1. **Problem Statement:** We are given zero coupon bond prices for different maturities and asked to calculate the 10-year yield to maturity ($Y_{10}$) and the 5-year forward rate starting at year 10 ($F_{5,10}$). 2. **Given Data:** - Price of 1-year zero coupon bond, $P_1 = 0.94$ - Price of 5-year zero coupon bond, $P_5 = 0.70$ - Price of 10-year zero coupon bond, $P_{10} = 0.47$ - Price of 15-year zero coupon bond, $P_{15} = 0.30$ 3. **Formulas:** - Yield to maturity for $n$ years zero coupon bond is given by: $$Y_n = \left(\frac{1}{P_n}\right)^{\frac{1}{n}} - 1$$ - The forward rate for a period starting at year $t$ for $m$ years, $F_{m,t}$, is given by: $$F_{m,t} = \left(\frac{P_t}{P_{t+m}}\right)^{\frac{1}{m}} - 1$$ 4. **Calculate $Y_{10}$:** $$Y_{10} = \left(\frac{1}{0.47}\right)^{\frac{1}{10}} - 1$$ Calculate inside the power: $$\frac{1}{0.47} \approx 2.1277$$ Then: $$Y_{10} = 2.1277^{0.1} - 1$$ Using a calculator: $$2.1277^{0.1} \approx 1.0781$$ So: $$Y_{10} = 1.0781 - 1 = 0.0781 = 7.81\%$$ 5. **Calculate $F_{5,10}$:** $$F_{5,10} = \left(\frac{P_{10}}{P_{15}}\right)^{\frac{1}{5}} - 1 = \left(\frac{0.47}{0.30}\right)^{0.2} - 1$$ Calculate inside the power: $$\frac{0.47}{0.30} \approx 1.5667$$ Then: $$F_{5,10} = 1.5667^{0.2} - 1$$ Using a calculator: $$1.5667^{0.2} \approx 1.0947$$ So: $$F_{5,10} = 1.0947 - 1 = 0.0947 = 9.47\%$$ **Final answers:** - 10-year yield to maturity, $Y_{10} = 7.81\%$ - 5-year forward rate starting at year 10, $F_{5,10} = 9.47\%$