1. **State the problem:** We are given the spot rate function $$s(n) = 0.09 - 0.03e^{-0.1n}$$ and asked to find the one-year forward rate at time 10, denoted as $$f(10,11)$$. 2. **Recall the formula for forward rates:** The one-year forward rate starting at time $$n$$ is given by $$ f(n,n+1) = \frac{(n+1)s(n+1) - ns(n)}{1} $$ where $$s(n)$$ is the spot rate for maturity $$n$$ years. 3. **Calculate $$s(10)$$:** $$ s(10) = 0.09 - 0.03e^{-0.1 \times 10} = 0.09 - 0.03e^{-1} $$ Using $$e^{-1} \approx 0.3679$$, $$ s(10) = 0.09 - 0.03 \times 0.3679 = 0.09 - 0.011037 = 0.078963 $$ 4. **Calculate $$s(11)$$:** $$ s(11) = 0.09 - 0.03e^{-0.1 \times 11} = 0.09 - 0.03e^{-1.1} $$ Using $$e^{-1.1} \approx 0.3329$$, $$ s(11) = 0.09 - 0.03 \times 0.3329 = 0.09 - 0.009987 = 0.080013 $$ 5. **Calculate the forward rate $$f(10,11)$$:** $$ f(10,11) = (11 \times 0.080013) - (10 \times 0.078963) = 0.880143 - 0.78963 = 0.090513 $$ 6. **Interpretation:** The one-year forward rate at time 10 is approximately $$0.0905$$ or 9.05%. **Final answer:** $$\boxed{0.0905}$$