(c) A ten-year bond with half yearly coupons of 8% pa has just been issued with a redemption yield of 12% pa effective. It is redeemable at par. What price would an investor paying 20% tax on income pay for the bond? Tax payments are due three months after each coupon is received. [4 marks]

To solve this problem, we need to calculate the price of the bond for an investor who pays a 20% tax on income. The bond has a redemption yield of 12% per annum effective, which means that the bond will be redeemed at par value. The bond has a coupon rate of 8% per annum, which means that the bond will pay semi-annual coupons of $8\% \times \frac{1}{2} = 4\% $ per semi-annual period.<br /><br />First, let's calculate the present value of the coupon payments. Since the coupons are paid semi-annually, we need to discount them to the present value using the redemption yield of 12% per annum effective. The redemption yield is equivalent to a semi-annual yield of $\frac{12\%}{2} = 6\% $.<br /><br />The present value of the coupon payments can be calculated using the following formula:<br /><br />$PV_{\text{coupons}} = C \times \left(1 - (1 + r)^{-n}\right) / r$<br /><br />where $C$ is the coupon payment, $r$ is the semi-annual yield, and $n$ is the number of semi-annual periods.<br /><br />$PV_{\text{coupons}} = 4\% \times \left(1 - (1 + 0.06)^{-10}\right) / 0.06$<br /><br />$PV_{\text{coupons}} = 4\% \times \left(1 - (1.06)^{-10}\right) / 0.06$<br /><br />$PV_{\text{coupons}} = 4\% \times \left(1 - 0.5664\right) / 0.06$<br /><br />$PV_{\text{coupons}} = 4\% \times 0.4336 / 0.06$<br /><br />$PV_{\text{coupons}} = 4\% \times 7.2267$<br /><br />$PV_{\text{coupons}} = 0.28907$<br /><br />Next, let's calculate the present value of the redemption value. Since the bond is redeemable at par, the redemption value is equal to the face value of the bond, which is $100. We need to discount the redemption value to the present value using the redemption yield of 12% per annum effective.<br /><br />$PV_{\text{redemption}} = F / (1 + r)^n$<br /><br />where $F$ is the face value of the bond.<br /><br />$PV_{\text{redemption}} = 100 / (1 + 0.06)^{10}$<br /><br />$PV_{\text{redemption}} = 100 / (1.06)^{10}$<br /><br />$PV_{\text{redemption}} = 100 / 1.7908$<br /><br />$PV_{\text{redemption}} = 55.88$<br /><br />Now, let's calculate the price of the bond for the investor who pays a 20% tax on income. The price of the bond for the investor is equal to the sum of the present value of the coupon payments and the present value of the redemption value, both adjusted for the tax rate.<br /><br />$P_{\text{investor}} = PV_{\text{coupons}} \times (1 - \text{tax rate}) + PV_{\text{redemption}} \times (1 - \text{tax rate})$<br /><br />$P_{\text{investor}} = 0.28907 \times (1 - 0.20) + 55.88 \times (1 - 0.20)$<br /><br />$P_{\text{investor}} = 0.28907 \times 0.80 + 55.88 \times 0.80$<br /><br />$P_{\text{investor}} = 0.23128 + 44.70$<br /><br />$P_{\text{investor}} = 44.93$<br /><br />Therefore, an investor paying 20% tax on income would pay approximately $44.93 for the bond.