Spot Rate

Study guide: Exam FM-2

The $n$ -year spot rate $s_n$ is the yield to maturity on a zero-coupon bond that matures in exactly $n$ years, representing the market's required return for a single cash flow received at time $n$ with no intermediate payments. A coupon bond is priced by discounting each cash flow at the spot rate for that maturity:

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

Spot rates are read from the Yield Curve or bootstrapped from observed bond prices
The relationship between spot rates and forward rates is $(1+s_n)^n = (1+s_{n-1})^{n-1}(1+f_{n-1,n})$ , where $f_{n-1,n}$ is the one-year forward rate from year $n-1$ to $n$

Pricing a Bond with Spot Rates Spot rates are $s_1 = 4\%$ , $s_2 = 5\%$ , $s_3 = 6\%$ . Find the price of a $ $1{,}000$ face value 3-year bond paying annual coupons of $ $60$ .

Answer Discount each cash flow at the corresponding spot rate: $P = \frac{60}{1.04} + \frac{60}{1.05^2} + \frac{1060}{1.06^3}$ $P = 57.69 + 54.42 + 890.00 = \$1{,}002.11$ Note this differs from using a single flat yield, because the spot curve is upward-sloping (longer maturities have higher rates).