Yield Curve

Study guides: Exam FM-2, Exam INV-201

The yield curve is a graph plotting the Spot Rate s_t — the yield rate on a zero-coupon bond maturing at t — against time to maturity t, summarizing the term structure of interest rates at a given point in time. Bonds are priced by discounting each cash flow at the spot rate for its maturity:

P = \sum_{t=1}^{n} \frac{C_t}{(1+s_t)^t}

• A normal (upward-sloping) yield curve has long rates above short rates; its shape reflects the market's expectation that interest rates will rise
• An inverted (downward-sloping) curve has short rates above long rates
• A flat yield curve has the same rate at all maturities
• The yield curve is bootstrapped from observed market prices of coupon bonds to extract the implied spot rates s_1, s_2, \ldots, s_n

Reading and Using a Yield Curve The yield curve gives s_1 = 3\%, s_2 = 4\%, s_3 = 4.5\%. Price a 3-year $1{,}000 bond paying annual coupons of $50.

Answer

\begin{align*}
P &= \frac{50}{1.03} + \frac{50}{1.04^2} + \frac{1{,}050}{1.045^3} \\
&= 48.54 + 46.23 + 920.11 \\
&= 1{,}014.88
\end{align*}

This is an upward-sloping (normal) yield curve. The bond prices above par because the coupon rate (5\%) exceeds even the longest spot rate (4.5\%), so the bond is a premium bond.