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Yield to Maturity
Outline and Suggested Reading
- Outline
- 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
Definition of Yield
Suppose a bond (or portfolio of bonds) has price P and positive fixed cash flows K 1 , K 2 ,..., Kn at times t 1 , t 2 ,..., tn. 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 1.043066 per dollar par value.
- That implies a yield of 5.4704%:
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=rT 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 is an average of zero rates...
Compare the formula with zero rates and the formula with yield: Notice that the single yield y must be a kind of average of the € different zero rates rt associated with the cash flows. K (^) j j = 1 n ∑ ×^ 1 ( 1 + rt (^) j / 2 ) 2 t (^) j^ =^ K^ j j = 1 n ∑ ×^ 1 ( 1 + y /2) 2 t (^) j
Example
Compare the two formulas for the 1.5-year 8.5%-coupon bond: The yield of 5.4704% is a kind of average of the zero rates 5.54%, 5.45%, and 5.47%.
- 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. The Coupon Effect
Upward Sloping Yield Curve
2.00% 3.00% 4.00% 5.00% 6.00% 7.00% 0 2 4 6 8 10 Yield Maturity Zero coupon 10% coupon 40% coupon Annuity
Downward Sloping Yield Curve
5.20% 5.40% 5.60% 5.80% 6.00% 6.20% 6.40% 0 2 4 6 8 10 Yield Maturity Zero coupon 10% coupon 40% coupon Annuity
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.
Example of Zero and Annuity Yield Curves Maturity Zero rate Zero price Annuity price Annuity yield 0.5 2% 0.9901 0.9901 2% 1.0 3% 0.9707 1.9608 2.66% 1.5 4% 0. 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.
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.
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 cT is the coupon rate that sets the bond price equal to par, i.e., so in terms of zero prices dt , the T -year par cT is
Solve for the 2-year par rate in the term structure below: Class Problem Maturity Zero Rate Zero Price 0.5 5.54% 0. 1.0 5.45% 0. 1.5 5.47% 0. 2.0 5.50% 0. Yield Curves for Zeroes and Par Bonds
