Chapter 19, Problem 5PS

Now suppose the investor in Problem $\text{3}$ also sells forward $\pounds \text{5},000$ at a forward exchange rate of $\$\text{2}.\text{1}0/\pounds$ .
a. Recalculate the dollar-denominated returns for each scenario.
b. What happens to the standard deviation of the dollar-denominated return?
Compare it to both its old value and the standard deviation of the pound-denominated return. $\left(\text{LO 19}-\text{2}\right)$

Summary Introduction

(a)

To calculate:

The rates of return in dollar-denominated in different scenarios including a forward sale by the investor of $\text{Pound }5000$ at a rate of $\$2/\text{pound}$

Introduction:

The dollar-denominated return is the return earned by the U.S. investor by investing a particular amount in foreign currency.

Answer to Problem 5PS

Below table highlights the values:

Per share price (£)	Dollar-Denominated Return (%)
	$\$2.10/\pounds$
£ $35$	$8.125\%$
£ $40$	$-5\%$
£ $45$	$-18.125\%$

Explanation of Solution

Given:

  $\begin{array}{c}\text{Current price per share}=\pounds 40\text{ per share}\\ \text{Year-end market prices}=\pounds \text{35,}\pounds \text{40and}\pounds \text{45}\\ \text{Current exchange rate}=\$2/\pounds \\ \text{Forward exchange rate}=\$2.1/\pounds \end{array}$

Let's calculate the dollar-denominated return and for that, the following formula is used:

  $\text{r}=\frac{\left({\text{E}}_0\times {\text{P}}_0\right)-\left({\text{F}}_0\times {\text{P}}_1\right)}{\left({\text{E}}_0\times {\text{P}}_0\right)}$

  $\begin{array}{c}\text{Here,}\\ {\text{E}}_0=\text{current exchange rate}\\ {\text{F}}_0=\text{forward exchange rate}\\ {\text{P}}_0=\text{current market price}\\ {\text{P}}_1=\text{year-end market price}\end{array}$

By using the above formula, Dollar-denominated return is computed as follows:

Per share price (£)	Dollar-Denominated Return (%)
$\pounds 35$	$\begin{array}{c}\text{Return}=\frac{\left(\left(\pounds 40\times \$2\right)-\left(\pounds 35\times \$2.1\right)\right)}{\left(\pounds 40\times \$2\right)}\\ =\frac{\$80-\$73.5}{\$80}\\ =8.125\%\end{array}$
$\pounds 40$	$\begin{array}{c}\text{Return}=\frac{\left(\left(\pounds 40\times \$2\right)-\left(\pounds 40\times \$2.1\right)\right)}{\left(\pounds 40\times \$2\right)}\\ =\frac{\$80-\$84}{\$80}\\ =-5\%\end{array}$
$\pounds 45$	$\begin{array}{c}\text{Return}=\frac{\left(\left(\pounds 40\times \$2\right)-\left(\pounds 45\times \$2.1\right)\right)}{\left(\pounds 40\times \$2\right)}\\ =\frac{\$80-\$94.5}{\$80}\\ =-18.125\%\end{array}$

Summary Introduction

(b)

To calculate:

Ascertain the standard deviation of the dollar-denominated return.

Introduction:

Standard deviation is a measure to calculate the deviation from the mean which is also called as a measure of dispersion. It helps in analyzing the performance of the fund.

Answer to Problem 5PS

The standard deviation for the dollar-denominated return is $3.79\%$.

Explanation of Solution

Given:

The table showing outcomes:

Per share price (£)	Dollar-Denominated Return (%)
	$\$2.10/\pounds$
£ $35$	$8.125\%$
£ $40$	$-5\%$
£ $45$	$-18.125\%$

For calculating the standard deviation, the following formula is to be used:

  $\text{Standard deviation}=\sqrt{\text{Variance}}$

For computing standard deviation of dollar-denominated return, following calculation is needed:

Per share price (£)	Probability	Dollar-Denominated Return (%)	$\text{Return}\times \text{Probability }$	$\left({\text{x}}_{\text{i}}-\text{X}\right)$	${\left({\text{x}}_{\text{i}}-\text{X}\right)}^2\times \text{P}$
£ $35$	$0.33$	$8.125\%$	$2.6812$	$4.3312$	$7.1891$
£ $40$	$0.33$	$-5\%$	$-1.65$	$0.0$	$0.0\%$
£ $45$	$0.33$	$-18.125\%$	$-5.9812$	$-4.3312$	$7.1891$
			$\text{Mean}=-1.65$		${\displaystyle \sum =}14.3728$

Variance computed as above is $14.3728$

Now, for standard deviation of dollar-denominated return at the rate of $\$2.1/\pounds$ , the calculation is:

  $\begin{array}{c}\text{Standard deviation}=\sqrt{\text{Variance}}\\ =\sqrt{14.3728}\\ =3.79\end{array}$