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3.5 THE EMPIRICAL BEHAVIOR OF EXCHANGE RATES Changes in Exchange Rates
rate is 1/($0.5839/SFr)≈SFr1.7126/$ and the ending rate is 1/($0.5725/SFr)≈ SFr1.7467/$. The percentage rise in the dollar (in the denominator) is then
[(SFr1.7467/$−SFr1.7126/$)]/(SFr1.7126/$)≈ +0.0199 That is, the dollar appreciated 1.99 percent over the 6-month period.
Percentage changes in direct and indirect FX rates are related, as an appreciation in one currency must be offset by a depreciation in the other. Applying the equality Sdt/f=1/Sf/dt and simplifying the result yields
Sd/f1 /Sd/f0 =1/(Sf/d1 /Sf/d0 )
Alternatively, we can let (Sd/f1 /Sd/f0 )=(1+sd/f), where sd/f is the percentage change in the d-per-f spot rate during the period. This can then be rewritten as
(1+sd/f)=1/(1+sf/d) (3.5) For a+1.99 percent change in the dollar that is offset by a−1.95 percent change in the Swiss franc, the algebra looks like this.
(1+s$/SFr)=(1−0.0195)≈1/(1+0.0199)=1/(1+sSFr/$)
Note that an appreciation in one currency is offset by a depreciation of smaller magnitude in the other currency. This asymmetry is an unfortunate but essential part of the algebra of holding period returns.3
A Reminder: Always Follow Rule #2
The intuition ‘‘buy low and sell high’’ works only for the currency in the denominator of a foreign exchange quote. Thus, there is a simple remedy for keeping things straight — just follow Rule #2. If the currency that you would like to reference is in the numerator, simply move it to the denominator according to Sd/f=1/Sf/d. Following this convention will help you avoid needless confusion. (Actually, this rule is entirely self-serving. If you conscientiously follow Rule #2, your teachers — me included — will be spending less time on the phone answering your questions!)
3.5 THE EMPIRICAL BEHAVIOR OF EXCHANGE RATES
FX changes are close to a random walk.
For daily measurement intervals, nominal spot rate changes are close to a random walk with a nearly equal probability of rising or falling. Because of this behavior, the best guess of tomorrow’s exchange rate is simply today’s exchange rate. The current spot rate outperforms most other exchange rate forecasts for forecasting horizons of up to one year in most currencies. At forecast horizons of longer than one year, forecasts derived from the international parity conditions (see Chapter 4) begin to outperform spot rates as predictors of future exchange rates.
Time-Varying Exchange Rate Volatility
Empirical studies of exchange rates reject the simplest form of the random walk model. In its place, researchers have modeled exchange rates as a process in which the following is true.
FX volatility is predictable.
■ Spot rate changes are approximately normally distributed at each point in time.
■ Exchange rate volatility (or standard deviation) changes over time in a pre- dictable way.
A time series exhibiting this behavior is frequently modeled as a GARCH process.
GARCHstands forgeneralized autoregressive conditional heteroskedasticityand is a statistician’s way of saying ‘‘variance (heteroskedasticity) depends (is conditional) on previous (autoregressive) variances.’’ That is, today’s variance depends on the recent history of exchange rate changes.
A GARCH(1,1) Model The conditional variance of a GARCH(1,1) process at time t is4
σt
2=a0+a1σt−12+b1s
t−1
2 (3.6)
where a0, a1, and b1are constants constrained so that the process is stable, and σt−1
2=the conditional variance estimate from period t−1, st−1
2=the square of the percentage change in the spot rate during period t−1. At each point in time, this GARCH process is normally distributed with conditional (time-varying) varianceσt
2. The GARCH variance is called anautoregressive condi- tional variancebecause it depends on last period’s variance (σt−12) and the square of the most recent change in the spot exchange rate (s
t−1 2).
The GARCH process includes the random walk as a special case in which the parameters a1 and b1 are zero. In this case, variance σ2t is a constant equal to a0. Empirical studies of nominal FX rates have rejected the random walk model in favor of GARCH specifications for yearly, monthly, weekly, daily, and intra-day measurement intervals. The particular form of GARCH is not as important as the recognition that volatility is autoregressive; that is, exchange rate volatility depends on recent market history.
RiskMetrics
’ Conditional Volatility Model The most widely known model for pro- ducing conditional volatility estimates is fromRiskMetrics. TheRiskMetricssystem was created in 1992 by J.P.Morgan (www.jpmorgan.com) to assist clients in assess- ing and managing exposures tofinancial price risks,including currency, interest rate, and commodity price risk.RiskMetricswas spun off from J.P.Morgan in 2008 and then acquired by Morgan Stanley (www.msci.com) in 2010. The system provides users with daily data on more than 300 financial price indices including interest rates, exchange rates, commodity prices, and equity market indices. RiskMetrics uses a restricted form of Equation 3.6,σt
2=aσt−12+(1−a)s
t−1
2 (3.7)
in which the intercept term is omitted, the autoregressive parameter ‘‘a’’ is bounded by 0 < a < 1 to ensure that the process is stable, and the parameter weights sum to one: a+(1−a)=1. For monthly intervals, the standardRiskMetricsmodel assigns a weight of a=0.97 on the most recent conditional variance and a weight of (1−a)=0.03 on the most recent squared spot rate change. For daily intervals, the model assigns weights of a=0.95 and (1−a)=0.05. RiskMetrics’ model is an exponentially weighted moving average in which the impact of past spot rate changes on conditional variance decays at a rate of (1−a)(at).
Figure 3.8 illustrates the RiskMetrics model. The left graph displays monthly spot rates S¥t/$in the floating rate era since 1971. The dollar tended to fall during this period because dollar inflation was higher than yen inflation. The right-hand graph displays absolute changes in the spot rate|s¥t/$|along with theRiskMetricsestimate
0 100 200 300 400
Dec-70 Dec-80 Dec-90 Dec-00 Dec-10 0%
5%
10%
15%
Dec-70 Dec-80 Dec-90 Dec-00 Dec-10
|st¥/$| and RiskMetrics’ conditional volatility st¥/$
FIGURE 3.8 Yen-per-Dollar Spot Rates and Volatilities.
Source:Exchange rates from www.oanda.com.
of conditional volatility as a black line. RiskMetrics conditional volatility rises in periods of high absolute monthly changes in the value of the dollar, such as when the dollar depreciated by 15 percent in October 1998. Conditional volatility falls during less volatile periods, such as in the early years of the 21st century. Conditional volatility estimates are sensitive to market conditions, and that is a useful attribute for a volatility measure that is used to manage exposures to currency risk.