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Lecture 5
Managing Bond Portfolios
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Chapter Overview
Examine various fixed-income portfolio strategies.
• Distinguish between passive and active approaches.
Discuss sensitivity of bond prices to interest rates
fluctuations.
• Sensitivity is measured by duration.
Consider refinements in the way interest rate sensitivity is
measured, focusing on bond convexity.
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Figure 16.1 Change in Bond Price as a Function
of Change in Yield to Maturity
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Malkiel’s Bond–Pricing Relationships
1
1. Bond prices and yields are inversely related.
2. An increase in a bond’s yield to maturity results in a
smaller price change than a decrease in yield of equal
magnitude.
3. Prices of long-term bonds tend to be more sensitive to
interest rate changes than prices of short-term bonds.
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Malkiel’s Bond–Pricing Relationships
2
4. Interest rate risk (that is, sensitivity of the price to interest
rates) is less than proportional to bond maturity.
5. Interest rate risk is inversely related to the bond’s coupon
rate.
6. The sensitivity of a bond’s price to a change in its yield is
inversely related to the Y TM at which the bond is currently
selling (Homer and Liebowitz).
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Duration
A measure of the average maturity of a bond’s promised
cash flows.
Macaulay’s duration equals the weighted average of the
times of each coupon or principal payment.
• The weight given to each payment time is the proportion of
total value of the bond accounted for by that payment (i.e.,
the PV of the payment divided by the bond price).
Duration = Maturity for zero coupon bonds.
Duration < Maturity for coupon bonds.
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Duration Calculation
Duration calculation:
T
D = t wt
t =1
CFt / (1 + y )t
wt =
P
CFt = Cash Flow at Time t
P = Price of Bond
y = Yield to Maturity
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Spreadsheet 16.1 Calculating the Duration of a
Two-Year Coupon Bond and a Two Year Zero
Spreadsheet 16.1. The coupon bond makes semiannual payments, so each period is six
months.
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Interest Rate Risk 1
Duration as a measure of interest rate sensitivity.
• Sensitive of price to interest rate changes is proportional to
duration. Approximately, can show that:
P (1 + y )
= −D
P 1 + y
• D* = D/(1+y) = Modified duration, commonly used by
practitioners.
P
= − D* y
P
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Interest Rate Risk 2
Consider the 2-year maturity, 8% coupon bond in
Spreadsheet 16.1 making semiannual coupon payments and
selling at $964.54, with a yield to maturity of 10% and
duration of 1.8852. What is the approximate price change if
the yield to maturity increases to 10.5%?
P
= − D* y →
P
P = − D* y P
1.8852
=− .005 $964.54 = $8.27
1.10
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Duration Rules 1
Rule 1
• The duration of a zero-coupon bond equals its time to
maturity.
Rule 2
• Holding maturity constant, a bond’s duration is lower when
the coupon rate is higher.
Rule 3
• Holding the coupon rate constant, a bond’s duration
generally increases with its time to maturity.
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Duration Rules 2
Rule 4
• Holding other factors constant, the duration of a coupon
bond is higher when the bond’s yield to maturity is lower
Rule 5
• The duration of a level perpetuity is equal to:
1+ y
y
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Figure 16.2 Bond Duration Versus Bond Maturity
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Convexity 1
Relationship between bond prices and yields is not linear.
Duration rule is a good approximation for only small changes
in bond yields.
Bonds with higher convexity exhibit higher curvature in the
price–yield relationship.
• Convexity is measured as the rate of change of the slope
of the price–yield curve, expressed as a fraction of the
bond price.
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Figure 16.3 Bond Price Convexity (30-Year
Maturity; 8% Coupon; Initial YTM = 8%)
Access the text alternative for slide images.
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Convexity 2
1 T
CFt
Convexity =
P (1 + y ) 2
(1 + y )t (t + t )
t =1
2
Accounting for convexity changes the equation:
P 1
= − D y + Convexity (y ) 2
*
P 2
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Why Do Investors Like Convexity?
Bonds with greater curvature gain more in price when yields
fall than they lose when yields rise.
• The more volatile interest rates, the more attractive this
asymmetry.
• It increases the expected return on the bond
But then in equilibrium, investors must pay higher prices and
accept lower yields to maturity on bonds with greater
convexity.
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Passive Bond Management
Passive managers take bond prices as fairly set and seek to
control only the risk of their fixed-income portfolio.
Two classes of passive management:
• Bond Indexing strategy: attempt to replicate the
performance of a given bond index.
• Immunization techniques: shield portfolio from exposure to
interest rate fluctuations.
Both classes accept market prices as being correct (that is,
they’re not trying to profit from mispricing) but differ greatly in
terms of risk exposure.
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Passive Management: Immunization
Immunization techniques are used to shield overall financial
status from interest rate risk.
• Widely used by pension funds, insurers, and banks.
Suppose an insurance company sells for $10,000 a
guaranteed investment contract (GIC) to a customer, which is
basically a zero-coupon bond.
• Assume GIC has five year maturity, insurance company
promises 8% interest, so will owe 10,000 x 1.085 =
14,693.28 in five years.
Also suppose insurance company chooses to fund the future
payment by buying $10,000 of 8% annual coupon bonds,
selling at par value, with six (not five) years to maturity.
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Table 16.4.A Value of bonds in 5 years if
interest rates stay at 8%.
Years Remaining Accunmulated Value
Payment Number Until obligation of Invested Payment
A. Rates Remain at 8%
800 (1.08 ) =
4
1 4 $ 1,088.39
2 3 800 (1.08 )
3
= 1,007.77
3 2 800 (1.08 )
2
= 933.12
800 (1.08 ) =
1
4 1 864.00
800 (1.08 ) =
0
5 0 800.00
Sale of bond 0 10,800/1.08 = $10,000.00
$14,693.28
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Passive Management: Immunization
If interest rates fall to 7%, will insurance company still be
able to pay the GIC obligation?
Two effects:
• Reinvestment risk: Reinvested coupons now earn lower
interest.
• Price risk: But the resale price of the bond at the end of
year 5 will be higher.
These two effects go in opposite directions.
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Table 16.4.B Value of bonds in 5 years if
interest rates fall to 7%.
Years Remaining Accunmulated Value
Payment Number Until obligation of Invested Payment
B. Rates Fall to 7%
800 (1.07 ) =
4
1 4 $ 1,048.64
2 3 800 (1.07 )
3
= 980.03
3 2 800 (1.07 )
2
= 915.92
800 (1.07 ) =
1
4 1 856.00
800 (1.07 ) =
0
5 0 800.00
Sale of bond 0 10,800/1.07 = $ 10,093.46
$14,694.05
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Table 16.4.C Value of bonds in 5 years if
interest rates rise to 9%.
Years Remaining Accunmulated Value
Payment Number of Invested Payment
Until obligation
C. Rates Increase to 9%
800 (1.09 ) =
4
1 4 $ 1,129.27
2 3 800 (1.09 )
3
= 1,036.02
3 2 800 (1.09 )
2
= 950.48
800 (1.09 ) =
1
4 1 872.00
800 (1.09 ) =
0
5 0 800.00
Sale of bond 0 10,800/1.09 = $ 9,908.26
$14,696.02
Table 16.4
Terminal value of a 6-year maturlty bond portfolio after 5 years (all proceeds reinvested)
Note: The sale price of the 6-year maturlty bond portfolio equals the portfolio's final payment ($ 10.800) divided by 1 + r
because the time to maturity of the bonds will be 1 year at the time of sale.
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Passive Management: Immunization
When can we guarantee that price risk and reinvestment risk
cancel out?
For a horizon T equal to the portfolio’s duration D, price
risk and reinvestment risk are approximately offsetting.
In other words,
At time T = D, the accumulated value of the bond
portfolio is approximately independent of the interest
rate.
So if you are funding a future time T payment by investing in
a bond portfolio, choosing a portfolio with duration D=T will
immunize you from small changes in interest rate.
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Figure 16.9 Growth of Invested Funds
Figure 16.9 Growth of invested funds. The solid-colored curve represents the
growth of portfolio value at the original interest rate. If interest rates increase at
time t*, the portfolio value initially falls but increases thereafter at the faster rate
represented by the broken curve. At time D (duration), the curves cross.
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Figure 16.10 Immunization
Figure 16.10 Immunization. The coupon bond fully funds the obligation at
an interest rate of 8%. Moreover, the present value curves are tangent at
8%, so the obligation will remain fully funded even if rates change by a
small amount.
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