1. **Problem Statement:** An investor buys a 20-year government bond with a 10% annual coupon rate paid semiannually, priced at 875 for a 1000 par value. We need to find the yield to maturity (YTM) if held to maturity, and the holding-period yield (HPY) if sold after 10 years for 950. 2. **Yield to Maturity (YTM) Definition:** YTM is the internal rate of return (IRR) on the bond's cash flows, equating the present value of all future coupon payments and the par value at maturity to the current price. 3. **Given Data:** - Par value $F = 1000$ - Coupon rate $c = 10\%$ annually - Coupon payment per period $C = \frac{10\% \times 1000}{2} = 50$ (semiannual) - Number of periods $N = 20 \times 2 = 40$ - Current price $P = 875$ 4. **YTM Calculation:** The price equals the sum of present values of coupons and par: $$ P = \sum_{t=1}^{N} \frac{C}{(1 + r)^t} + \frac{F}{(1 + r)^N} $$ where $r$ is the semiannual YTM. We solve for $r$ such that: $$ 875 = 50 \times \frac{1 - (1 + r)^{-40}}{r} + \frac{1000}{(1 + r)^{40}} $$ This requires numerical methods (trial, interpolation, or financial calculator). 5. **Approximate YTM:** Using trial and error or a financial calculator, the semiannual YTM $r \approx 0.061$ (6.1%). Annual YTM = $2 \times 6.1\% = 12.2\%$. 6. **Holding-Period Yield (HPY) after 10 years:** The investor sells after 10 years (20 periods). The HPY is the total return over 10 years. - Total coupons received in 10 years: $20 \times 50 = 1000$ - Sale price after 10 years: $950$ - Initial price: $875$ Total cash inflow = $1000 + 950 = 1950$ HPY over 10 years: $$ \text{HPY} = \frac{\text{Total inflow} - \text{Initial price}}{\text{Initial price}} = \frac{1950 - 875}{875} = 1.2286 = 122.86\% $$ Annualized HPY: $$ \left(1 + 1.2286\right)^{\frac{1}{10}} - 1 = 0.0835 = 8.35\% \text{ per year} $$ **Final answers:** - Yield to maturity (annual) $\approx 12.2\%$ - Holding-period yield (annualized) $\approx 8.35\%$