Chapter 23, Problem 21PS

Summary Introduction

To compute: The swap rate on an agreement to exchange currency over a 3-year period supposing the U.S. yield curve is flat at 3% and the current exchange rate is $1.20 per euro.

Introduction:

Swap rate: Swap rate refers to that rate of interest which is applicable during the exchange of currency between two different countries. By swapping, the companies are benefitted by hedging against interest rate exposure. This is possible only when the uncertainty of cash flows is reduced.

Answer to Problem 21PS

The swap rate per year would be $1223100.

Explanation of Solution

U.S. yield curve = 4%

Euro yield curve =3%

Current exchange rate= $1.20 per euro

Amount to be exchanged= $1 million

We are asked to calculate the swap rate to exchange 1 million euros for a given number of dollars in each year.

The formula used is as follows-

  $\frac{F_1}{(1+y_1)}+\frac{F_2}{{(1+y_2)}^2}+\frac{F_3}{{(1+y_3)}^3}=\frac{F*}{(1+y_1)}+\frac{F*}{{(1+y_2)}^2}+\frac{F*}{{(1+y_3)}^3}$

Where

F₁, F₂, F₃ = Price to be paid the period 1,2 and 3.

F*= Constant rate to be paid in 3 years

y₁,y₂,and y₃ = Yields

The formula to calculate forward price is as follows:

  $F_0=E_0{\left(\frac{1+r_{US}}{1+r_{EU}}\right)}^T$

Let us assign values to the variables in the formula.

E₀= $1.20 per euro

U.S. Yield r_US=4% (when we convert it by dividing it by 100 we get 0.04)

Euro yield r_EU=3% (when we convert it by dividing it by 100 we get 0.03)

Now, we have to consider each contract as a separate contract and then calculate the forward price for each period.

Calculation of forward price for 1^styear:

Let F₁ be the forward price of first year.

  $\begin{array}{l}{F}_1=\$1.20{\left(\frac{1+0.04}{1+0.03}\right)}^1\\ \Rightarrow \$1.20\left(\frac{1.04}{1.03}\right)\\ \Rightarrow \$1.20\times 1.009=\$1.2108\end{array}$

Therefore the forward price is $1.2108

  $\begin{array}{l}Dollarstobedelivered=F_1\times \$1million\\ =\$1.2108\times \$1million=\$1.2108million\end{array}$

Therefore, the dollars to be delivered in first year will be $1.2108 million.

Calculation of forward price for 2nd year:

Let F₂ be the forward price of second year.

  $\begin{array}{l}{F}_2=\$1.20{\left(\frac{1+0.04}{1+0.03}\right)}^2\\ \Rightarrow \$1.20{\left(\frac{1.04}{1.03}\right)}^2\\ \Rightarrow \$1.20\left(\frac{1.0816}{1.0609}\right)\\ \Rightarrow \$1.20\times 1.0195=\$1.2234\end{array}$

Therefore the forward price for second year is $1.2234.

  $\begin{array}{l}Dollarstobedelivered=F_2\times \$1million\\ =\$1.2234\times \$1million=\$1.2234million\end{array}$

Therefore, the dollars to be delivered in second year will be $1.2234 million.

Calculation of forward price for 3rd year:

Let F₃ be the forward price of third year.

  $\begin{array}{l}{F}_3=\$1.20{\left(\frac{1+0.04}{1+0.03}\right)}^3\\ \Rightarrow \$1.20{\left(\frac{1.04}{1.03}\right)}^3\\ \Rightarrow \$1.20\left(\frac{1.1248}{1.0927}\right)\\ \Rightarrow \$1.20\times 1.0293=\$1.2351\end{array}$

Therefore the forward price is $1.2351.

  $\begin{array}{l}Dollarstobedelivered=F_3\times \$1million\\ =\$1.2351\times \$1million=\$1.2351million\end{array}$

Therefore, the dollars to be delivered in third year will be $1.2351 million.

Let us now calculate the swap rate. But before that we have to calculate the constant number of dollars F* to be delivered each year.

  $\begin{array}{l}\frac{F*}{(1+y_1)}+\frac{F*}{{(1+y_2)}^2}+\frac{F*}{{(1+y_3)}^3}=\frac{F_1}{(1+y_1)}+\frac{F_2}{{(1+y_2)}^2}+\frac{F_3}{{(1+y_3)}^3}\\ \frac{F*}{(1.04)}+\frac{F*}{{(1.04)}^2}+\frac{F*}{{(1.04)}^3}=\frac{F_1}{(1.04)}+\frac{F_2}{{(1.04)}^2}+\frac{F_3}{{(1.04)}^3}\\ \frac{F*}{(1.04)}+\frac{F*}{{(1.04)}^2}+\frac{F*}{{(1.04)}^3}=\frac{1.2108}{(1.04)}+\frac{1.2234}{{(1.04)}^2}+\frac{1.2351}{{(1.04)}^3}\\ \Rightarrow \frac{1.2108+1.2234+1.2351}{3}\\ \Rightarrow \frac{3.6693}{3}=\$1.2231\end{array}$

  $\begin{array}{l}Dollarstobedelivered=F*\times \$1million\\ =\$1.2231\times \$1million=\$1223100\end{array}$

Therefore, the swap rate per year would be $1223100.