Practice Quiz 3 Solutions — Forward Rates & Credit Spreads
Setup: Spot rates r(1) = 5%, r(2) = 6%. Risk-free 2-year zero-coupon bond (face $100). Corporate 2-year zero-coupon bond trades at $86.
(a) Compute the 1-year forward rate f(1,2). Is the market expecting rates to rise or fall?
The forward rate satisfies $(1 + r_1)(1 + f_{1,2}) = (1 + r_2)^2$:
\[f_{1,2} = \frac{(1.06)^2}{1.05} - 1 = \frac{1.1236}{1.05} - 1 = 7.01\%\]Since f(1,2) = 7.01% > r(1) = 5%, the market is expecting short-term rates to rise next year.
(b) Price of risk-free zero? YTM of corporate bond? Positive or negative spread?
Risk-free zero price:
\[P_{rf} = \frac{100}{(1.06)^2} = \frac{100}{1.1236} = \$89.00\]Corporate bond YTM: solve $86 = \frac{100}{(1+y)^2}$:
\[y = \left(\frac{100}{86}\right)^{1/2} - 1 = (1.1628)^{0.5} - 1 = 7.84\%\]Credit spread = 7.84% − 6.00% = +184 bp. The corporate bond trades at a positive spread to Treasuries.
(c) What does a positive credit spread compensate the investor for?
A positive credit spread compensates the investor for bearing the risk that the issuer may default on its promised payments — it reflects expected default losses, a liquidity premium, and a risk premium for the systematic component of credit risk.