You purchase a bond for $875. It pays $60 a year (semiannual coupon is 3%),
&the bond matures after 10 years. What is the yield to maturity?
Calculation of yield to maturity:
Yield to maturity = (C + ((F-P)/n))*2 / ((F+P)/2)
F = Face value = $ 1000
C = Semiannual Coupon Payment = Face value * Coupon rate = $1000*3% = $30
P = Price = $875
n = number of semiannual = 10 years *2 = 20 Semiannual
Yield to maturity = (30 + ((1000-875)/20))*2 / ((1000+875)/2)
= (30 + 6.25)*2 / 937.50
= 72.50 / 937.50
= 0.077333
= 7.73 %
thank you very much
To calculate the yield to maturity (YTM) of a bond, we need to follow a formula and make certain assumptions. The YTM is the annualized return an investor can expect to earn if they hold the bond until maturity. Here's how you can calculate it:
1. Calculate the periodic coupon payment: The coupon payment is given as $60 per year, and since it is paid semiannually, we need to divide it by 2 to get the semiannual coupon payment. Therefore, the semiannual coupon payment is $30.
2. Determine the number of periods: Since the bond matures after 10 years, and coupons are paid semiannually, the total number of periods is 20 (10 years x 2 coupons per year).
3. Determine the present value of the bond: The present value of the bond is the sum of the present value of all the coupon payments plus the present value of the face value (the amount received at maturity). To calculate the present value, we need to use the YTM as the discount rate. We'll have to make an initial assumption for the YTM and iteratively adjust it until we find the correct value.
4. Start with an initial assumption for the YTM: Let's assume an initial YTM of 4%. This is just an arbitrary value to start the calculation.
5. Calculate the present value of each coupon payment: Using the YTM of 4%, we can calculate the present value of each semiannual coupon payment using the present value of an ordinary annuity formula. In this case, we have an ordinary annuity because the bondholder receives equal payments at regular intervals. The formula is:
PV = C * (1 - (1 + r)^(-n)) / r
Where:
PV = Present value of the coupon payment
C = Coupon payment per period
r = Discount rate (YTM divided by the number of periods in a year)
n = Number of periods
Applying this formula, for the first semiannual coupon payment:
PV = $30 * (1 - (1 + 0.04/2)^(-1)) / (0.04/2)
PV = $30 * (1 - (1.02)^(-1)) / (0.02)
PV = $30 * (1 - 0.9804) / 0.02
PV = $29.412
Similarly, you can calculate the present value of each subsequent semiannual coupon payment.
6. Calculate the present value of the face value: The face value is the amount received at maturity, which in this case is $875. The present value of the face value can be calculated using the formula:
PV = F / (1 + r)^n
Where:
PV = Present value of the face value
F = Face value of the bond
r = Discount rate (YTM divided by the number of periods in a year)
n = Number of periods
Applying this formula:
PV = $875 / (1 + 0.04/2)^20
PV = $875 / (1.02)^20
PV = $608.31
7. Calculate the total present value of the bond: The total present value of the bond is the sum of the present value of all the coupon payments and the present value of the face value. In this case, it would be the sum of the present values calculated in steps 5 and 6:
Total PV = Sum of PV of coupon payments + PV of face value
Total PV = $29.412 + $608.31
Total PV = $637.722
8. Adjust the YTM iteratively: With the initial assumption of 4% for the YTM, we obtained a total present value of $637.722. Now, compare this value with the actual purchase price of $875. If the total present value is less than the purchase price, you need to increase the YTM assumption and repeat steps 5-7 until the total present value approaches the purchase price.
For example, let's increase the YTM assumption to 5% and repeat the calculations:
- Calculate the present value of each semiannual coupon payment using the new YTM.
- Calculate the present value of the face value using the new YTM.
- Calculate the total present value of the bond using the new YTM.
Compare this new total present value with the purchase price. If it is still less, continue increasing the YTM guess and repeating the calculations until you reach a total present value close to the purchase price.
Repeat this iteration process until you find the YTM that results in a total present value of the bond close to the purchase price of $875. The YTM value you find will be the yield to maturity of the bond.