Spot Rates and Forward Rates

Spot rates and forward rates are two descriptions of the same term structure of interest rates. The Spot Rate $s_t$ is the annual effective rate earned on an investment made now and repaid in one sum at time $t$ ; the Forward Rate $f_{n,n+k}$ is the annual effective rate for an investment that starts $n$ years from now and lasts $k$ years. Either set determines the other, because investing for $n$ years and then at the forward rate for $k$ more must accumulate to the same amount as investing at the $(n+k)$ -year spot rate:

$(1+s_n)^n\,(1+f_{n,n+k})^k = (1+s_{n+k})^{n+k}$

Chaining one-year forward rates builds the spot rates, and the spot rates discount any set of cash flows $C_t$ :

$(1+s_t)^t = (1+f_{0,1})(1+f_{1,2})\cdots(1+f_{t-1,t})$

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

$f_{n,n+k}$ is what SOA calls the " $k$ -year forward rate, deferred $n$ years" (or "starting in $n$ years"); the one-year forward rate starting now is the one-year spot rate, $f_{0,1} = s_1$
Plotted against term, the spot rates form the Yield Curve. Exam FM asks for the present value of a set of cash flows using a yield curve developed from forward and spot rates: convert whichever rates are given to spot rates (or to a product of forward-rate factors), then discount each cash flow at the rate for its own time
For the one-step formulas — pricing a bond from spot rates, or a one-year forward rate from two adjacent spot rates — see Spot Rate and Forward Rate

From Forward Rates to Spot Rates and a Present Value The one-year forward rates are $f_{0,1} = 3\%$ , $f_{1,2} = 4\%$ and $f_{2,3} = 5.5\%$ . An insurer must pay $1{,}000$ at the end of each of the next three years. Find the 2- and 3-year spot rates and the present value of the payments.
Answer Chain the forward-rate factors:
 $\begin{align*} (1+s_2)^2 &= (1.03)(1.04) \\ &= 1.0712 \\ s_2 &= 1.0712^{1/2} - 1 \\ &= 3.50\% \\ (1+s_3)^3 &= (1.03)(1.04)(1.055) \\ &= 1.130116 \\ s_3 &= 1.130116^{1/3} - 1 \\ &= 4.16\% \end{align*}$ 
Discount each payment by its own factor:
 $\begin{align*} PV &= \frac{1{,}000}{1.03} + \frac{1{,}000}{1.0712} + \frac{1{,}000}{1.130116} \\ &= 970.874 + 933.532 + 884.865 \\ &= 2{,}789.27 \end{align*}$ 

A Two-Year Forward Rate, Deferred Two Years The 2-year spot rate is $4.5\%$ and the 4-year spot rate is $5.5\%$ . A company will borrow at the end of year 2 for two years. Find the 2-year forward rate, deferred 2 years, implied by these spot rates.
Answer With $n = 2$ and $k = 2$ :
 $\begin{align*} (1.045)^2\,(1+f_{2,4})^2 &= (1.055)^4 \\ (1+f_{2,4})^2 &= \frac{1.238825}{1.092025} \\ &= 1.134429 \\ f_{2,4} &= 1.134429^{1/2} - 1 \\ &= 6.51\% \end{align*}$ 
The forward rate is above both spot rates. The 4-year spot rate is a geometric average of the rate for years 1–2 and the rate for years 3–4, so for $s_4$ to exceed $s_2$ the later two years must earn more than $s_4$ .