To calculate the bond price given its face value, coupon rate, maturity, and yield to maturity (YTM), we’ll use the present value (PV) formula for bond pricing. In this case, the bond has a face value (FV) of $1,000, coupon rate (C) of 5%, and matures in 10 years (N). The yield to maturity (YTM) is 4%.
The bond pricing formula is:
$$PV = \sum\limits_{i=1}^{N} \cfrac{C \cdot FV}{(1 + YTM)^i} + \cfrac{FV}{(1 + YTM)^N}.$$
In this example:
$$PV = \sum\limits_{i=1}^{10} \cfrac{0.05 \cdot 1000}{(1 + 0.04)^i} + \cfrac{1000}{(1 + 0.04)^{10}}.$$
Using this formula, we can calculate the present value of the bond by calculating the PV for each cash flow and then adding them up. This gives:
$$\begin{aligned}
PV & = \frac{50}{1.04} + \frac{50}{1.04^2} + \frac{50}{1.04^3} + \ldots + \frac{50}{1.04^{10}} + \frac{1000}{1.04^{10}} \\
& \approx 48.08 + 46.23 + 44.45 + 42.74 + 41.09 + 39.51 \\
& + 38.00 + 36.54 + 35.13 + 33.78 + 675.56 \\
& \approx 1081.11\end{aligned}$$
So, the price of the bond given a yield to maturity of 4% is approximately $1081.11.