Chapter 19, Problem 13PS

Summary Introduction

(A)

1. According to the given result, he should lend in the UK market.
2. According to the explanation, he should borrow in the US market.

Answer to Problem 13PS

Hence, in total, the cash flow today would be nullified, but would give a profit of 0.0079 at the time when this swap would be made.

Explanation of Solution

Given Information:

Rus = 5%

rUK = 7%

E0 = 2.0 dollars per pound

F0 = $1.97/E (one-year delivery)

Data can be summarized as follows:

Risk free rate of US$\left(r_{US}\right)=5\%$

Risk free rate of UK$\left(r_{UK}\right)=7\%$

Original exchange rate$\left(E_o\right)=\$2/\pounds$

Forward exchange rate$\left(F_o\right)=\$1.97/\pounds$

Determine where to Lend:

Suppose an investor lends $10,000 in the UK market. Thus, according to the exchange rate, he will give £5,000, calculated by multiplying with the current exchange rate, that is,$E_o=\$2/\pounds$ the risk free rate in UK is 7%, so the interest payment he receives in one year would be as follows:

$\begin{array}{l}Interestamount=\pounds 5,000\times 7\%\\ =\pounds 350\end{array}$

Hence, the amount he gets after one year, which includes the total interest and the principle amount, is$\pounds 5,350\left[\pounds 5,000+\pounds 350=\pounds 5,350\right].$

The exchange rate after one year is,$F_o=\$1.97/\pounds$. Hence, to obtain the return in dollar, multiply$\pounds 5,350$ with the exchange rate as follows:

  $\pounds 5,350\times \$1.97=\$10,539.50$

Thus, it can be seen that initially, he had $10,000 to invest in return, and later, he gets total of $10,593.50 as returns. Therefore, his net profit is $539.50.

$\left[\$10,539.50-\$10,000=\$539.50\right]$

If he would have lent the same amount in the US market, he would have got interest according to the rate of 5%, which is the risk free rate in US market as follows:

$\begin{array}{l}Interestamount=\$10,000\times 5\%\\ =\$500\end{array}$

Thus, in total, he would have got $10,500 by the year end, which is less than what he is getting in the UK market, that is, $10,539.50.

Thus, according to the above result, he should lend in the UK market.

Conclusion

Thus, according to the above result, he should lend in the UK market.

Summary Introduction

(B)

Adequate information:

  $\begin{array}{l}rus=5\%\\ rUK=7\%\\ E0=2.0dollarsperpound\\ F0=\$1.97/E\left(one-yeardelivery\right)\end{array}$

To compute:

Where would you borrow?

Introduction:

Answer to Problem 13PS

According to the explanation, he should borrow in the US market.

Explanation of Solution

Determine where to borrow from:

Suppose an investor borrows £10,000 in the US market. According to the exchange rate, he will get $20,000, which is calculated by multiplying with the current exchange rate, that is,

$E_0=\$2/\pounds .$

The risk free rate in US is 5%. Hence, the interest payment for one year is as follows:

$\begin{array}{l}Interestamount=\$20,000\times 5\%\\ =\$1,000\end{array}$

After one year, the total amount including the principle that he gives, in addition to the interest, is $21,000. However, the exchange rate after one year is,$F_0=\$1.97/\pounds$. Hence, in pounds, the amount payable would be $21,000 divided by the exchange rate as follows:

$\frac{\$21,000}{1.97}=\pounds 10,659$

If he would have borrowed the same amount in the UK market, he would have paid interest according to the rate of 7%, that is, the risk free rate in UK market. The interest is calculated as follows:

$\begin{array}{l}Interestamount=\pounds 10,000\times 7\%\\ =\pounds 700\end{array}$

Hence, he would have paid £10,700 in total by the year end, which is more that what is more that what he is paying in the US market.

Thus, according to the explanation, he should borrow in the US market.

Conclusion

Thus, according to the explanation, he should borrow in the US market.

Summary Introduction

(C)

Adequate information:

  $\begin{array}{l}rus=5\%\\ rUK=7\%\\ E0=2.0dollarsperpound\\ F0=\$1.97/E\left(one-yeardelivery\right)\end{array}$

To compute:

How could you arbitrage?

Introduction:

Answer to Problem 13PS

According to the explanation, he should borrow in the US market.

Explanation of Solution

Calculate the forward rate according to the interest rate parity relationship.

Interest rate parity relationship: It is the relation between the forward exchange rate and spot exchange rate that do not allow arbitrage to take place. Forward price can be calculated using the formula as follows:

$\frac{F_0}{E_0}=\frac{1+r_f\left(US\right)}{1+r_f\left(UK\right)}$

Here,

Original exchange rate =$\left(E_o\right)$

Forward rate =$\left(F_o\right)$

Risk free rate of US=$\left(r_{US}\right)$

Risk free rate of UK=$\left(r_{UK}\right)$

Calculate forward rate as follows:

$\begin{array}{l}\frac{F_0}{2}=\frac{1+0.05}{1+0.07}\\ F_0=2\times \frac{1.05}{1.07}\\ F_0=1.9626\end{array}$

The following steps can be taken to take advantage of the arbitrage position:

Action	Initial cash flow	Cash flow at Time T
Enter a contract to sell   $\pounds 1.07$ at a (future price) of$F_0=\$1.97$	0.0	$1.07\times \left(1.97-E_1\right)$
Borrow $2.00 in the US	2.00	$-2.00\times 1.05$
Convert the borrowed dollars to pounds, and lend the UK at a 7% interest rate	-2.00	$1.07\times E_1$
Total	0	0.0079

Enter into a contract to sell £1.07$\left[1+r_f\left(UK\right)\right]$ at a future price of 1.96 as calculated. At this time, the initial cash flow would be nil. At the future date, the cash flow would be

$1.07\times \left(1.97-E_1\right)$

Here,$E_1$ is the forward exchange rate.

The borrowed dollars are converted to pounds and given in the UK market at 7% interest rate thus, initial cash flow here would be -2.00, but in future, the cash flow would become$1.07\times E_1$ .

Hence, in total, the cash flow today would be nullified, but would give a profit of 0.0079 at the time when this swap would be made.

Conclusion

Hence, in total, the cash flow today would be nullified, but would give a profit of 0.0079 at the time when this swap would be made.