Assessing Exchange Rate Assessing Exchange Rate Risk: Part IRisk: Part I

Forecasting Exchange Forecasting Exchange RatesRates

Three econometricians went turkey hunting. The first took a shot and missed to the left. The second missed to the right. The third shouted “We got it!!”

“There. Look at this. See? See? I'm right again. Nobody could've predicted that Dr. Grant would suddenly, suddenly jump out

of a moving vehicle.”

See, here I'm now by myself, uh, er, talking to myself. That's chaos theory.

Econometricians believe that there is “true” relationship between all things on our planet. If we run enough tests, we can eventually figure is out!

More importantly, this relationship is stable and can be used for prediction

Probability

EventMean

Probability distributions identify the chance of each possible event occurring

1 SD

2 SD

3 SD

-1 SD

-2 SD

-3 SD

65%

95%

99%

Mean = 1

Variance = 4

Std. Dev. = 2

Probability distributions are scaleable

22

2

σ,kkNy

kxy

μ,σNx

3 X =

Mean = 3

Variance = 36 (3*3*4)

Std. Dev. = 6

Mean = 1

Variance = 1

Std. Dev. = 1

Probability distributions are additive

xyyxyx

yy

xx

σ,σNyx

,σNy

,σμNx

cov222

2

2

+Mean = 2

Variance = 9

Std. Dev. = 3

Cov = 2

=Mean = 3

Variance = 14 (1 + 9 + 2*2)

Std. Dev. = 3.7

Mean = 6

Variance = 4

Std. Dev. = 2

Mean = $ 32,000

Variance = 16,000,000

Std. Dev. = $ 4,000

Suppose we know that your salary is based on your shoe size:

Salary = $20,000 +$2,000 (Shoe Size)

Shoe Size Salary

We could also use this to forecast:

Salary = $20,000 +$2,000 (Shoe Size)

If Bigfoot had a job…how much would he make?

Size 50!!!

Salary = $20,000 +$2,000 (50) = $120,000

Searching for the truth….

You believe that there is a relationship between shoe size and salary, but you don’t know what it is….

1. Collect data on salaries and shoe sizes

2. Estimate the relationship between them

Note that while the true distribution of shoe size is N(6,2), our collected sample will not be N(6,2). This sampling error will create errors in our estimates!!

0

10000

20000

30000

40000

50000

60000

70000

0 2 4 6 8 10 12 14

Shoe Size

Sala

ry

Salary = a +b * (Shoe Size) + error

a

20,σNerror

Slope = b

We want to choose ‘a’ and ‘b’ to minimize the error!

Regression Results

Variable Coefficients Standard Error t Stat

Intercept 45415.65 1650.76 27.51

Shoe 1014.75 257.21 3.94

Salary = $45,415 + $1,014 * (Shoe Size) + error

We have our estimate of “the truth”

Intercept (a)

Mean = $45,415

Std. Dev. = $1,650

Shoe (b)

Mean = $1,014

Std. Dev. = $257

T-Stats bigger than 2 are considered statistically significant!

Regression Results

Variable P-value Lower 95% Upper 95%

Intercept 5.2E-102 42172.33 48658.97

Shoe 9.12E-05 509.40 1520.10

Intercept (a) Shoe (b)

$42,172 - $48,658 $509 - $1,520

The P-value tells you the probability that the coefficient is equal to zero

Regression Statistics

Multiple R 0.17

Standard Error 11673.01

Observations 500

Error Term

Mean = 0

Std, Dev = $11,673

Percentage of income variance explained by shoe size

Regression Results

Variable Coefficients Standard Error t Stat

Intercept 20,000 0 Infinite

Shoe 2,000 0 Infinite

Regression Results

Variable P-value Lower 95% Upper 95%

Intercept 0 20,000 20,000

Shoe 0 2,000 2,000

Regression Statistics

Multiple R 0

Standard Error 0

Observations 500

If we ever found “the truth”, it would look something like this!

Using regressions to forecast….

Salary = $45,415 + $1,014 * (Shoe Size) + error

50

Mean = $45,415

Std. Dev. = $1,650

Mean = $1,014

Std. Dev. = $ 257

Mean = $0

Std. Dev. = $11,673

Salary Forecast

Mean = $96,115

Std. Dev. = $17,438

438,17$)673,11()257()50()650,1( 2222 StdDev

Given his shoe size, you are 95% sure Bigfoot will earn between $61,239 and $130,991

We’ve looked at several currency pricing models that have potential for being “the truth”

Uncovered Interest Parity

% Change in e = Inflation – Inflation*

Purchasing Power Parity

% Change in e = Interest Rate – Interest Rate*Covered Interest Parity

% Change in e = Forward Premium/Discount

Currency Fundamentals

% Change in e = (%M - %M*) + (%Y - %Y*) + (i - i*)

Technical Analysis

% Change in e = Past Behavior of exchange rate

Any combination of these could be “the truth”!!

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10.0 -5.0 0.0 5.0 10.0 15.0

Inflation Differential

% C

han

ge in

Exch

an

ge R

ate

tttt bae *% Note: PPP implies that a = 0 and b = 1

PPP and the Swiss Franc

Regression Results

Variable Coefficients Standard Error t Stat

Intercept .027 .231 .12

Inflation 1.40 .742 1.89

Regression Results

Variable P-value Lower 95% Upper 95%

Intercept .910 -.49 .43

Inflation .06 -.065 2.86

Regression Statistics

R Squared .02

Standard Error 2.69

Observations 155

For every 1% increase in US inflation over Swiss inflation, the dollar depreciates by 1.40%

-10

-8

-6

-4

-2

0

2

4

6

8

10

-0.7

-0.1

-0.6

-0.4

-0.2 -0

-0.9

-0.2

0.02

-0.2 -0

0.01

-0.2

-0.3

-0.5

-0.4

0.03

0.34

0.48

0.83

0.13

0.41 0.6

0.1

0.4

-0.5

Predicted

Actual

Obviously, we have not explained very much of the volatility in the CHF/USD exchange rate

tttt iibae *% Note: UIP implies that a = 0 and b = 1

UIP and the Swiss Franc

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-8

-6

-4

-2

0

2

4

6

8

10

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

Interest Differential

% C

han

ge in

e

Regression Results

Variable Coefficients Standard Error t Stat

Intercept .55 .31 1.77

Interest Rate -2.87 1.53 -1.87

Regression Results

Variable P-value Lower 95% Upper 95%

Intercept .07 -.06 1.18

Interest Rate .06 -5.89 .15

Regression Statistics

R Squared .02

Standard Error 2.69

Observations 155

For every 1% increase in US interest rates over Swiss interest rates, the dollar appreciates by 2.87%

We still have not explained very much of the volatility in the CHF/USD exchange rate

-10

-8

-6

-4

-2

0

2

4

6

8

10

-0.1

-0.2 -0

0.03

0.13

0.24

0.21

0.29

0.33

0.39

0.32

0.28

0.28

0.26

0.22

0.19

0.08

0.04

0.06

0.07

0.07

0.11

0.16

Exchange Rate

Predicted Exchange Rate

Using regressions to forecast….

= .55 – 2.87 * (i-i*) + error

(3 – 1.5) = 1.5

Mean = .55

Std. Dev. = .31

Mean = -2.87

Std. Dev. = 1.53

Mean = $0

Std. Dev. = 2.69

Salary Forecast

Mean = -3.755%

Std. Dev. = 3.58%

%58.3)69.2()53.1()5.1()31(. 2222 StdDev

Given current interest rates, you are 95% sure that the % change in the exchange rate will be between -10.91% and 3.40%!!

% Change in e

Technical Analysis Uses prior movements in the exchange rate to predict the future

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-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

%Change (t-1)

% C

han

ge (t)

ttt ebae 1%%

Regression Results

Variable Coefficients Standard Error t Stat

Intercept .12 .21 .57

Prior Change .29 .07 3.86

Regression Results

Variable P-value Lower 95% Upper 95%

Intercept .56 -.29 .53

Prior Change .0001 .14 .45

Regression Statistics

R Squared .09

Standard Error 2.59

Observations 154

A 1% depreciation of the dollar is typically followed by a .29% depreciation