Serial bond price

Question

On August 15, 2015 a corporation issues a 10% serial bond with face amount 50,000,000. The redemption is scheduled to take place at 5,000,000 every August 15 from 2025 to 2029 and 25,000,000 on August 15, 2030. Find the price of the entire issue on the issue date at a yield of i(2) = 0.125.

I don't want to use Makeham's bond price formula and have been trying to use P= F(Vj^n) + Frā|j (j is a subscript, denoting yield rate)

I cant seem to get the same answer with this formula. The price is $41,995,392

Answer 1

You can use your bond price formula to compute the price of each bond in the serial bond separately, and then add the prices.

Let $P_1$ be the price of the bond maturing in 2025, $P_2$ the price of the bond maturing in 2026, ..., and $P_6$ the price of the bond maturing in 2030. The semiannual coupons for the first five bonds are $(5\%)(5,000,000)=250,000$ and the semiannual coupons of the last bond are $(5\%)(25,000,000) = 1,250,000$. The 6-month interest rate is $6.25\%$, so the prices are given by \begin{align*} P_1 & = 250,000 a_{\overline{20}|6.25\%} + 5,000,000 \nu_{6.25\%}^{20}=4,297,455\\ P_2 &=250,000 a_{\overline{22}|6.25\%} + 5,000,000 \nu_{6.25\%}^{22}=4,263,490\\ P_3 &=250,000 a_{\overline{24}|6.25\%} + 5,000,000 \nu_{6.25\%}^{24}=4,233,402\\ P_4 &=250,000 a_{\overline{26}|6.25\%} + 5,000,000 \nu_{6.25\%}^{26}=4,206,751\\ P_5 &=250,000 a_{\overline{28}|6.25\%} + 5,000,000 \nu_{6.25\%}^{28}=4,183,143\\ P_6 &= 1,250,000 a_{\overline{30}|6.25\%} + 25,000,000 \nu_{6.25\%}^{30}=20,811,151 \end{align*} Summing these gives the answer $41,995,392$.

Answer 2

Rather than using some pricing formula, I suggest you build a table in excel or similar software package that looks something like the following.

$\begin {array}{} \text{Time} &\text{Principal outstanding} &\text{Principal paydown} &\text{Interest} &\text{Total payment} &\text{discount factor} &\text{discounted cash flow}\\ 15&\$50,000,000\\ 15.5&\$50,000,000 &&\$2,500,000 &\$2,500,000 & 1.063 &\$2,352,941.18\\ 16&\$50,000,000 &&\$2,500,000&\$2,500,000 &1.129 &\$2,214,532.87\\ 16.5&\$50,000,000 &&\$2,500,000 &\$2,500,000 &1.199 &\$2,084,266.23\\ 17& \$50,000,000 && \$2,500,000 & \$2,500,000 &1.274 & \$1,961,662.34\\ \vdots\\\ 25& \$50,000,000&\$5,000,000&\$2,500,000&\$7,500,000&3.362&\$2,230,912.25\\ 25.5&\$45,000,000&&\$2,250,000&\$2,250,000&3.572&\$629,904.64\\ 26&\$45,000,000&\$5,000,000 &\$2,250,000&\$7,250,000&3.795&\$1,910,299.03\\ 26.5&\$40,000,000&&\$2,000,000& \$2,000,000& 4.032& \$495,980.27\\ \vdots\\ 29.5&\$25,000,000&&\$1,250,000&\$1,250,000&5.801&\$215,462.05\\ 30& \$25,000,000 &\$25,000,000& \$1,250,000&\$26,250,000&6.164 &\$4,258,544.07\\ \\ &&&&&& \$41,995,391.95\end {array}$

The interest is 5% paid semi annually.

The discount factor is $(1+\frac {0.125}{2})^{2t}$ where $t$ is the time in years since the initial investment.

And the discounted cashflow is the periodic cashflow divided by the discount factor.