Bond Prices and Yields





Kerry Back

Coupons vs Yields

  • The coupon rate of a bond is set at the time of its issue.

  • However, what one anticipates earning on a bond varies with the market price.

    • Price < par \(\Rightarrow\) coupon + capital gain
    • Price > par \(\Rightarrow\) coupon - capital loss

  • What one would earn per year on a bond if held to maturity (assuming no default) is called the bond yield.

Definition of bond yield

  • Buy a bond at price \(P\), receive coupons \(c_1, \ldots, c_n\) and face value \(F\) at date \(n\).
  • The rate of return \(r\) for which an investment of \(P\) would exactly finance the cash flows \(c_1, \ldots, c_n\) and \(F\) is called the bond yield.
  • It is the rate at which the present value of the promised cash flows \(c_1, \ldots, c_n\) and \(F\) equals the bond price.

Calculating Yields

  • Bonds usually pay coupons semi-annually, so it is conventional to use semi-annual discounting.
  • Calculate a six-month rate. Double it to annualize.
  • Suppose the next coupon payment is six months away. Solve the following for a six-month rate \(r\). Yield is \(y=2r\).

\[P=\frac{c}{1+r}+\frac{c}{(1+r)^2}+\cdots+\frac{c+F}{(1+r)^n}\]

Example

5 year bond, 6% coupon, $100 face. The price \(P\) and six-month rate \(r\) satisfy

\[P=\frac{3}{1+r}+\frac{3}{(1+r)^2}+\cdots+\frac{3+100}{(1+r)^{10}}\]

The yield is

\[y = 2r\]

Premium and Discount Bonds

  • Premium bond: price > face value
    • Implies yield < coupon rate
  • Discount bond: price < face value
    • Implies yield > coupon rate

Price sensitivity to market rates

  • When market rates rise, a bond price falls until its yield is commensurate with market rates.
  • When market rates fall, a bond price rises until its yield is commensurate with market rates.

So,

  • rising rates \(\Rightarrow\) negative bond returns
  • falling rates \(\Rightarrow\) positive bond returns

History of Treasury rates

Credit ratings

  • Issuers are rated by S&P, Moody’s and Fitch
    • AAA=best quality
    • AA
    • A
    • BBB=worst investment grade
    • BB=best non-investment grade (= junk = high-yield)
    • B, etc. (D is default)
  • And +’s and -’s. Moody’s uses baa instead of BBB, etc.

Credit spreads

  • Investors require higher yields on riskier bonds
    • default \(\rightarrow\) low return
    • no-default \(\rightarrow\) high return (= yield)
    • yield must be high enough so that expected return = Treasury yield + risk premium
  • Usually think of yield as Treasury yield + spread
    • Part of spread offsets expected loss due to default
    • Remainder provides a risk premium

History of credit spreads