There is a fundamental connection between spot rates and forward rates. The accumulation amount at time $k$ of an investment of `$1` at time$t=0$is given by$(1 + s_k)^k$. If we had agreed to invest this amount at time$k$for$n-k$years, then the accumulation at time$n$would be:$(1 + s_k)^k \cdot (1 + f_{[k,n]})$.

We are also aware that `$1` invested at time 0 for$n$years will be$(1 + s_n)^n$. This gives us:

$(1 + s_k)^k \cdot (1 + f_{[k,n]}) = (1 + s_n)^n$

For a one-period rate, $f_k = f_{k,1}$. Also, define $f_0 = s_1$. Using the same arguments:

$(1 + s_n)^n = (1 + f_0)(1 + f_1)(1 + f_2)\ldots(1 + f_{n-1})$

Therefore,

$(1 + s_3)^3 = [(1.03)(1.04)(1.05)]$

$s_3 = [(1.03)(1.04)(1.05)]^{\frac{1}{3}} - 1 = (1.12476)^{1/3} - 1 = 1.03997 - 1 = 3.997\% \approx 4\%$

The 3-year spot rate is approximately 4%, which makes intuitive sense as it represents the geometric average of the forward rates over the three periods.