Question

Foreign Exchange Rates Volatility
You are kindly requested to collect the foreign exchange rates for the EUR, GBP, CAD, AUD, NZD and
JPY, on monthly average bases, for the time period: 1/1/2016 – 31/12/2019.
Question:
Analyze the volatility of the six currencies over the time horizon identified and determine which
currency has witnessed the highest volatility.

Answer 1

Exchange rates are puzzling in many aspects. First, exchange rates are disconnected from eco-

nomic fundamentals, especially relative consumption growth rate, which is in sharp contrast to

implications of most international macro-finance models (Backus and Smith, 1993). Second, high-

interest-rate currencies do not depreciate as the uncovered interest parity suggests. On the contrary,

they often appreciate in subsequent periods (Hansen and Hodrick, 1980; Fama, 1984), known as

the “forward premium puzzle”. As a result, excess returns of currency investment can be predicted

by interest rate differentials. Third, it is hard to obtain exchange rate volatility close to data in stan-

dard international macro-finance models (Chari et al., 2002; Brandt et al., 2006). Lastly, covered

interest rate parity, a classic no-arbitrage condition in the currency market, is violated for a decade

after the global financial crisis (Du et al., 2018). In this paper, we attempt to resolve these puzzles

by focusing on the role of leveraged financial intermediaries in exchange rate determination.

Financial intermediaries are major participants in the foreign exchange (FX) market. More than

85% of turnovers in the FX market have financial institutions involved, according to the recent

BIS triennial surveys. Moreover, non-dealer financial institutions account for more than half of

turnovers. With respect to aggregate portfolio holding, the BIS reporting banks hold about half

of the countries’ total external claims and more than 40% of total external liabilities in 21 OECD

countries1

.

The recent intermediary asset pricing literature has shown the importance of intermediaries on a

broad class of asset returns (for example, Brunnermeier and Pedersen, 2009; Adrian et al., 2014; He

and Krishnamurthy, 2013; He et al., 2017). It is natural to study exchange rates through the lens of

an intermediary-based model. An essential feature of financial intermediaries is the constraint on

taking leverage. The financial constraint is tightly linked to the volatility in the economy because

of the value-at-risk (VaR) rule adopted by major financial institutions (Adrian and Shin, 2014). The

VaR rule states that the size of the balance sheet shrinks with the rise of volatility in the economy.

The Model

There are two ex-ante identical countries in the economy, home and foreign, each populated with

a unit measure of households and endowed with a Lucas tree. The home tree delivers good X, and

the foreign tree delivers good Y, both of which are tradable. In both countries, each household

owns an intermediary and sends out a manager to operate it. Households make deposits in local

intermediaries. Intermediaries combine deposits and their own net worth to invest in risky assets.

There are two available risky assets, a claim to the local Lucas tree and an international bond.

Intermediation is imperfect, in the form that the intermediaries in each country face a financial

constraint, whose tightness is determined by the volatility in the local economy. Every period,

a fixed fraction of intermediaries exit the market and rebate back their net worth to their owners,

Households in the home and foreign countries are endowed with a Lucas tree with different goods,

X for home and Y for foreign. They follow cointegrated processes:

logXt+1 −logXt = µ +τ(logYt −logXt) +σX,tεX,t+1

logYt+1 −logY = µ −τ(logYt −logXt) +σY,tεY,t+1 (1)

Volatilities are stochastic, following:

log(σX,t+1) = (1−ρσ )logσ¯ +ρσ log(σX,t) +σσ ηX,t+1

log(σY,t+1) = (1−ρσ )logσ¯ +ρσ log(σY,t) +σσ ηY,t+1 (2)

The four shocks follow the standard normal distribution. The two goods aggregate into a consump-

tion basket. The aggregator takes the form of constant elasticity of substitution:

C = [(1−α)C

σ−1

σ

X +αC

σ−1

σ

Y

]

σ

σ−1 ,C

∗ = [(1−α)C

∗

σ−1

σ

Y +αC

∗

σ−1

σ

X

]

σ

σ−1

CX,CY are home households’ consumption of X and Y, while variables with an asterisk refer to the

foreign counterpart. Households in the home and foreign countries put different weights on X and

Y with consumption home bias, i.e., α <

1

2

. σ is the price elasticity of substitution between X and