Chapter
Managing Bond
11 Portfolios
Makoto Suzuki
Rikkyo Univ.
11.1 Interest Rate Risk
• Interest Rate Sensitivity
1. Bond prices and yields are inversely related
2. Increase in bond’s yield to maturity results in smaller
price change than yield decrease of equal magnitude
3. Long-term bond prices more sensitive to interest rate
changes than short-term bonds
4. As maturity increases, sensitivity of bond prices to
changes in yields increases at decreasing rate
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 2
11.1 Interest Rate Risk
• Interest Rate Sensitivity
5. As maturity increases, sensitivity of bond prices to
changes in yields increases at decreasing rate
6. Interest rate risk is inversely related to bond’s coupon
rate; low-coupon bonds are more sensitive to interest
rates
7. Sensitivity of bond’s price-to-yield change is inversely
related to current yield to maturity
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 3
Figure 11.1 Change in Bond Prices as a Function of Change in
Yield to Maturity
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 4
11.1 Interest Rate Risk: Coupons Rate and Sensitivity
Prices of 8% annual coupon bonds
Prices of zero-coupon bonds
P −P
*==YTM 9%
= YTM 8%
PYTM =8%
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 5
11.1 Interest Rate Risk
• Macaulay’s Duration
• Measures effective bond maturity
• Weighted average of the times until each
payment, with weights proportional to the present
value of payment
𝐶𝐶𝐶𝐶𝑡𝑡 /(1+𝑦𝑦)𝑡𝑡
• 𝑤𝑤𝑡𝑡 =
𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
• 𝐷𝐷 = ∑𝑇𝑇
𝑡𝑡=1 𝑡𝑡 × 𝑤𝑤𝑡𝑡
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 6
Spreadsheet 11.1 Calculation of Duration of Two Bonds
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 7
Concept of the Macaulay’s Duration
Dmac T=1 T=2 T=3
950.263 80 80 1080
Sum of the PV
72.727 66.116 811.420
Present Value
950.263 × Dmac = 1× 72.727 + 2 × 66.116 + 3 × 811.420
72.727 66.116 811.420
Dmac = 1× + 2× + 3×
950.263 950.263 950.263
= 1× 0.0765 + 2 × 0.0696 + 3 × 0.8539
= 2.774
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 8
11.1 Interest Rate Risk
• Change in Bond Price to Yield to Maturity
Δ𝑃𝑃 Δ 1+𝑦𝑦
• = 𝐷𝐷 ×
𝑃𝑃 1+𝑦𝑦
• Modified Duration
∗ 𝐷𝐷
• 𝐷𝐷 =
1+𝑦𝑦
Δ𝑃𝑃
• = −𝐷𝐷 ∗ Δ𝑦𝑦
𝑃𝑃
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 9
11.1 Interest Rate Risk
• What Determines Duration?
• Zero-coupon bond’s duration is time to maturity
• Time/yield to maturity constant, bond’s duration
and interest-rate sensitivity higher when coupon
price lower
• Coupon rate constant, bond’s duration and
interest-rate sensitivity generally increase with
time to maturity; duration always increases with
maturity for bonds at or above par
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 10
11.1 Interest Rate Risk
• What Determines Duration?
• Other factors constant, duration and interest rate
sensitivity of coupon bond higher when bond’s
yield to maturity lower
1+𝑦𝑦
• Duration of a perpetuity =
𝑦𝑦
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 11
Figure 11.2 Duration as Function of Maturity
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 12
Table 11.3 Annual Coupon Bond Duration
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 13
11.2 Passive Bond Management
• Immunization
• Strategy to shield net worth from interest rate
movements
• Rebalancing
• Realigning proportions of assets in portfolio as
needed
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 14
Table 11.4 Terminal Value of Bond Portfolio after Five Years
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 15
Figure 11.3 Growth of Invested Funds
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 16
Figure 11.4 Immunization
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 17
11.2 Passive Bond Management
• Cash Flow Matching and Deduction
• Cash flow matching
• Matching cash flows from fixed-income
portfolio with those of obligation
• Deduction strategy
• Multi-period cash flow matching
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 18
11.3 Convexity
• Convexity
• Curvature of price-yield relationship of bond
Δ𝑃𝑃
• = −𝐷𝐷 ∗ Δ𝑦𝑦 + 1⁄2 × 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 × (Δ𝑦𝑦)2
𝑃𝑃
• Why Do Investors Like Convexity?
• More convexity = greater price increases,
smaller price decreases when interest rates
fluctuate by larger amounts
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 19
Figure 11.5 Bond Price Convexity
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 20
11.4 Active Bond Management: Strategies
Sources of Potential Profit Strategy
Exchange of one bond for bond with similar
Substitution swap attributes and better price
Switching from one segment of bond
Intermarket swap
market to another
Switch made in response to forecasts of
Rate anticipation swap
interest rate changes
Moving to higher yield bonds, usually with
Pure yield pickup swap
longer maturities
Swapping two similar bonds to receive tax
Tax swap
benefit
Forecast of bond returns based largely on
Horizon analysis prediction of yield curve at end of
investment horizon
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 21
11.4 Active Bond Management
• Fixed-Income Investment Strategy
• Key features
• Firms respect market prices
• To have value, information cannot already be
reflected in prices
• Interest rate movements extremely hard to predict
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 22
