As an example, if you bought a $1,000 bond at face value that pays six percent interest twice per year for five years, you’d receive 10 coupon payments of $30 each and get your original $1,000 investment back at the end of the five year maturity term. In this case, the coupon rate also represents your annual yield.
However, things get more complicated when an alternative investment exists that offers a different interest rate. In such a competitive market, an investor would be foolish to purchase bonds at face value if the alternative investment yields a greater return on investment, assuming the risk is the same. Likewise, institutes might be unwilling to sell bonds at face value if the bond’s coupon rate is better than the alternative investment.
Determining the Maximum Theoretical Value
When other options exist, the purchase price of the bond might be lower when a comparable investment offers a higher interest rate or higher when the bond’s coupon rate exceeds the alternative investment’s rate. Thus, the bond’s annual yield is a product of the coupon rate and the difference between the purchase price and the face value. In such cases, the annual yield might be lower or higher than the bond’s coupon rate.
The price at which the bond offers the same annual yield as the alternative investment, or any required rate of return, is called the maximum theoretical value and represents the highest price you should pay for the bond. This figure is calculated by discounting each payment, including the ending face value payment, by the alternative investment’s interest rate, or the required rate of return.
Because each coupon payment represents the same, regular payment, you can use an annuity formula to ease these calculations and then simply add the discounted face-value payment. Thus, the formula is
Where:
- MTV = Maximum theoretical value, the highest price you should pay
- C = Amount of one coupon payment, calculated as (Coupon Rate * Par Value)/Number of Payments per Year)
- i = Required rate of return, calculated as (Alternative Interest Rate)/(Number of Payments per Year)
- n = Number of coupon payments, calculated as (Number of Payments per Year) * (Number of Years)
- P = Par or face value of the bond
An Example Calculation
1. Plug in the data from the scenario described at the beginning of this article in the original formula, as shown below:
Where:
- (0.06 * 1000)/2 calculates the periodic coupon payment
- 0.10/2 converts the alternative interest rate into a periodic rate
- 5 * 2 calculates the number of payment periods until maturity
2. After converting the coupon payment, alternative interest rate and maturity term to periodic values, the formula should look more like the original formula:
3. Add both parenthesized portions of the formula and raise the result to the n^{th} power:
4. Divide the par value by the result to discount the ending, par value payment to its present value:
This tells you that $1,000 five years from now is only worth $613.50 at the present time, given the alternative investment option.
5. Calculate the bracketed portion of the formula and divide by the periodic alternative rate. The result is the multiplier for discounting the coupon payments:
MTV = $30 * 7.73 + $613.50
6. Multiply the result by the coupon payment to discover the total of all future coupon payments in present day dollars:
MTV = $231.90 + $613.50
This tells you that the total of all 10 coupon payments of $30 only has a present value of $231.90.
7. Add the two figures together to get the maximum theoretical value:
MTV = $845.40
This means you should pay $845.40 for the bond to get a comparable investment to the alternative. If you pay a lower price, you’ll enjoy an even better yield, but if the purchase price is higher, you’re better off investing in the alternative.