$4.$ 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 Oliver wants to earn a return of $10\mathrm{.}50\%$ and is offered the opportunity to purchase a $$1,000$ par value bond that pays a $8\mathrm{.}75\%$ coupon rate $($distributed semiannually$)$ with three years remaining to maturity. The following formula can be used to compute the bond$$s intrinsic value:

Intrinsic 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$)6$

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 Oliver$$s potential bond investment is currently exhibiting an intrinsic value less than $$1,000.$

Now, consider the situation in which Oliver wants to earn a return of $11\mathrm{.}75\%,$ but the bond being considered for purchase offers a coupon rate of $8\mathrm{.}75\%.$ 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?

When the coupon rate is less than Oliver$$s required return, the bond should trade at a discount.

When the coupon rate is less than Oliver$$s required return, the bond should trade at a premium.

When the coupon rate is less than Oliver$$s required return, the intrinsic value will be greater than its par value.

A bond should trade at par when the coupon rate is less than Oliver$$s required return.