Final answer:
The value of a bond with a coupon rate less than the YTM is the present value of its future cash flows at the required YTM. For example, an 8% coupon rate bond would be worth less than $1,000 if YTM rises to 12%, as an investor would only pay $964 to achieve a similar return with prevailing market rates.
Step-by-step explanation:
The value of a bond is determined by the present value of its future cash flows, which are the coupon payments and the principal repayment at maturity, discounted at the required yield to maturity (YTM). When the coupon rate is less than the YTM, the bond's value will be less than its par value because investors require a higher return for holding the bond.
If you have the opportunity to purchase a x-year, $1,000 par value bond that has an annual coupon rate of <%y%> and you require a YTM of <%z%>, the bond's worth to you would be the present value of the future cash flows discounted at the YTM of <%z%>. This means it would not necessarily be $1,000, $1,000 plus $y, $1,000 minus $y, or $1,000 plus $z specifically without knowing the values of x, y, and z. You would need to calculate the present value of the annual coupon payments plus the present value of the $1,000 to be received at maturity.
For example, if the coupon rate is 8% (meaning annual payments of $80 on a $1,000 bond) and the market interest rate rises to 12%, as illustrated in the example given, then an investor would only be willing to pay $964 for the bond, which is less than its face value, to achieve an equivalent return. This is because $964 invested at 12% for one year would grow to $1,080, the same amount that the bond would pay in total at the end of the year ($80 in coupon plus $1,000 at maturity).