# Yield to Maturity Outline and Suggested Reading

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1 Yield to Maturity Outline Outline and Suggested Reading Yield to maturity on bonds Coupon effects Par rates Buzzwords Internal rate of return, Yield curve Term structure of interest rates Suggested reading Veronesi, Chapter 2 Tuckman, Chapter 3 Yield to Maturity 1

2 Definition of Yield Suppose a bond (or portfolio of bonds) has price P and positive fixed cash flows K 1, K 2,..., K n at times t 1, t 2,..., t n. Its yield to maturity is the single rate y that solves: Note that the higher the price, the lower the yield. Example Recall the 1.5-year, 8.5%-coupon bond. Using the zero rates 5.54%, 5.45%, and 5.47%, the bond price is per dollar par value. That implies a yield of %: Yield to Maturity 2

3 Yield of a Bond on a Coupon Date For an ordinary semi-annual coupon bond on a coupon date, the yield formula is where c is the coupon rate and T is the maturity of the bond in years. Annuity Formula Math result: 2T 1 = 1 (1+ y /2) s y /2 (1 1 (1+ y /2) ) 2T s=1 Finance application: This formula gives the present value of an annuity of \$1 to be received every period for n periods at a simply compounded rate of r per period. Yield to Maturity 3

4 Yield-to-Price Formula for a Coupon Bond Value the coupon stream using the annuity formula: The closed-form expression simplifies computation. Note that if c=y, P=1 (the bond is priced at par). If c>y, P>1 (the bond is priced at a premium to par). If c<y, P<1 (the bond is priced at a discount). The yield on a zero is the zero rate: c=0 ; y=r T Class Problem: Suppose the 1.5-year 8.5%-coupon bond is priced to yield 9%. What is its price per \$1 par? Bond Yields and Zero Rates Recall that we can construct coupon bonds from zeroes, and we can construct zeroes from coupon bonds. So in the absence of arbitrage, zero prices imply coupon bond prices and coupon bond prices imply zero prices. Therefore, zero rates imply coupon bonds yields and coupon bond yields imply zero yields. Yield to Maturity 4

5 Yield is an average of zero rates... Compare the formula with zero rates and the formula with yield: n K j j=1 1 (1+ r t j /2) 2t j n 1 = K j (1+ y /2) 2t j j=1 Notice that the single yield y must be a kind of average of the different zero rates r t associated with the cash flows. Example Compare the two formulas for the 1.5-year 8.5%-coupon bond: The yield of % is a kind of average of the zero rates 5.54%, 5.45%, and 5.47%. Yield to Maturity 5

6 Term Structure and Yield Curves The phrase term structure of interest rates refers to the general relation between yield and maturity that exists in a given bond market. A yield curve is a plot of a specific set of bond yields as a function of their maturity. The yield curve for zeroes is typically different than the yield curve for coupon bonds. In principle, we can derive one curve from the other. Yield Curves for Zeroes and 6% Bonds Here are the yield curves based on the historical STRIPS rates used for most of the course examples. Why does the coupon bond yield curve lie below the zero curve? Yield to Maturity 6

7 The Coupon Effect Proposition 1 If the yield curve is not flat, then bonds with the same maturity but different coupons will have different yields. Proposition 2 If the yield curve is upward-sloping, then for any given maturity, higher coupon bonds will have lower yields. Proposition 3 If the yield curve is downward-sloping, then for any given maturity, higher coupon bonds will have higher yields. Upward Sloping Yield Curve (10/30/1992) 7.00% 6.00% Yield 5.00% 4.00% 3.00% Zero coupon 10% coupon 40% coupon Annuity 2.00% Maturity Yield to Maturity 7

8 Downward Sloping Yield Curve (11/30/2000) 6.40% Yield 6.20% 6.00% 5.80% 5.60% 5.40% Zero coupon 10% coupon 40% coupon Annuity 5.20% Maturity Why the coupon effect? Start by comparing zero rates and annuity yields An annuity for a given maturity pays \$1 each period until maturity, let s say every six months. The annuity yield is an average of the zero rates associated with each of its cash flows. If the zero yield curve is upward sloping, the annuity yield curve will be upward sloping too, because each time we extend the annuity maturity, we introduce another, higher, zero rate into the average. Also, the annuity yield for a given maturity will be lower than the zero rate for that maturity, because it is the average of the zero rates associated with its cash flows. So it s lower than maximium, which is zero rate for that maturity. Yield to Maturity 8

9 Example of Zero Rates and Annuity Rates For nice round numbers, use the zero rates and zero prices below for an example. What are the corresponding annuity prices and rates? Maturity Zero rate Zero price Annuity price Annuity yield 0.5 2% % 1.0 3% % 1.5 4% A 0.5-year annuity pays \$1 at time 0.5, so it is the same as the 0.5-year zero. A 1-year annuity pays \$1 at time 0.5 and \$1 at time 1, so it is the sum of the 0.5-year zero and the 1-year zero. Its price is = Its yield is y such that 1 1+ y /2 + 1 = => y = 2.66% 2 (1+ y /2) The annuity yield is a kind of average of the zero rates corresponding to its cash flows, in this case, a kind of average of the 2% and the 3%. Class Problem: Zero and Annuity Rates Take the zero rates and zero prices below as given. What is the 1.5-year annuity rate? Maturity Zero rate Zero price Annuity price Annuity yield 0.5 2% % 1.0 3% % 1.5 4% ?? Yield to Maturity 9

10 Example of Zero and Annuity Yield Curves Maturity Zero rate Zero price Annuity price Annuity yield 0.5 2% % 1.0 3% % 1.5 4% Zero rates and annuity yields: Downward-sloping yield curve By the same logic, if the zero yield curve is downward sloping, the annuity curve will be downward sloping and the annuity yield will be higher than the zero rate of the same maturity, because the average is less than the minimum. Yield to Maturity 10

11 Now coupon bonds and the coupon effect.. Every coupon bond consists of a coupon stream and a par payment. So a coupon bond of a given maturity is a combination of an annuity and a zero with that same maturity. So the yield on the coupon bond of a given maturity is an average of the annuity yield and the zero rate for that same maturity. The higher the coupon, the closer the bond s yield is to the annuity rate. The lower the coupon, the closer the bond s yield is to the zero rate. The coupon effect in upward or downward sloping yield curves In an upward-sloping yield curve, zero rates are higher than annuity rates for the same maturity, so lower coupon bonds have higher yields. In a downward-sloping yield curve, zero rates are lower than annuity rates, so lower coupon bonds have lower yields. Yield to Maturity 11

12 Humped Yield Curve (3/31/2000) 6.45% 6.30% 6.15% Yield 6.00% 5.85% 5.70% Zero coupon 10% coupon 40% coupon Annuity Maturity Must all curves intersect at the same point? Inverted Humped Yield Curve (7/31/1989) 8.00% Yield 7.90% 7.80% 7.70% 7.60% 7.50% Zero coupon 10% coupon 40% coupon Annuity 7.40% Maturity Yield to Maturity 12

13 Par Rates The par rate for a given maturity T is the coupon rate that makes a T-year coupon bond sell for par. Of course, the yield on the bond will also be the par rate. Since coupon bonds are usually issued at par, par rates are yields on newly issued bonds. Par Rate in Terms of Zero Prices In practice, bond pricing data usually comes in the form of par rates yields on newly issued bonds that are sold at par. In other cases, we might want to compute par rates from zero prices. This is one yield computation that is explicit: For each maturity T, the par rate c T is the coupon rate that sets the bond price equal to par, i.e., so in terms of zero prices d t, the T-year par c T is Yield to Maturity 13

14 Class Problem Solve for the 2-year par rate in the term structure below: Maturity Zero Rate Zero Price % % % % Yield Curves for Zeroes and Par Bonds Yield to Maturity 14

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1 You estimate that you will have \$24,500 in student loans by the time you graduate. The interest rate is 6.5%. If you want to have this debt paid in full within five years, how much must you pay each

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