we obtain a more accurate expression, given by

dB 1 d2 B
∆B = ∆y + (∆y)2
dy 2 dy 2

This leads to
∆B 1
= −D∆y + C(∆y)2
B 2

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Key Idea: Convexity improves the accuracy of price change estimates beyond duration.

For combined use with duration, define modified convexity:

Convexity
Modified Convexity =  .
y 2
1+
m

By choosing a portfolio of assets and liabilities with a net duration of zero and a net convexity of
zero, a financial institution can make itself immune to relatively large parallel shifts in the zero curve.
However, it is still exposed to nonparallel shifts.

References

(i) Berk, J. B., & DeMarzo, P. M. (2007). Corporate finance. Pearson Education.

(ii) Hull, John C. Options, futures, and other derivatives. Pearson Education India, Eleventh &
Global Edition.

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