Question

Bond valuation

The process of bond valuation is based on the fundamental concept that the current price of a security can be determined by calculating the present value of the cash flows that the security will generate in the future.

There is a consistent and predictable relationship between a bond’s coupon rate, its par value, a bondholder’s required return, and the bond’s resulting intrinsic value. Trading at a discount, trading at a premium, and trading at par refer to particular relationships between a bond’s intrinsic value and its par value. This also results from the relationship between a bond’s coupon rate and a bondholder’s required rate of return.

Remember, a bond’s coupon rate partially determines the interest-based return that a bond   pay, and a bondholder’s required return reflects the return that a bondholder   to receive from a given investment.

The mathematics of bond valuation imply a predictable relationship between the bond’s coupon rate, the bondholder’s required return, the bond’s par value, and its intrinsic value. These relationships can be summarized as follows:

•	When the bond’s coupon rate is equal to the bondholder’s required return, the bond’s intrinsic value will equal its par value, and the bond will trade at par.
•	When the bond’s coupon rate is greater than the bondholder’s required return, the bond’s intrinsic value will     its par value, and the bond will trade at a premium.
•	When the bond’s coupon rate is less than the bondholder’s required return, the bond’s intrinsic value will be less than its par value, and the bond will trade at   .

For example, assume Noah wants to earn a return of 15.75% and is offered the opportunity to purchase a $1,000 par value bond that pays a 13.50% coupon rate (distributed semiannually) with three years remaining to maturity. The following formula can be used to compute the bond’s intrinsic value:

Intrinsic ValueIntrinsic Value

=  =

A(1+C)1+A(1+C)2+A(1+C)3+A(1+C)4+A(1+C)5+A(1+C)6+B(1+C)6A1+C1+A1+C2+A1+C3+A1+C4+A1+C5+A1+C6+B1+C6

Complete the following table by identifying the appropriate corresponding variables used in the equation.

Unknown	Variable Name	Variable Value
A
B		$1,000
C	Semiannual required return

Based on this equation and the data, it is   to expect that Noah’s potential bond investment is currently exhibiting an intrinsic value greater than $1,000.

Now, consider the situation in which Noah wants to earn a return of 11.50%, but the bond being considered for purchase offers a coupon rate of 13.50%. Again, assume that the bond pays semiannual interest payments and has three years to maturity. If you round the bond’s intrinsic value to the nearest whole dollar, then its intrinsic value of    (rounded to the nearest whole dollar) is   its par value, so that the bond is   .

Given your computation and conclusions, which of the following statements is true?

A bond should trade at a par when the coupon rate is greater than Noah’s required return.

When the coupon rate is greater than Noah’s required return, the bond should trade at a discount.

When the coupon rate is greater than Noah’s required return, the bond’s intrinsic value will be less than its par value.

When the coupon rate is greater than Noah’s required return, the bond should trade at a premium.

Answer 1

Answer a.

Remember, a bond’s coupon rate partially determines the interest-based return that a bond will pay, and a bondholder’s required return reflects the return that a bondholder would like to receive from a given investment.

Answer b.

When the bond’s coupon rate is greater to the bondholder’s required return, the bond’s intrinsic value will exceed its par value, and the bond will trade at a premium.

When the bond’s coupon rate is less than the bondholder’s required return, the bond’s intrinsic value will be less than its par value, and the bond will trade at a discount.

Answer c.

Unknown Variable A:

Annual Coupon Rate = 13.50%
Semiannual Coupon Rate = 6.75%

Bond’s Semiannual Coupon Payment = 6.75% * $1,000
Bond’s Semiannual Coupon Payment = $67.50

Unknown Variable B:

Bond’s Par Value = $1,000

Unknown Variable C:

Annual Required Return = 15.75%

Semiannual Required Return = 7.875%

Answer d.

Based on this equation and the data, it is unreasonable to expect that Noah’s potential bond investment will exhibit an intrinsic value greater than $1,000.

Answer e.

Annual Required Return = 11.50%
Semiannual Required Return = 5.75%

Time to Maturity = 3 years
Semiannual Period to Maturity = 6

Price of Bond = $67.50/1.0575 + $67.50/1.0575^2 + $67.50/1.0575^3 + $67.50/1.0575^4 + $67.50/1.0575^5 + $67.50/1.0575^6 + $1,000/1.0575^6
Price of Bond = $67.50 * (1 - (1/1.0575)^6) / 0.0575 + $1,000 * (1/1.0575)^6
Price of Bond = $1,050

Now, consider the situation in which Noah wants to earn a return of 11.50%, but the bond being considered for purchase offers a coupon rate of 13.50%. Again, assume that the bond pays semiannual interest payments and has three years to maturity. If you round the bond's intrinsic value to the nearest whole dollar, then its intrinsic value of $1,050 is greater than its par value, so that the bond is trading at a premium.

Answer f.

When the coupon rate is greater than Noah’s required return, the bond should trade at a premium.