CHAPTER 9-1
Interest Rate Risk 2
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1
Overview
This chapter discusses a market valuebased model for managing interest rate risk,
the duration gap model.
• Duration.
• Computation of duration.
• Economic interpretation.
• Immunization using duration.
• Problems in applying duration.
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Introduction
In most countries, FIs report their balance
sheets using book value accounting.
To reflect current market conditions:
• Market value accounting.
• Marking to market.
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Price Sensitivity and Maturity
• In general, the longer the term to maturity,
the greater the sensitivity to interest rate
changes.
• The longer maturity bond has the greater
drop in price because the payment is
discounted a greater number of times.
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Duration
Duration:
• Weighted-average time to maturity using the
relative present values of the cash flows as
weights.
• More complete measure of interest rate sensitivity
than is maturity.
• The units of duration are years.
• To measure and hedge interest rate risk, FI should
manage duration gap rather than maturity gap.
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Duration
A simple example
• A loan of $100
• Interest rate 0.15 (=15%)
• Required repayment of half the $100 in principal at
the end of six months and the other half at the end
of the year.
What are the values of CF1/2 and CF1?
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Duration
• CF1/2 = $57.50
• The $50 promised repayment of principal plus $7.50 promised
interest payment ($100*(1/2)*15%) received after six months.
• CF1 = $53.75
• The promised cash flow at the end of the year (= the second $50
promised principal repayment plus $3.75 promised interest
($50*(1/2)*15%).
• What are the PV(CF1/2) and PV(CF1)?
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Duration
• What are the PV(CF1/2) and PV(CF1)?
• PV(CF1/2) = CF1/2 / (1+R/2) = $57.50 / (1+0.075) = $53.49
• PV(CF1) = CF1 / (1+R/2)2 = $53.75 / (1+0.075)2 = $53.49
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Duration
• Duration: the weighted-average time to maturity on the loan
using the relative present values of the cash flows as
weights.
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Duration
• Duration: the weighted-average time to maturity on the loan
using the relative present values of the cash flows as
weights.
D = X1/2(1/2) + X1(1)
= 0.5349*(1/2) + 0.4651*(1) = 0.7326 years
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Duration
• The duration of the previous bond
DL = X1/2(1/2) + X1(1)
= 0.5349*(1/2) + 0.4651*(1) = 0.7326 years
• What about the duration of the one-year, $100, 15 percent interest
certificate of deposit?
•
Since all cash flows are received in one payment at the end of the year, X1 =
1
DD = X1(1) = 1 year
• The maturities are the same (maturity gap = 0)
•
ML – M D = 1 – 1 = 0
• The durations are not the same (duration gap = -0.2674)
•
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DL – DD = 0.7326 – 1 = -0.2674 years
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Macaulay’s Duration
σ𝑁
𝑡=1 𝐶𝐹𝑡 × 𝐷𝐹𝑡 × 𝑡
𝐷=
σ𝑁
𝑡=1 𝐶𝐹𝑡 × 𝐷𝐹𝑡
σ𝑁
𝑡=1 𝑃𝑉𝑡 × 𝑡
=
σ𝑁
𝑡=1 𝑃𝑉𝑡
where
D = Duration measured in years
𝐶𝐹𝑡
= Cash flow received at end of period t
N = Last period in which cash flow is received
𝐷𝐹𝑡 = Discount factor = 1 𝑡
)
(1 + 𝑅
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Semiannual Cash Flows
• For semiannual cash flows, Macaulay’s
duration, D, is equal to:
𝐶𝐹𝑡 × 𝑡
1 + 𝑅/2 2𝑡
𝐷=
𝐶𝐹𝑡
𝑁
σ𝑡=1/2
1 + 𝑅/2 2𝑡
σ𝑁
𝑡=1/2
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Macaulay’s Duration
• Key assumptions
• The yield curve (the term structure of interest
rates) is flat.
• When rates change, the yield curve shifts in a parallel
fashion.
• No default risk
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Duration of Zero-Coupon Bond
• Zero-coupon bonds sell at a discount from face
value on issue, pay the face value upon maturity,
and have no intervening cash flows between
issue and maturity.
• Duration equals the bond’s maturity since there
are no intervening cash flows between issue and
maturity.
• For all other bonds, duration < maturity because
here are intervening cash flows between issue
and maturity.
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Duration of Consol Bonds
• A consol bond pays a fixed coupon each
year indefinitely.
• Have yet to be issued in the U.S.
• Maturity of a consol (perpetuity):
M𝑐 = ∞
• Duration of a consol (perpetuity):
D𝑐 = 1 + 1/R
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Features of Duration
Duration and maturity:
• Duration increases with maturity of a fixedincome asset/liability, but at a decreasing rate.
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Features of Duration
Duration and yield:
• Duration decreases as yield increases.
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Features of Duration
Duration and coupon interest:
• Duration decreases as coupon increases.
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Economic Interpretation
1
• Duration is a direct measure of interest
elasticity, or sensitivity, of an asset or
liability:
ΔP/P ÷ ΔR/(1+R) = −D
• Or equivalently,
ΔP/P = −D ΔR/(1+R) = −MD 𝑑𝑅
where MD is modified duration
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Economic Interpretation
2
• To estimate the change in price, we can
rewrite this as:
ΔP = −D ΔR/(1+R) P = − MD × ΔR × P
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Dollar Duration
• Dollar value change in the price of a
security to a 1 percent change in the return
on the security.
Dollar duration = MD × Price
• Using dollar duration, we can compute the
change in price as.
Δ𝑃 = −Dollar duration × Δ𝑅
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Semiannual Coupon Bonds
• With semi-annual coupon payments, the
percentage change in price is calculated
as:
ΔP
ΔR
= −D
R
P
1+
2
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CHAPTER 9-2
Interest Rate Risk 2
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1
Macaulay’s Duration
σ𝑁
𝑡=1 𝐶𝐹𝑡 × 𝐷𝐹𝑡 × 𝑡
𝐷=
σ𝑁
𝑡=1 𝐶𝐹𝑡 × 𝐷𝐹𝑡
σ𝑁
𝑡=1 𝑃𝑉𝑡 × 𝑡
=
σ𝑁
𝑡=1 𝑃𝑉𝑡
where
D = Duration measured in years
𝐶𝐹𝑡
= Cash flow received at end of period t
N = Last period in which cash flow is received
𝐷𝐹𝑡 = Discount factor = 1 𝑡
)
(1 + 𝑅
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2
Semiannual Cash Flows
• For semiannual cash flows, Macaulay’s
duration, D, is equal to:
𝐶𝐹𝑡 × 𝑡
1 + 𝑅/2 2𝑡
𝐷=
𝐶𝐹𝑡
𝑁
σ𝑡=1/2
1 + 𝑅/2 2𝑡
σ𝑁
𝑡=1/2
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Duration of Zero-Coupon Bond
• Zero-coupon bonds sell at a discount from face
value on issue, pay the face value upon maturity,
and have no intervening cash flows between
issue and maturity.
• Duration equals the bond’s maturity since there
are no intervening cash flows between issue and
maturity.
• For all other bonds, duration < maturity because
here are intervening cash flows between issue
and maturity.
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Duration of Consol Bonds
• A consol bond pays a fixed coupon each
year indefinitely.
• Have yet to be issued in the U.S.
• Maturity of a consol (perpetuity):
M𝑐 = ∞
• Duration of a consol (perpetuity):
D𝑐 = 1 + 1/R
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Economic Interpretation
1
• Duration is a direct measure of interest
elasticity, or sensitivity, of an asset or
liability:
ΔP/P ÷ ΔR/(1+R) = −D
• Or equivalently,
ΔP/P = −D ΔR/(1+R) = −MD 𝑑𝑅
where MD is modified duration
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Economic Interpretation
2
• To estimate the change in price, we can
rewrite this as:
ΔP = −D ΔR/(1+R) P = − MD × ΔR × P
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Dollar Duration
• Dollar value change in the price of a
security to a 1 percent change in the return
on the security.
Dollar duration = MD × Price
• Using dollar duration, we can compute the
change in price as.
Δ𝑃 = −Dollar duration × Δ𝑅
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Semiannual Coupon Bonds
• With semi-annual coupon payments, the
percentage change in price is calculated
as:
ΔP
ΔR
= −D
R
P
1+
2
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Example 9-1
• A bond with
•
•
•
•
•
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Maturity of 6 years
Annual coupon rate of 8 percent
Face value of $1,000
Current yield to maturity of 8 percent
Find D
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Example 9-1
• A bond with
•
•
•
•
•
Maturity of 6 years
Annual coupon rate of 8 percent
Face value of $1,000
Current yield to maturity of 8 percent
Find D
D = (1*80/1.08 + 2*80/1.08^2 + 3*80/1.08^3 + 4*80/1.08^4 +
5*80/1.08^5 + 6*1080/1.08^6) / (80/1.08 + 80/1.08^2 + 80/1.08^3 +
80/1.08^4 + 80/1.08^5 + 1080/1.08^6) = 4.993
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Example 9-3
• A bond with
•
•
•
•
•
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Maturity of 6 years
Annual coupon rate of 8 percent
Face value of $1,000
Current yield to maturity of 8 percent
Find MD, DD, and ΔP when interest rate increases by 1%
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Example 9-3
• A bond with
•
•
•
•
•
Maturity of 6 years
Annual coupon rate of 8 percent
Face value of $1,000
Current yield to maturity of 8 percent
Find MD, DD, and ΔP when interest rate increases by 1%
MD = D/(1+R) = 4.993/1.08 = 4.623
DD = MD*Price = 4.623*1000 = 4623
ΔP = -MD*ΔR*P = - 4.623*0.01*1000 = - 46.23
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Example 9-4
• A consol bond with
• Annual coupon rate of 8 percent
• Current yield to maturity of 8 percent
• Find D, MD, DD, and ΔP/P when interest rate increases by 0.01%
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Example 9-4
• A consol bond with
• Annual coupon rate of 8 percent
• Current yield to maturity of 8 percent
• Find D, MD, DD, and ΔP/P when interest rate increases by 0.01%
D = 1 + 1/R = 1+1/.08 = 13.5
MD = D/(1+R) = 13.5/1.08 = 12.5
ΔP/P = - MD*ΔR = - 12.5*0.0001 = - 0.00125 = - 0.125%
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Duration and Interest Rate Risks
• Duration as a measure of interest rate risk
exposure.
• How do Fis use duration to
reduce/eliminate interest rate risk?
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Duration and Interest Rate Risks
• Insurance Company
• Collects premiums and invest them for returns
• Pays the holder some lump sum on reaching
retirement age.
• Risk: if the interest rate falls on the funds
generated from investing the holder’s
premium, the insurance company may not
have enough money to pay out.
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Duration and Interest Rate Risks
Suppose
• The FI must make a guaranteed payment of $1,469 in five years.
(equivalent to investing $1,000 at an annually compounding rate of 8
percent over five years)
• Case 1: the FI purchases discount bonds with 5 years of maturity,
$1,000 face value and 8 percent yield.
•
P = PV = 1000 / 1.08^5 = 680.58
•
The FI buys 1.469 of these bonds at a total cost of $1,000
•
D = maturity = 5 (because it is a zero-coupon bond)
•
Duration of the asset = Duration of the liability
•
Unaffected by intervening interest rate changes
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Duration and Interest Rate Risks
Suppose
• Case 2: the FI purchases coupon bonds from Example 9-1.
•
Maturity: 6 years, FV = $1,000, 8% coupon, 8% current yield to maturity
•
Duration = 4.993 ≈ 5 years
•
Case 2-1: if the interest rate stays at 8%
•
Case 2-2: if the interest rate falls to 7%
•
Case 2-3: if the interest rate increases to 9%
• It does not matter! Why?
•
Two risks – Price risk (from bond sale proceeds) and reinvestment risk.
•
They offset each other when duration is matched!
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Immunization
• Matching the maturity of an asset with a
future payout responsibility does not
necessarily eliminate interest rate risk.
• Matching the duration of a fixed-interest
rate instrument (that is, loan, mortgage,
etc.) to the FI’s target or investment
horizon will immunize the FI against
shocks to interest rates.
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