Chapter 14, Problem 31PS

Summary Introduction

(a)

To calculate:

The rate of return for a one year investment if the bond is selling at a yield to maturity of$7\%$by the end of the year where bond has$5\%$coupon rate with$8\%$yield to maturity and maturity life is$20$years.

Introduction:

Return is a profit on an investment. So the rate of return is a profit over a period of time of investment. It expressed as a percentage of the original investment.

Answer to Problem 31PS

The rate of return for holding one year is$19.62\%$

Explanation of Solution

Given:

Coupon on bond$\left(\text{PMT}\right)$is$\$50$

Maturity of bond$\left(\text{n}\right)$is$20$years

Yield to maturity of bond$\left(\text{r}\right)$is$8\%$

Face value of bond$\left(\text{FV}\right)$is$\$1000$

Bond price is${\displaystyle \sum _{\text{t=1}}^{20}\frac{\text{PMT}}{{\left(1+\text{r}\right)}^1}+\frac{\text{FV}}{{\left(1+\text{r}\right)}^{20}}}$

For starting year price is

For price for the next year is

  $\begin{array}{c}\text{Price}={\displaystyle \sum _{\text{t=1}}^{19}\frac{\$50}{{\left(1.07\right)}^1}+\frac{\$1000}{{\left(1.07\right)}^{19}}}\\ =\frac{\$50}{\left(1.07\right)}+\frac{\$50}{{\left(1.07\right)}^2}+\frac{\$50}{{\left(1.07\right)}^3}+\frac{\$50}{{\left(1.07\right)}^4}+\frac{\$50}{{\left(1.07\right)}^5}+\frac{\$50}{{\left(1.07\right)}^6}+\frac{\$50}{{\left(1.07\right)}^7}+\frac{\$50}{{\left(1.07\right)}^8}+\frac{\$50}{{\left(1.07\right)}^9}+\frac{\$50}{{\left(1.07\right)}^{10}}+\\ \frac{\$50}{{\left(1.07\right)}^{11}}+\frac{\$50}{{\left(1.07\right)}^{12}}+\frac{\$50}{{\left(1.07\right)}^{13}}+\frac{\$50}{{\left(1.07\right)}^{14}}+\frac{\$50}{{\left(1.07\right)}^{15}}+\frac{\$50}{{\left(1.07\right)}^{16}}+\frac{\$50}{{\left(1.07\right)}^{17}}+\frac{\$50}{{\left(1.07\right)}^{18}}+\frac{\$50}{{\left(1.07\right)}^{19}}+\frac{\$1000}{{\left(1.07\right)}^{19}}\\ =\$46.73+\$43.67+\$40.81+\$38.14+\$35.65+\$33.31+\$31.14+\$29.10+\$27.20+\$25.42+\$23.75+\\ \$22.20+\$20.75+\$19.39+\$18.12+\$16.94+\$15.83+\$14.79+\$13.83+\$276.51\\ =\$793.28\end{array}$

The return on holding period is

  $\begin{array}{c}\text{Return on Holding Period}=\left(\frac{\text{Coupon income +}\left(\text{End of year value - Beginning price}\right)}{\text{Beginning price}}\right)\times 100\\ =\frac{\$50+\left(\$793.28-\$705.44\right)}{\$705.44}\\ =19.54\%\end{array}$

Summary Introduction

(b)

To calculate:

The tax on income from bond and capital gain if the bond is selling at a yield to maturity of$7\%$by the end of the year where bond has$5\%$coupon rate with$8\%$yield to maturity and maturity life is$20$years.

Introduction:

Tax to be levied on income of coupon bond as per regular rate of tax and capital gain shall be levied on change in price if there is change in yield to maturity from constant yield to maturity.

Answer to Problem 31PS

Total tax on income is$\$46.99$

Explanation of Solution

Given:

Coupon on bond$\left(\text{PMT}\right)$is$\$50$

Maturity of bond$\left(\text{n}\right)$is$20$years

Yield to maturity of bond$\left(\text{r}\right)$is$8\%$

Face value of bond$\left(\text{FV}\right)$is$\$1000$

Tax rate on income is$40\%$

Tax on capital gain is$30\%$

For calculation income from bond if yield to maturity is$8\%$

Price for initial year is$\$705.44$

  $\left(\text{as per calculated above}\right)$

Price for first year is

  $\text{Price}={\displaystyle \sum _{\text{t=1}}^{19}\frac{\text{PMT}}{{\left(1+\text{r}\right)}^1}+\frac{\text{FV}}{{\left(1+\text{r}\right)}^{19}}}$

  $\begin{array}{c}\text{Price}={\displaystyle \sum _{\text{t=1}}^{19}\frac{\$50}{{\left(1.08\right)}^1}+\frac{\$1000}{{\left(1.08\right)}^{19}}}\\ =\frac{\$50}{\left(1.08\right)}+\frac{\$50}{{\left(1.08\right)}^2}+\frac{\$50}{{\left(1.08\right)}^3}+\frac{\$50}{{\left(1.08\right)}^4}+\frac{\$50}{{\left(1.08\right)}^5}+\frac{\$50}{{\left(1.08\right)}^6}+\frac{\$50}{{\left(1.08\right)}^7}+\frac{\$50}{{\left(1.08\right)}^8}+\frac{\$50}{{\left(1.08\right)}^9}+\frac{\$50}{{\left(1.08\right)}^{10}}+\\ \frac{\$50}{{\left(1.08\right)}^{11}}+\frac{\$50}{{\left(1.08\right)}^{12}}+\frac{\$50}{{\left(1.08\right)}^{13}}+\frac{\$50}{{\left(1.08\right)}^{14}}+\frac{\$50}{{\left(1.08\right)}^{15}}+\frac{\$50}{{\left(1.08\right)}^{16}}+\frac{\$50}{{\left(1.08\right)}^{17}}+\frac{\$50}{{\left(1.08\right)}^{18}}+\frac{\$50}{{\left(1.08\right)}^{19}}+\frac{\$1000}{{\left(1.08\right)}^{19}}\\ =\$46.30+\$42.87+\$39.69+\$36.75+\$34.03+\$31.51+\$29.17+\$27.01+\$25.01+\$23.16+\$21.44+\\ \$19.86+\$18.38+\$17.02+\$15.76+\$14.59+\$13.51+\$12.51+\$11.59+\$231.71\\ =\$711.87\end{array}$

Implicit interest for the first year is

  $\begin{array}{c}\text{Interest Rate}=\$711.87-705.44\\ =\$6.43\end{array}$

Tax on bond is

  $\begin{array}{c}\text{Tax}=\left(\text{interest income+Implicit Interest}\right)\times \text{tax rate}\\ \text{=}\left(\text{\$50+\$6}\text{.43}\right)\times 40\%\\ =\$22.57\end{array}$

Tax on capital gain is as follows:

  $\begin{array}{c}\text{Tax}=\left(\text{Bond price under 7\% YTM- price under 8\% YTM}\right)\times \text{capital gain tax rate}\\ =\left(\$793.28-\$711.88\right)\times 30\%\\ =\$24.42\end{array}$

So total tax on income is

  $\begin{array}{c}\text{Tax}=\$22.57+\$24.42\\ =\$46.99\end{array}$

Summary Introduction

(c)

To calculate:

The rate of return after tax for a one year investment if the bond is selling at a yield to maturity of$7\%$by the end of the year where bond has$5\%$coupon rate with$8\%$yield to maturity and maturity life is$20$years.

Introduction:

Return is a profit on an investment. So the rate of return is a profit over a period of time of investment. It expressed as a percentage of the original investment.

Answer to Problem 31PS

Total return on income after tax for holding period is

  $=12.88\%$

Explanation of Solution

Given:

Coupon on bond$\left(\text{PMT}\right)$is$\$50$

Total tax on income is$\$46.99$

End of period value is$\$793.28$

Initial year price is$\$705.44$

Holding period return is

  $\begin{array}{c}\text{Holding Period Return}=\frac{\text{Coupon Income+}\left(\text{Value at End of period-Initial price}\right)-\text{Tax}}{\text{Initial price}}\\ =\frac{\$50+\left(\$793.28-\$705.44\right)-\$46.99}{\$705.44}\\ =12.88\%\end{array}$

Summary Introduction

(d)

To calculate:

The realized compound yield before taxes for two-years holding period where bond is selling after two years and having$7\%$YTM at the end of second year and coupon amount is reinvested at$3\%$interest rate for one-year.

Introduction:

Return is a profit on an investment. So the rate of return is a profit over a period of time of investment. It expressed as a percentage of the original investment.

Answer to Problem 31PS

The realized compound yield is$12.95\%$

Explanation of Solution

Here, the formula to calculate and given are added followed by a detailed explanation / working out the complete problem

Initial price of bond$\left(8\%\text{YTM}\right)=\$705.44$

Price on end of second year under$\left(\text{7\%YTM}\right)$is

  $\begin{array}{c}\text{Price}={\displaystyle \sum _{\text{t=1}}^{18}\frac{\text{PMT}}{{\left(1+\text{r}\right)}^1}+\frac{\text{FV}}{{\left(1+\text{r}\right)}^{18}}}\\ ={\displaystyle \sum _{\text{t=1}}^{18}\frac{\$50}{{\left(1.07\right)}^1}+\frac{\$1000}{{\left(1.07\right)}^{18}}}\\ =\$798.82\end{array}$

We invest coupon amount for one-year so total income including coupon amount in end of second year is

  $\begin{array}{c}\text{Total Income}=\$50+\left(\$50\times 1.03\right)\\ =\$101.15\end{array}$

So, total fund after two year including coupon income is

  $\begin{array}{c}\text{Total Fund}=\$798.82+\$101.15\\ =\$899.97\end{array}$

So, realized yield is

  $\begin{array}{c}\text{Yield}=\sqrt{\frac{\text{Total proceeds}}{\text{Inital price}}}-1\\ =\sqrt{\frac{\$899.97}{\$705.44}}-1\\ =12.95\%\end{array}$

Summary Introduction

(e)

To calculate:

The realized compound yield after taxes for two-years where bond is selling after two years and having$7\%$YTM at the end of second year and coupon amount is reinvested at$3\%$interest rate for one-year

Introduction:

Return is a profit on an investment. So the rate of return is a profit over a period of time of investment. It expressed as a percentage of the original investment.

Answer to Problem 31PS

The realized compound yield after tax is$8.47\%$

Explanation of Solution

Here, the formula to calculate and given are added followed by a detailed explanation / working out the complete problem

Price for second year is

  $\begin{array}{c}\text{Price}={\displaystyle \sum _{\text{t=1}}^{18}\frac{\$50}{{\left(1.08\right)}^1}+\frac{\$1000}{{\left(1.08\right)}^{18}}}\\ =\frac{\$50}{\left(1.08\right)}+\frac{\$50}{{\left(1.08\right)}^2}+\frac{\$50}{{\left(1.08\right)}^3}+\frac{\$50}{{\left(1.08\right)}^4}+\frac{\$50}{{\left(1.08\right)}^5}+\frac{\$50}{{\left(1.08\right)}^6}+\frac{\$50}{{\left(1.08\right)}^7}+\frac{\$50}{{\left(1.08\right)}^8}+\frac{\$50}{{\left(1.08\right)}^9}+\frac{\$50}{{\left(1.08\right)}^{10}}+\\ \frac{\$50}{{\left(1.08\right)}^{11}}+\frac{\$50}{{\left(1.08\right)}^{12}}+\frac{\$50}{{\left(1.08\right)}^{13}}+\frac{\$50}{{\left(1.08\right)}^{14}}+\frac{\$50}{{\left(1.08\right)}^{15}}+\frac{\$50}{{\left(1.08\right)}^{16}}+\frac{\$50}{{\left(1.08\right)}^{17}}+\frac{\$50}{{\left(1.08\right)}^{18}}+\frac{\$1000}{{\left(1.08\right)}^{18}}\\ =\$46.30+\$42.87+\$39.69+\$36.75+\$34.03+\$31.51+\$29.17+\$27.01+\$25.01+\$23.16+\$21.44+\\ \$19.86+\$18.38+\$17.02+\$15.76+\$14.59+\$13.51+\$12.51+\$250.25\\ =\$718.82\end{array}$

So implicit interest for second year is

  $\begin{array}{c}\text{Interest}=\$718.82-\$711.88\\ =\$6.94\end{array}$

Total income after tax in first year after considering tax on imlicit interest is

  $\begin{array}{c}\text{Income after tax}=\left(\text{coupon income}\right)\left(1-\text{tax rate}\right)-\text{tax on implicit interesrt}\\ =\left(\$50\right)0.6-\left(\$6.43\right)0.4\\ =\$27.43\end{array}$

We invested it for one-year then total proceed after tax is

  $\begin{array}{c}\text{Proceeds after tax}=\$27.43\times 1.018\\ =\$27.92\end{array}$

So total proceeds up to the second year is

Price of bond in second year is $798.82

Tax on imputed interest in second year is

  $\begin{array}{c}\text{Tax}=\$6.95\times 0.4\\ =\$2.78\end{array}$

Coupon amount after tax in second year is

  $\begin{array}{c}\text{Coupon}=\$50\times 0.6\\ =\$30\end{array}$

Capital gain tax on second year sale price

  $\begin{array}{c}\text{Capital Gain Tax}=\left(\$798.82-\$718.82\right)\times 30\%\\ =\$24\end{array}$

So total cash flow in second year is

  $\begin{array}{c}\text{Cash Flow}=\$798.82+\$30-\$2.78-\$24+\$27.92\\ =\$829.96\end{array}$

Realized coupon yield in second year is

  $\begin{array}{c}\text{Coupon Yield}=\sqrt{\frac{\text{Total proceeds}}{\text{Inital price}}}-1\\ =\sqrt{\frac{\$829.96}{\$705.44}}-1\\ =8.47\%\end{array}$