Chapter 21, Problem 18QP

Using the Exact International Fisher Effect [LO2] From our discussion of the Fisher effect in Chapter 7, we know that the actual relationship between a nominal rate, R, a real rate, r, and an inflation rate, h, can be written as:

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

This is the domestic Fisher effect.

a. What is the nonapproximate form of the international Fisher effect?

b. Based on your answer in (a), what is the exact form for UIP? (Hint: Recall the exact form of IRP and use UFR.)

c. What is the exact form for relative PPP? (Hint: Combine your previous two answers.)

d. Recalculate the NPV for the Kihlstrom drill bit project (discussed in Section 21.5) using the exact forms for UIP and the international Fisher effect. Verify that you get precisely the same answer either way.

Summary Introduction

To find: The inappropriate form of international fisher effect.

Introduction:

The theory that states that the original rate of interest is equivalent across the countries is an international fisher effect (IFE).

Explanation of Solution

Given information:

The original relationship between the nominal rate “R”, the real rate “r”, and the inflation rate “h” is written as follows:

$1+r=\frac{\left(1+R\right)}{\left(1+h\right)}$

This is the domestic fisher effect.

Formula of the domestic fisher effect:

$\begin{array}{l}1+R_U{}_S=\left(1+r_U{}_S\right)\left(1+h_U{}_S\right)\\ 1+r_U{}_S=\frac{\left(1+R_U{}_S\right)}{\left(1+h_U{}_S\right)}\end{array}$

Note:

• $R_F{}_C$ is the risk-free interest rate of the foreign country
• $R_U{}_S$  is the nominal risk-free interest rate in Country U
• $h_F{}_C$ is the inflation rate of the foreign country
• $h_U{}_S$  is the inflation rate in Country U
• r is the real rate
• $r_U{}_S$ is the real rate of Country U
• $r_{FC}$ is the real rate of the foreign country

The relationship that must hold for the country is:

$1+r_{FC}=\frac{\left(1+R_F{}_C\right)}{\left(1+h_F{}_C\right)}$

The international fisher effect states that the real rates are equivalent throughout the countries:

$1+r_U{}_S=\frac{\left(1+R_U{}_S\right)}{\left(1+h_U{}_S\right)}=\frac{\left(1+R_F{}_C\right)}{\left(1+h_F{}_C\right)}=1+r_{FC}$

Summary Introduction

To find: The uncovered interest parity

Introduction:

The condition that states the expected change in percentage in the exchange rate equivalent to the variations in the interest rate is the uncovered interest parity.

Explanation of Solution

The appropriate form of the unbiased interest rate parity is as follows:

$E\left[S_t\right]=F_t=S_0\times {\left(\frac{1+R_F{}_C}{\left(1+R_U{}_S\right)}\right)}^t$

Note:

• $S_0$ is the current spot rate of exchange (foreign currency for a dollar)
• $R_F{}_C$ is the risk-free interest rate of the foreign country
• $R_U{}_S$  is the nominal risk-free interest rate in Country U
• t is the period
• $F_t$ is the forward rate of exchange for the settlement at the time

Summary Introduction

To find: The relative purchasing power parity.

Introduction:

The relative purchasing power parity depends on the frictions and inadequacies of the market in order to be hold strictly.

Explanation of Solution

The exact form of the relative purchasing power parity is as follows:

Formula of the relative purchasing power parity:

$\text{E}\left(S_t\right)=S_0\times {\left[1+\left(h_{FC}-h{R}_{US}\right)\right]}^t$

Note:

• $\text{E}\left(S_t\right)$ is the expected rate of exchange
• $S_0$ is the spot exchange rate
• $h_F{}_C$ is the inflation rate of the foreign country
• $R_U{}_S$ is the nominal risk-free interest rate in Country U

Summary Introduction

To find: The net present value of the Project K (discussed in Section 21.5) using the uncovered fisher interest and the international fisher effect.

Introduction:

The present value of the cash outflows and cash inflows is known as the net present value.

Answer to Problem 18QP

The net present value is $316,230.72

Explanation of Solution

Computation of the net present value using the home country approach:

$\begin{array}{c}E\left[S_t\right]=F_t=S_0\times {\left(\frac{1+R_F{}_C}{\left(1+R_U{}_S\right)}\right)}^t\\ =€0.5{\left(\frac{1+0.07}{1+0.05}\right)}^t\\ =€0.5{\left(1.019\right)}^t\end{array}$

The euro cash flows are converted using the above equation and the net present value is determined as follows:

Formula to calculate the net present value:

$\begin{array}{c}\text{NPV}=-\text{Outflow}+\text{Inflow}\\ =-\text{Year 0 cash flow}+\left(\begin{array}{l}\frac{\text{Year 1 cash flow}}{{\left(1+\text{r}\right)}^1}+\frac{\text{Year 2 cash flow}}{{\left(1+\text{r}\right)}^2}+\\ \frac{\text{Year 3 cash flow}}{{\left(1+\text{r}\right)}^3}\end{array}\right)\end{array}$

Computation of the net present value:

$\begin{array}{c}\text{NPV}=-\text{Outflow}+\text{Inflow}\\ =-\text{Year 0 cash flow}+\left(\begin{array}{l}\frac{\text{Year 1 cash flow}}{{\left(1+\text{r}\right)}^1}+\frac{\text{Year 2 cash flow}}{{\left(1+\text{r}\right)}^2}+\\ \frac{\text{Year 3 cash flow}}{{\left(1+\text{r}\right)}^3}\end{array}\right)\\ \text{Net present value}=\left[\begin{array}{l}–\left[\frac{€2M}{€.5}\right]+\frac{\left\{\frac{€.9M}{\left[1.019\left(€.5\right)\right]}\right\}}{1.1}+\frac{\left\{\frac{€.9M}{\left[{1.019}^2\left(€.5\right)\right]}\right\}}{{1.1}^2}\\ +\frac{\left\{\frac{€.9M}{\left[{1.019}^3\left(€.5/\$1\right)\right]}\right\}}{{1.1}^3}\end{array}\right]\\ =\$316,230.72\end{array}$

Hence, the net present value is $316,230.72.

Computation of the net present value using the foreign country approach:

The return in euros:

$\begin{array}{c}{R}_{FC}=1.10\left(\frac{1.07}{1.05}\right)-1\\ =0.121\end{array}$

Computation of the net present value:

$\begin{array}{c}\text{NPV}=-€2M+\left(\frac{€0.9M}{1.121}\right)+\left(\frac{€0.9M}{{\left(1.121\right)}^2}\right)+\left(\frac{€0.9M}{{\left(1.121\right)}^3}\right)\\ =€158,115.36\end{array}$

Computation of the euros to dollars:

$\begin{array}{c}\text{NPV}=\frac{€158,115.36}{€0.5\text{ per \$}}\\ =\$316,230.72\end{array}$

Hence, the net present value is $316,230.72.