1 Learning Objectives

By the end of this chapter, the student should be able to:

  1. explain why extreme financial risk measurement became important in modern finance;
  2. define risk as a random variable representing possible profits and losses;
  3. explain exchange rate risk and distinguish transaction, translation, and economic risk;
  4. explain why Value-at-Risk and Expected Shortfall are important risk measures;
  5. describe the stylized facts commonly observed in financial return series;
  6. explain why normality-based models may fail to capture extreme events;
  7. explain the basic motivation for using Extreme Value Theory in financial risk measurement;
  8. compute simple returns, log returns, losses, empirical VaR, and empirical Expected Shortfall using R.

2 Introduction

The last several decades have been characterized by significant instability in financial markets. Financial institutions, investors, regulators, and policymakers have repeatedly faced events that were rare in probability but large in impact. Examples include the stock market crash of 1987, the Asian financial crisis of 1997–1998, and the global financial crisis of 2007–2008. Such events motivated the development of tools that can assess the probability and severity of rare financial extreme events.

The central problem in this course is therefore the measurement of financial risk when losses are not ordinary, small, or normally behaved. A financial institution is not only interested in what happens on an average day. It is also interested in what can happen during unusual market conditions, when losses are large enough to threaten capital, profitability, liquidity, or even survival.

In this course, risk will be treated as a random variable that transforms unforeseen future states of the world into numerical values representing profits and losses. If the future state is favourable, the outcome may be a profit. If the future state is unfavourable, the outcome may be a loss. Since the future is uncertain, the profit or loss is random.

Let \(X\) represent the profit-and-loss position of a firm or portfolio. If \(X>0\), the position represents a profit. If \(X<0\), it represents a loss. In many parts of extreme risk measurement, however, it is more convenient to define losses as positive values. If \(R\) is a return, then a loss variable may be written as

\[ L=-R. \]

Large financial losses then appear as large positive values of \(L\). This convention is natural for Extreme Value Theory because EVT is mainly concerned with large observations in the tail of a distribution.

3 Exchange Rate Risk

Exchange rates play a crucial role in a country’s level of trade. Exchange rate fluctuations affect importers, exporters, multinational firms, banks, investors, governments, and households. Because of this, exchange rates are among the most watched and analysed economic measures.

Exchange rate risk is the financial risk that arises from exposure to unanticipated changes in the exchange rate between two currencies. It is related to the effect of unexpected exchange rate changes on the value of a firm. Such changes can influence cash flows, profitability, share prices, competitiveness, and the value of foreign-currency assets and liabilities.

A firm that is exposed to exchange rate movements must first identify the specific type of currency risk it faces. The three main types of exchange rate risk are:

  1. Transaction risk, which is associated with the effect of exchange rate changes on future cash flows from already agreed transactions.
  2. Translation risk, which is associated with the effect of exchange rate changes on the accounting value of foreign-currency assets, liabilities, income, or expenses when financial statements are consolidated.
  3. Economic risk, which reflects the effect of exchange rate movements on the present value of future operating cash flows and the long-term value of the firm.

Identifying the type of exchange rate exposure is important because measurement and management depend on the nature of the risk. A short-term foreign-currency invoice, a foreign subsidiary, and a long-term change in international competitiveness do not create the same type of risk.

4 Financial Returns and Losses

Most financial risk measurement begins with returns. Suppose the price of an asset at time \(t\) is \(P_t\), and the price at the previous time is \(P_{t-1}\). The simple return is

\[ R_t=\frac{P_t-P_{t-1}}{P_{t-1}}=\frac{P_t}{P_{t-1}}-1. \]

The log return is

\[ r_t=\log\left(\frac{P_t}{P_{t-1}}\right). \]

Log returns are frequently used in financial modelling because they are additive over time. For example,

\[ \log\left(\frac{P_2}{P_0}\right) = \log\left(\frac{P_2}{P_1}\right) + \log\left(\frac{P_1}{P_0}\right). \]

For risk measurement, returns are often converted into losses. If \(r_t\) is a log return, then the corresponding loss is

\[ L_t=-r_t. \]

A negative return therefore becomes a positive loss. This is useful because the right tail of the loss distribution contains the largest losses.

prices <- c(100, 102, 101, 98, 99)

simple_returns <- prices[-1] / prices[-length(prices)] - 1
log_returns <- log(prices[-1] / prices[-length(prices)])
losses <- -log_returns

tibble(
  Day = 2:5,
  Price = prices[-1],
  Simple_Return = simple_returns,
  Log_Return = log_returns,
  Loss = losses
)

In the table above, positive returns correspond to negative losses, while negative returns correspond to positive losses. The values of greatest interest for extreme risk measurement are the largest positive losses.

5 Measuring Financial Risk

Measuring risk means summarising the distribution of a risk variable using a numerical risk measure. Simple risk summaries include the mean, variance, and standard deviation. These measures are useful, but they do not fully describe extreme risk.

The mean describes the average outcome. The variance and standard deviation describe typical dispersion around the mean. However, a portfolio may have a moderate standard deviation and still be exposed to severe tail losses. For this reason, extreme financial risk measurement focuses on the tail of the loss distribution.

If \(L\) is a loss random variable with distribution function \(F_L(x)=P(L\leq x)\), then the right-tail probability is

\[ P(L>x)=1-F_L(x). \]

This probability is important because it measures the chance that losses exceed a specified level \(x\). Extreme risk is concerned with the behaviour of \(P(L>x)\) when \(x\) is large.

Two risk measures are especially important in this course: Value-at-Risk and Expected Shortfall.

Value-at-Risk at confidence level \(q\) is the \(q\)-quantile of the loss distribution:

\[ VaR_q=F_L^{-1}(q). \]

For example, a one-day 99% VaR of KSh 10 million means that, under the model being used, the loss will not exceed KSh 10 million on 99% of days. Equivalently, there is a 1% probability that the loss will exceed KSh 10 million. VaR is therefore a loss threshold, not a maximum possible loss.

Expected Shortfall measures the expected size of the loss once VaR has been exceeded:

\[ ES_q=E[L\mid L>VaR_q]. \]

VaR answers the question: Where does the dangerous tail begin? Expected Shortfall answers the question: Once we are in that tail, how large is the loss on average?

set.seed(123)
returns <- rt(1000, df = 4) / 100
losses <- -returns

VaR_95 <- quantile(losses, 0.95)
VaR_99 <- quantile(losses, 0.99)
ES_95 <- mean(losses[losses > VaR_95])
ES_99 <- mean(losses[losses > VaR_99])

tibble(
  Confidence_Level = c("95%", "99%"),
  Empirical_VaR = c(as.numeric(VaR_95), as.numeric(VaR_99)),
  Empirical_ES = c(ES_95, ES_99)
)

The 99% VaR is usually larger than the 95% VaR because a higher confidence level looks further into the loss tail. Expected Shortfall is usually larger than VaR because it averages only the losses that exceed the VaR threshold.

6 Stylized Facts of Financial Return Series

Empirical financial return series often display patterns that are not well captured by simple normal models. These empirical regularities are commonly referred to as stylized facts.

Important stylized facts include volatility clustering, heavy-tailedness, heteroskedasticity, skewness, excess kurtosis, and non-linearity. Volatility clustering means that large movements tend to be followed by large movements, while calm periods tend to be followed by calm periods. Heavy-tailedness means that extreme observations occur more frequently than they would under a normal distribution. Heteroskedasticity means that the variance of returns changes over time.

These features matter because risk models based on constant variance and conditional normality may underestimate the probability of extreme events.

set.seed(123)

n <- 10000
normal_data <- rnorm(n)
t_data <- rt(n, df = 3)

tibble(
  Distribution = c("Normal", "Student t with 3 degrees of freedom"),
  Mean = c(mean(normal_data), mean(t_data)),
  Standard_Deviation = c(sd(normal_data), sd(t_data)),
  Skewness = c(skewness(normal_data), skewness(t_data)),
  Kurtosis = c(kurtosis(normal_data), kurtosis(t_data)),
  Excess_Kurtosis = c(kurtosis(normal_data) - 3, kurtosis(t_data) - 3)
)
tibble(
  Normal = normal_data,
  Student_t = t_data
) %>%
  pivot_longer(cols = everything(), names_to = "Distribution", values_to = "Value") %>%
  ggplot(aes(x = Value)) +
  geom_histogram(bins = 80) +
  facet_wrap(~ Distribution, scales = "free_y") +
  labs(
    title = "Normal Distribution and Heavy-Tailed Student t Distribution",
    x = "Simulated value",
    y = "Frequency"
  )

The Student’s t distribution produces more extreme observations than the normal distribution. This illustrates why heavy-tailed models are important in financial risk measurement.

A direct comparison of tail probabilities also shows the danger of assuming normality:

tibble(
  Distribution = c("Standard normal", "Student t with 3 degrees of freedom"),
  Probability_Greater_Than_3 = c(
    pnorm(3, lower.tail = FALSE),
    pt(3, df = 3, lower.tail = FALSE)
  )
)

The heavy-tailed distribution assigns a larger probability to observations greater than 3. If losses are heavy-tailed but a normal model is used, the probability of severe losses may be underestimated.

7 Normality-Based Models and Their Limitation

Several traditional models of financial risk rely either directly or indirectly on normality. For example, the variance-covariance method of calculating VaR assumes that returns are normally distributed and therefore requires only an expected return and a standard deviation.

This approach is attractive because it is simple. Once the mean and standard deviation are known, quantiles can be obtained from the normal distribution. However, the simplicity comes at a cost. The normal distribution is symmetric and thin-tailed. It may not represent financial return series with skewness, excess kurtosis, volatility clustering, and extreme losses.

GARCH-family models were introduced to deal with changing volatility. They are useful because they allow the current volatility background to affect risk estimates. However, if they are used with a conditional normality assumption, they may still have difficulty capturing extreme tail events.

This motivates the use of methods that focus directly on the tail of the distribution. Since VaR and Expected Shortfall are tail-based measures, Extreme Value Theory becomes a natural framework for estimating them.

8 Motivation for Extreme Value Theory

Extreme Value Theory provides a formal statistical framework for studying the behaviour of extreme observations. Instead of modelling the entire distribution, EVT focuses on the tail of the distribution. This is important because VaR and Expected Shortfall are concerned with extreme quantiles and losses beyond such quantiles.

EVT is built around two main approaches. The first is the Block Maxima approach, where data are divided into blocks and the maximum observation from each block is modelled. This approach leads to the Generalized Extreme Value distribution.

The second is the Peaks Over Threshold approach, where a high threshold is selected and all observations above that threshold are modelled. This approach leads to the Generalized Pareto Distribution.

The POT approach is often preferred in practical financial risk work because it uses more extreme observations than the block maxima approach. Instead of selecting only one maximum per block, it keeps all observations above a sufficiently high threshold.

The course will later develop these ideas in detail, including maxima, block maxima, domains of attraction, POT methods for estimating VaR, threshold selection, stress testing, dynamic risk horizons, square-root-of-time scaling, and back-testing.

9 Chapter R Application

The following short application brings together the basic ideas of this chapter using simulated data. We generate returns, convert them into losses, examine the distribution, and estimate VaR and Expected Shortfall.

set.seed(2426)

n <- 1500
returns <- rt(n, df = 5) / 100
losses <- -returns

risk_summary <- tibble(
  Mean_Loss = mean(losses),
  Standard_Deviation = sd(losses),
  Skewness = skewness(losses),
  Kurtosis = kurtosis(losses),
  Excess_Kurtosis = kurtosis(losses) - 3,
  VaR_95 = as.numeric(quantile(losses, 0.95)),
  VaR_99 = as.numeric(quantile(losses, 0.99)),
  ES_95 = mean(losses[losses > quantile(losses, 0.95)]),
  ES_99 = mean(losses[losses > quantile(losses, 0.99)])
)

risk_summary
threshold_95 <- quantile(losses, 0.95)
threshold_99 <- quantile(losses, 0.99)

tibble(Loss = losses) %>%
  ggplot(aes(x = Loss)) +
  geom_histogram(bins = 70) +
  geom_vline(xintercept = threshold_95, linetype = "dashed") +
  geom_vline(xintercept = threshold_99, linetype = "dotted") +
  labs(
    title = "Simulated Loss Distribution with VaR Thresholds",
    x = "Loss",
    y = "Frequency"
  )

The dashed and dotted vertical lines represent high loss quantiles. These are not worst-case losses. They are thresholds that separate ordinary losses from the upper tail of the loss distribution.

10 Exercises

10.1 Conceptual Questions

  1. Explain why financial crises motivated the development of tools for measuring rare financial extreme events.
  2. Define risk as a random variable in the context of profit and loss.
  3. Explain the meaning of exchange rate risk.
  4. Distinguish between transaction risk, translation risk, and economic risk.
  5. Explain why identifying the type of currency risk is important before measuring or managing it.
  6. Define simple return and log return.
  7. Explain why losses are often defined as negative returns.
  8. Explain why the right tail of the loss distribution is important in extreme financial risk measurement.
  9. Explain why the mean and variance may not provide enough information about extreme risk.
  10. Define Value-at-Risk and explain its three elements: confidence level, time horizon, and loss amount.
  11. Explain why VaR should not be interpreted as the maximum possible loss.
  12. Define Expected Shortfall and explain why it gives more information about tail losses than VaR.
  13. List four stylized facts commonly observed in financial return series.
  14. Explain why normality-based VaR may underestimate extreme losses.
  15. Explain why GARCH models help with changing volatility but may still struggle with extreme events under conditional normality.
  16. Explain why EVT is suitable for estimating tail-based risk measures.
  17. Distinguish between Block Maxima and Peaks Over Threshold.
  18. Explain why POT may use data more efficiently than Block Maxima.

10.2 Computational Questions

  1. Given prices \(100, 104, 103, 97, 99\), compute simple returns, log returns, and losses.
  2. Simulate 1,000 observations from a normal distribution and 1,000 observations from a Student’s t distribution with 4 degrees of freedom. Compare their kurtosis and excess kurtosis.
  3. Using R, compute \(P(Z>3)\) for \(Z\sim N(0,1)\) and \(P(T>3)\) for \(T\sim t_3\). Comment on the difference.
  4. Simulate 2,000 daily returns from a Student’s t distribution with 5 degrees of freedom divided by 100. Convert them into losses and estimate the 95% and 99% VaR.
  5. Using the same simulated losses, estimate the 95% and 99% Expected Shortfall.

10.3 Exam-Style Question

Financial markets have experienced several major crises, including the 1987 stock market crash, the Asian financial crisis, and the global financial crisis. These events have made the estimation and forecasting of extreme financial losses a major concern for financial institutions and regulators.

Required:

  1. Explain why ordinary risk measures such as the mean and variance may be insufficient for measuring extreme financial risk.
  2. Define exchange rate risk and explain its three major forms.
  3. Define Value-at-Risk and explain how it summarizes market risk.
  4. Explain one major limitation of Value-at-Risk.
  5. Define Expected Shortfall and explain why it is useful in tail-risk measurement.
  6. Explain why financial return series may require methods beyond the normal distribution.
  7. Explain why Extreme Value Theory is appropriate for modelling rare but severe financial losses.