Semiannual coupon = 10%*$100/2 = $5
Since it is one-year semiannual bond, it pays two coupons, one at six months from now and the other at maturity when the bank account will be closed too. Hence only one coupon payment is reinvested in the bank account for six months. At maturity, we get $105 (principal and coupon payment).
Present value = $5/(1+(0.1/2)2 + $100/(1+(0.1/2)2 = $95.24
(a) When reinvestment rate is 10%, the $5 reinvested in bank account will be worth
Future value = $5*(1+(0.1/2)) = $5.25
Since the bank offers semiannual compounding, the reinvested coupon will be worth $5.25 when the account is closed. Thus after one year, I get $110.25. Hence the HPR is
HPR = ($110.25/$100) - 1 = 10.25%
Thus HPR in this part is 10.25%
(b) When reinvestment rate is 4%, the $5 reinvested in bank account will be worth
Future value = $5*(1+(0.04/2)) = $5.10
Since the bank offers semiannual compounding, the reinvested coupon will be worth $5.10 when the account is closed. Thus after one year, I get $110.10. Hence the HPR is
HPR = ($110.10/$100) - 1 = 10.10%
Thus HPR in this part is 10.10%
(c) When reinvestment rate is 16%, the $5 reinvested in bank account will be worth
Future value = $5*(1+(0.16/2)) = $5.40
Since the bank offers semiannual compounding, the reinvested coupon will be worth $5.40 when the account is closed. Thus after one year, I get $110.40. Hence the HPR is
HPR = ($110.40/$100) - 1 = 10.40%
Thus HPR in this part is 10.40%
YTM is the average return if the bond is held to maturity and HPR is the rate of return over a particular investment period. YTM is based on coupon rate, maturity and par value, whereas HPR is based on bond's price at the beginning and end of the holding period and other additional income from the bond, if any. When YTM is unchanged from its initial value, the HPR when the holding period is until maturity, is equal to the YTM.
Consider the one-year bond paying a semiannual coupon of $5 and selling at face value of $100. The bond's initial YTM is 10%, equal to the coupon rate, because the current price, $100 equals the face value, which means that the coupon rate = 10/100 = 10%, equals the YTM. If the YTM remains at 10% over the year, the bond price will remain at par (at face value), so the holding period return will also be 10%. This is the case, when the returns are not reinvested or reinvestments are ignored. However, in the above three cases, since the semiannual coupon is reinvested and an interest earned on the coupon, the HPR is slightly higher than the YTM, based on the interest rate paid by the bank and its mode.