Chapter 31, Problem 18QP

Using the Exact International Fisher Effect From our discussion of the Fisher effect in Chapter 8, we know that the actual relationship between a nominal rate, R, a real rate, r, and an inflation rate, h, can be written as follows:

1 + r =(1 + R)/(1 + h)

This is the domestic Fisher effect.

1. a. What is the nonapproximate form of the international Fisher effect?
2. b. Based on your answer in (a), what is the exact form for UIP? (Hint: Recall the exact form of IRP and use UFR.)
3. c. What is the exact form for relative PPP? (Him: Combine your previous two answers.)
4. d. Recalculate the NPV for the Kihlstrom drill bit project (discussed in Section 31.5) using the exact forms for the UIP and the international Fisher effect. Verify that you get precisely the same answer either way.

Summary Introduction

To explain: Non approximate form of international fisher effect.

Exchange Rate:

Exchange rate defines the value of one currency against the other currency. Exchange rate has two main components one is the currency used to compare that is domestic currency and other is the currency used to compare against that is foreign currency.

Explanation of Solution

The domestic fisher effect is:

$\begin{array}{c}1+R_{US}=\left(1+r_{US}\right)\left(1+h_{US}\right)\\ 1+r_{US}=\frac{\left(1+R_{US}\right)}{\left(1+h_{US}\right)}\end{array}$

Here,

• $R_{US}$ is country U interest rate.
• $h_{US}$ is country U inflation rate.

This relationship must hold for any country that is.

$1+r_{FC}=\frac{\left(1+R_{FC}\right)}{\left(1+h_{FC}\right)}$

As per the international Fisher effect the real rates are equal across countries so,

$\begin{array}{c}1+r_{US}=\frac{\left(1+R_{US}\right)}{\left(1+h_{US}\right)}\\ =\frac{\left(1+R_{FC}\right)}{\left(1+h_{FC}\right)}\\ =1+r_{FC}\end{array}$

Conclusion

So, non approximate form of international fisher effect is equals to $1+r_{FC}$.

Summary Introduction

To explain: Exact form of unbiased interest rate parity.

Explanation of Solution

The exact form of unbiased parity rate is,

$\begin{array}{c}E\left[S_t\right]=F_t\\ =S_0{\left[\frac{\left(1+R_{FC}\right)}{1+R_{US}}\right]}^t\end{array}$

Here,

• $E\left[S_t\right]$ is parity rate.
• $S_0$ is the spot rate.
• $R_{FC}$ is foreign currency interest rate.
• $R_{US}$ is country U interest rate.
• t is the time period.

Conclusion

So the exact form of unbiased parity rate is $S_0{\left[\frac{\left(1+R_{FC}\right)}{1+R_{US}}\right]}^t$.

Summary Introduction

To explain: Exact form of relative purchasing power parity.

Explanation of Solution

Relative purchasing power parity,

$E\left[S_t\right]=S_0{\left[\frac{\left(1+h_{FC}\right)}{\left(1+h_{US}\right)}\right]}^t$

Here,

• $E\left[S_t\right]$ is purchasing power parity.
• $S_0$ is the spot rate.
• $h_{FC}$ is foreign currency inflation rate.
• $h_{US}$ is country U inflation rate.
• t is the time period.

Conclusion

So exact form of purchasing power parity is $S_0\left[\frac{\left(1+h_{FC}\right)}{\left(1+h_{US}\right)}\right]$

Summary Introduction

To calculate: Net present value by country U using the exact form of UIP and the international fisher effect.

Explanation of Solution

Calculation of expected currency spot rate:

Given,

Spot rate is €0.5.

Foreign currency inflation rate is 7%.

Inflation in rate in country U is 5%.

Formula to calculate expected currency spot rate at time,

$E\left[S_t\right]=S_0{\left[\frac{\left(1+h_{FC}\right)}{\left(1+h_{US}\right)}\right]}^t$

Here,

• $E\left[S_t\right]$ is purchasing power parity
• $S_0$ is the spot rate
• $h_{FC}$ is foreign currency inflation rate
• $h_{US}$ is country U inflation rate.
• t is the time period

Substitute €0.5 for Spot rate, 7%. for Foreign currency inflation rate and 5% for Inflation in rate in country U.

$\begin{array}{c}E\left[S_t\right]=\left(€0.5\right){\left[\frac{1+0.7}{1+0.5}\right]}^t\\ =\left(0.5\right){\left(1.019\right)}^t\\ \end{array}$

Calculation of Net present value by converting them in E currency cash flow:

Given,

Initial cash flow is €2 million.

Cash flow for 3 years after that is €0.9 million.

Formula to calculate NPV,

$NPV={\displaystyle \sum _{im}^n\frac{C{F}_i}{{\left(1+r\right)}^i}-C_0}$

Here,

• NPV is net present value.
• i is number of years.
• $C_0$  is initial cash flow.
• $C{F}_i$  is cash flow in particular year.
• r is discount rate.

Substitute €2 million for initial investment, €0.9 million Cash flow for 3 years and discount rate is $\frac{1.019}{1.1}$ for 1^st year $\frac{{1.019}^2}{{1.1}^2}$ for 2^nd year and $\frac{{1.019}^3}{{1.1}^3}$ for 3^rd year.

$\begin{array}{c}NPV=\left[\begin{array}{l}\left\{\frac{€900,000}{\frac{\left[1.019\left(€0.5\right)\right]}{1.1}}\right\}+\left\{\frac{€900,000}{\frac{\left[{1.019}^2\left(€0.5\right)\right]}{{1.1}^2}}\right\}+\\ \left\{\frac{€900,000}{\frac{\left[{1.019}^3\left(€0.5\right)\right]}{{1.1}^3}}\right\}-\left[\frac{€2,000,000}{\left(€0.5\right)}\right]\end{array}\right]\\ =\left[\left\{\frac{\$900,000}{0.56045}\right\}+\left\{\frac{\$900,000}{0.6282}\right\}+\left\{\frac{\$900,000}{0.7041}\right\}-\left\{\$4,000,000\right\}\right]\\ =\left[\$1,605,852.44+\$1,432,645.588+\$1,278,120.785-\$4,000,000\right]\\ =\$316,618.813\end{array}$

So value of NPV in dollars is $316,618.813.

Calculation of return in e currency,

$\begin{array}{c}{R}_{FC}=1.10\left(\frac{1.07}{1.05}\right)-1\\ =0.1209\end{array}$

Calculation of NPV in e currency,

$\begin{array}{c}NPV=\left[\begin{array}{l}\frac{\left(€900,000\right)}{1.1209}+\frac{\left(€900,000\right)}{{1.1209}^2}+\\ \frac{\left(€900,000\right)}{{1.1209}^3}-€2\text{,000,000}\end{array}\right]\\ =\left[\begin{array}{l}€802,926.22+€716,322.7942+\\ €639,060.3927-€2\text{,000,000}\end{array}\right]\\ =€158,309.4069\end{array}$

So, Net present value at in E currency is €158,309.4069.

Convert e currency to dollars at the current exchange rate,

$\begin{array}{c}NPV\left(\$\right)=\frac{€158,309.4069}{\frac{\left(€0.5\right)}{\$1}}\\ =\frac{€158,309.4069}{\$0.5}\\ =\$316,618.813\end{array}$

So value of NPV in dollars is $316,618.813.

Conclusion

So, the net present value in dollars is $316,618.813 which is same as computed above.