1 Not To Be Naïve about Duration 1.The duration D we have been discussing also known as Macaulay duration. 2.First derivative of price-yield curve is and is known as modified duration. Found in %ΔPB formula. 3.Convexity is second derivative of price-yield curve. Is a complicated expression (not studied here).

2 Managing Interest Rate Risk (a) Duration is the holding period for which reinvestment risk exactly offsets price risk. Designed to give investor the YTM that was in effect at time bond purchased. A way duration is used: If have a $5 million liability 7.5 years from now, buy a bond (or a portfolio of bonds) today that has a duration of 7.5 years. 10/31

3 Example 18: Rebalancing Bond Portfolio Consider the $20,000 portfolio ($4,000 in D = 5, $10,000 in D = 7, $6,000 in D = 9) of Example 17. How much in D = 9 bonds should be sold, and how much in D = 5 bonds should be purchased, to reduce Portfolio D to 6.80?

4 Eliminating Interest Rate Risk (b) Zero-coupon approach (best way). Buy high quality “zeros” with maturity equal to desired holding period. Locks in YTM. No reinvestment risk because no coupons payments, no price risk when held to maturity. Duration matching (next best way). Selecting a portfolio of bonds whose duration matches desired holding period. Theoretically perfect, but only approximately perfect in real world as per footnote 8 on p Maturity matching (don’t use). That is, selecting bonds with terms to maturity equal to desired holding period. Don’t use. Doesn’t work for eliminating interest rate risk.

5 Three Bond Yields 1.Yield-to-maturity. Assumes Issuer makes all payments as promised Coupon payments are reinvested at the rate that the bond yielded when purchased Investor holds bond to maturity 2.Realized yield. An ex post calculation of the bond’s yield while holding it. For instance, holder sells a bond before maturity. 3.Expected yield. An ex ante calculation of a bond’s expected yield based upon anticipated cash flows. All trial-and-error “number crunching” calculations

6 Example 19: Yield-to-Maturity Yield-to-maturity. The annual rate that causes all cash flows to discount back to the bond’s market price. Solve by trial-and-error. What is the yield-to-maturity on a 12-year, 8% coupon bond (semi-annual payments) whose price is $1,097.37? Find r-value that fits. 25 terms (lot of work). Then double to obtain yield-to-maturity answer.

7 Example 19: (con’t) Easier if we employ annuity formula. Only 2 terms.

8 Example 20: Realized Yield Realized yield. Rate that causes all cash flows to discount back to the purchase price. What did bond project yield (annual rate) now that it is over? Paid $995 for a new 6% coupon bond (semi-annual payments). Sold after 3 years for $1,068 (minutes after coupon payment). What was realized yield? Principle: If trial r makes RHS too low, decrease r If trial r makes RHS too high, increase r

9 Example 21: Expected Yield Calculation Expected yield -- the discount rate that causes the sum of the PV’s of all expected cash flows to equal purchase price. Solve by trial-and-error. Let’s do 6-month clock version of prob on p. 148 in book: Purchased a new 8% 10-yr, semiannual coupon payment bond at par. Plan to sell in 2 yrs when bond expected to yield 6% (at which point PB = 1,126). What is project’s expected yield?

10 Fixed Income Securities (a) Fixed income security are liabilities to their issuers. Assets Liabilities Capital Securities issued by governments and corporations that are designed to pay contractually a specified income over a specified time horizon. on issuer’s books in here