PTWhen the exchange rate takes a plunge
Anyone who exports, imports or invests abroad knows the dollar sometimes jumps in a single day. The thesis studies expected shortfall (ES), the measure of how large the loss tends to be on the worst days, and applies it to four currencies against the dollar. Across nine ways of computing this limit, only the Brazilian real crossed it in the test period. Explained in 12 screens.
Our illustration: how many days the exchange rate moved by how much. ES looks only at the red tail.
VaR says where the danger starts; ES says how deep it goes
Value at Risk (VaR) answers: "on 99 out of 100 days, what is the most I can lose?". It says nothing about the remaining day. Expected shortfall (ES) answers that question: it is the average of the losses beyond VaR.
Two portfolios can have the same VaR and very different tails. Switch the portfolio and the confidence level.
Our illustration, with 1,000 made-up days. The fat-tailed portfolio was scaled to have the same VaR as the thin-tailed one; the first column gathers everything below −6%. The definition of ES (average of the losses beyond VaR) is the thesis's, section 3.4.
Regulators switched yardsticks after the 2008 crisis
For almost two decades, VaR was the standard measure of market risk in banks. The subprime crisis showed what it missed: the losses beyond the limit. The third Basel accord adopted ES for market risk.
Sections 2.2 and 3.3 of the thesis, which cite Artzner and colleagues (1999), Yamai and Yoshiba (2002, 2004) and Koch-Medina and Munari (2015). The numbers in the diversification example are our illustration.
In the studies, ES appears as a complement to VaR
The first article of the thesis searched three academic databases for everything published on ES and risk management up to April 2021, and read in depth the 20 articles that dealt with the topic.
The most cited, by Yamai and Yoshiba (2004), with 142 citations, compares VaR and ES. It became the template for the second article of the thesis.
Systematic review of article 1 (section 2.4.2). Each article can have up to three codings per theme; the percentages are of the codings, as in the thesis.
Four currencies, nine ways to compute the limit
The author computed the ES of the daily change of four exchange rates against the dollar with data from 2000 to 2021. Then he looked at what happened from August 2021 to December 2023: on how many days did the exchange rate cross the limit?
Because ES is about rare days, there are many ways to estimate it. The thesis used nine, in five families. Tap a family.
Data from the Central Bank of Brazil; computations in Excel and Stata, following Mehlitz and Auer (2020). The real represents an emerging country; the euro, yen and yuan, economies with weight on the international stage (section 3.2).
Each method gives a different limit
Each column is the one-day ES, as a % change of the exchange rate: the average loss expected on the worst days, according to each method. In all four currencies, the historical methods give the lowest limits.
Table 12 of the thesis, converted to %. The normal limit for the real (21.7% a day) sits far above the other seven methods (2.4% to 3.3%); the thesis does not comment on this value. The average of the limits (MV) is the simple mean of the other eight, which is why it rises to 5.1%.
Only the real crossed the limit
From August 2021 to December 2023, on how many days did the exchange-rate change cross the ES limit? Before you look, guess: which currency crossed it most often?
Table 13 of the thesis. Tap a stack to see the number. The euro, yen and yuan did not cross any limit.
Lower limit, more alarms
For the real, the nine limits ranged from 2.4% to 21.7% a day. The lower the limit, the more days the exchange rate crossed it. Drag to switch limits.
Tables 12 and 13 of the thesis; the ordering of the limits is ours. Our reading: the thesis reads more days above the limit as "greater sensitivity" of the historical methods and concludes they are preferable. It does not compare this number with what each limit should produce by chance, so one cannot tell which of them was right.
Hedging currency risk means choosing how much to sell in the futures market
A company that will receive dollars loses if the dollar falls. It can sell dollar futures contracts: if the currency falls, the contract gains. The question is how much to sell, the so-called hedge ratio.
The third article proposes choosing this ratio so that the ES of the hedged position is as small as possible, with a model that forecasts the next day's volatility (the DCC-GJR, from Ubukata, 2018). Try the idea.
Our illustration: US$ 100k receivable, the dollar at R$ 5.00, and a futures price that moves together with the spot price. In real life the two do not move exactly together, so the best ratio may not be 100%: that is what the thesis's model tries to compute. The thesis describes the method (section 4.3) but does not report the ratios it found.
Bad news did not weigh more than good news
The GJR model asks whether a down day stirs the exchange rate more in the following days than an up day of the same size. With daily data from December 2017 to April 2024 (1,608 days), the answer was no, in all six series: pound, euro and yen, spot and futures.
What worked was the "memory" part: a shock to the exchange rate keeps stirring the following days and fades only gradually. Pick a series.
Coefficients from Tables 22 to 27 of the thesis (the "p" shows whether the effect is clear: below 0.05). The curve and the days until it halves are our approximate calculation, using the stability sum the thesis uses (alpha + beta + gamma).
The sophisticated model forecast a straight line
The models were stable, but the DCC-GJR forecast became a straight line: it did not follow the ups and downs of the exchange rate. The author switched tools and used a simpler model, ARIMA, to forecast the ratio between the futures price and the spot price of each currency.
What the research changes and what it does not say
For those with revenue or debt in dollars
Look beyond VaR: knowing how large the loss tends to be on the worst days helps decide how much to hedge. In the period tested, the real was the only one of the four currencies that crossed the limits.
Articles 2 and 3For risk managers
ES changes a lot depending on the method. It is worth computing it in more than one way and checking how often each limit is crossed, as the thesis did, instead of trusting a single number.
Tables 12 and 13; our readingFor researchers
Test other models to forecast the tail of exchange rates (the multivariate GARCH did not) and measure the effectiveness of the hedge the method proposes.
Conclusion, section 5What this research does not say
- The hedge was not simulated. Article 3 describes how to find the hedge ratio and measure its effectiveness, but reports neither ratios nor hedging results. · our note
- More alarms do not mean a better limit. The thesis does not compare the days above the limit with what each limit should produce by chance. · our note
- Unexplained values. The normal ES of the real (21.7% a day) and the yen's ARIMA error (0.0416, larger than the real's) do not fit the rest of the tables and charts; the thesis does not comment on them. · our note
- Short test period. About two years and four months, with four currencies; other currencies and other crises may paint a different picture. · our note
- Review limited to the databases. Article 1 depends on the criteria and the databases chosen (Scopus, Web of Science and Science Direct). · according to the author
- Not investment advice. The research measures risk; it does not recommend trades. · our note
Credits
FUKUI, Marcelo. O expected shortfall como mensurador de riscos em uma estratégia de dynamic hedging em casos de exposição cambial. Doctoral thesis (Business Administration), Universidade Presbiteriana Mackenzie, São Paulo, 2024. Advisor: Prof. Dr. Leonardo Fernando Cruz Basso. Co-advisor: Prof. Dr. Benilde Maria do Nascimento Oliveira.
Data from the Central Bank of Brazil, Bloomberg and Investing.com; analyses in Excel, Stata and OxMetrics. This version simplifies the work for a general audience; the formulas and full tables are in the original text.