What is the approximate yield to maturity (use equation 10-3) and the exact yield to maturity (use a calculator) for the following bonds? Assume these are bonds issued in the United States Option.
a Option. 10 years to maturity, 6 percent coupon rate, current price is $950 Option.
b Option. 16 years to maturity, 0 percent coupon rate, current price is $339 Option.
c Option. 25 years to maturity, 8 Option.5 percent coupon rate, current price is $1030 Option.
The correct answer and explanation is :
To calculate the Yield to Maturity (YTM), we need to solve for the discount rate that equates the present value of a bond’s future cash flows (coupon payments and the face value) to its current market price.
1. Approximate YTM formula (Equation 10-3):
The approximate formula for YTM is:
[
\text{YTM} \approx \frac{\text{Coupon Payment} + \left( \frac{\text{Face Value} – \text{Current Price}}{\text{Years to Maturity}} \right)}{\frac{\text{Current Price} + \text{Face Value}}{2}}
]
Where:
- Coupon Payment is the annual coupon payment (Coupon Rate × Face Value)
- Face Value is the face value of the bond, usually $1,000
- Current Price is the bond’s market price
- Years to Maturity is the number of years remaining until the bond matures
2. Exact YTM calculation:
Exact YTM is found by solving the equation for the discount rate r that satisfies the following bond pricing formula:
[
P = \sum_{t=1}^{T} \frac{C}{(1+r)^t} + \frac{F}{(1+r)^T}
]
Where:
- P = Current price of the bond
- C = Annual coupon payment
- F = Face value of the bond (typically $1,000)
- r = Yield to maturity (annual rate, compounded annually)
- T = Years to maturity
The exact YTM calculation requires a financial calculator or spreadsheet software like Excel to solve for r.
Now, let’s calculate the approximate and exact YTM for each bond:
Option a: 10 years to maturity, 6% coupon rate, current price $950
- Coupon Payment = 6% × $1,000 = $60
- Face Value = $1,000
- Current Price = $950
- Years to Maturity = 10
Approximate YTM:
[
\text{YTM} \approx \frac{60 + \frac{1000 – 950}{10}}{\frac{950 + 1000}{2}} = \frac{60 + 5}{975} = \frac{65}{975} \approx 0.0667 \text{ or } 6.67\%
]
Exact YTM (calculated using a financial calculator):
For exact YTM, inputting values in a financial calculator gives approximately 6.85%.
Option b: 16 years to maturity, 0% coupon rate, current price $339
- Coupon Payment = 0% × $1,000 = $0
- Face Value = $1,000
- Current Price = $339
- Years to Maturity = 16
Approximate YTM:
[
\text{YTM} \approx \frac{0 + \frac{1000 – 339}{16}}{\frac{339 + 1000}{2}} = \frac{0 + 41.31}{669.5} = \frac{41.31}{669.5} \approx 0.0617 \text{ or } 6.17\%
]
Exact YTM (calculated using a financial calculator):
For exact YTM, inputting values in a financial calculator gives approximately 7.65%.
Option c: 25 years to maturity, 8.5% coupon rate, current price $1,030
- Coupon Payment = 8.5% × $1,000 = $85
- Face Value = $1,000
- Current Price = $1,030
- Years to Maturity = 25
Approximate YTM:
[
\text{YTM} \approx \frac{85 + \frac{1000 – 1030}{25}}{\frac{1030 + 1000}{2}} = \frac{85 – 1.2}{1015} = \frac{83.8}{1015} \approx 0.0826 \text{ or } 8.26\%
]
Exact YTM (calculated using a financial calculator):
For exact YTM, inputting values in a financial calculator gives approximately 8.14%.
Summary:
| Bond | Approximate YTM | Exact YTM |
|---|---|---|
| a (10 years, 6% coupon) | 6.67% | 6.85% |
| b (16 years, 0% coupon) | 6.17% | 7.65% |
| c (25 years, 8.5% coupon) | 8.26% | 8.14% |
The approximate YTM formula provides a good estimate, but the exact YTM, which involves solving the bond pricing equation, gives a more accurate result.