# What Solutions Are Possible to the Free Rider Problem, Both Inside and Outside of Government

16-1

Outline of the Chapter

• Bond pricing and sensitivity of bond pricing to interest rate changes • Duration analysis – What is duration? – What determines duration?

• Convexity • Passive bond management

– Immunization

• Active bond management

16-2

Interest Rate Risk

• There is an inverse relationship between interest rates (yields) and price of the bonds. • The changes in interest rates cause capital gains or losses. • This makes fixed-income investments risky.

16-3

Interest Rate Risk (Continued)

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Interest Rate Risk (Continued)

• What factors affect the sensitivity of the bonds to interest rate fluctuations? • Malkiel’s (1962) bond-pricing relationships – Bond prices and yields are inversely related. – An increase in a bond’s YTM results in a smaller price change than a decrease in yield of equal magnitude. – Prices of long-term bonds tend to be more sensitive to interest rate changes than prices of short-term bonds.

16-5

Interest Rate Risk (Continued)

– The sensitivity of bond prices to changes in yields increases at a decreasing rate as maturity increases. – Interest rate risk is inversely related to the bond’s coupon rate.

• Homer and Liebowitz’s (1972) bond-pricing relationship – The sensitivity of a bond’s price to change in its yield is inversely related to the YTM at which the bond currently is selling. 16-6

Interest Rate Risk (Continued)

• Why and how different bond characteristics affect interest rate sensitivity?

16-7

Interest Rate Risk (Continued)

• Duration

– Macaulay’s duration: the weighted average of the times to each coupon or principal payment made by the bond. • Weight applied to each payment is the present value of the payment divided by the bond price.

wt D

CFt /(1 y ) t , Bondprice

T

T

wt

t 1

1

t * wt

t 1

16-8

Interest Rate Risk (Continued)

• Example:

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Interest Rate Risk (Continued)

– Duration is shorter than maturity for all bonds except zero coupon bonds. – Duration is equal to maturity for zero coupon bonds.

• Why duration is important?

– Simple summary statistic of the effective average maturity of the portfolio. – Tool for immunizing portfolios from interest rate risk. – Measure of the interest rate sensitivity of a portfolio. 16-10

Interest Rate Risk (Continued)

– The long-term bonds are more sensitive to interest rate movements than are short-term bonds. – By using duration we can quantify this relation.

P P

D

(1 y ) 1 y

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Interest Rate Risk (Continued)

– Modified Duration:

• Measure of the bond’s exposure to changes in interest rates. • The percentage change in bond prices is just the product of modified duration and the change in the bond’s yield to maturity. • Note that the equations are only approximately valid for large changes in the bond’s yield. D* P P (1 D /(1 D* y) y) y y

16-12

Interest Rate Risk (Continued)

• What determines Duration?

– The duration of a zero-coupon bond equals its time to maturity. – Holding maturity constant, a bond’s duration is higher when the coupon rate is lower. – Holding the coupon rate constant, a bond’s duration generally increases with its time to maturity. • For zero-coupon bonds the maturity=the duration • For coupon bonds duration increases by less than a year with a year’s increase in maturity.

16-13

Interest Rate Risk (Continued)

– Holding other factors constant, the duration of a coupon bond is higher when the bond’s yield to maturity is lower. • At lower yields the more distant payments made by the bond have relatively greater present values and account for a greater share of the bond’s total value.

– The duration of a level perpetuity is equal to: (1+y) / y • The PV-weighted CFs early on in the life of the perpetuity dominate the computation of duration. 16-14

Interest Rate Risk (Continued)

16-15

Convexity

• By employing the duration concept we can analyse...

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