Free sample chapter from FRM Level I: Risk Models.

Chapter 1: Measures of Financial Risk

Chapter 1: Measures of Financial Risk

Foundations of Risk Management gave Ravensmoor Financial Group a capital number for every risk it runs, each one a loss that would be exceeded only once in a thousand years. Financial Markets and Products I and II priced the instruments behind those numbers. This course asks the question the earlier courses took for granted: where does a risk number come from?

The answer, which every chapter of this course repeats in a different setting, is that a risk number is a percentile of a distribution, and somebody chose the distribution. This chapter makes the point on the simplest position the group owns. The $800 million Core Equity Fund has a volatility of 18% a year — that number has not changed since Foundations. But a volatility is not a distribution. Two return distributions with exactly the same 18% volatility give the fund a one-day 99.9% value at risk of $28.03 million and $46.01 million.

The chapter starts where risk measurement started, with standard deviation in the mean-variance framework, and shows why one number stopped being enough. It defines value at risk, finds the questions VaR cannot answer, and introduces the measure that answers them — expected shortfall. It then sets out the four properties any sensible risk measure should have, and shows with two small loans that VaR fails one of them.

Learning Objectives

1.1 Risk and Return in the Mean-Variance Framework

The first quantitative definition of investment risk, Harry Markowitz's in 1952, was the standard deviation of return. An investor cares about two things — the expected return of a portfolio and its standard deviation — and prefers more of the first and less of the second.

For a portfolio with weight $w$ in asset 1 and $1-w$ in asset 2, the two quantities are

The expected return is a weighted average. The standard deviation is not: unless the correlation $\rho$ is exactly 1, the portfolio's standard deviation is less than the weighted average of the two, and that difference is the benefit of diversification.

The efficient frontier

Ravensmoor's asset-allocation committee chooses between two asset classes:

Every mix of the two traces out a curve in the plane of volatility and expected return. The point on it with the smallest volatility is the minimum-variance portfolio. Points below it on the curve are inefficient — another mix has the same volatility and a higher return. The part above it is the efficient frontier: for each level of risk, the highest return available.

Efficient frontier The set of portfolios that offer the highest expected return for a given standard deviation — equivalently, the lowest standard deviation for a given expected return. No rational mean-variance investor holds a portfolio below it.

In chain mode, keying 0.18 [x²] [+] 0.045 [x²] [-] 2 [×] [RCL] 1 would compute (0.0324 + 0.002025 − 2) × 0.00081, not the denominator, which is why the doubled covariance is computed first and recalled.

The tangency portfolio and the capital market line

Add a risk-free asset yielding 3.0%. Any investor can now hold a mix of the risk-free asset and one risky portfolio, and those mixes lie on a straight line from the risk-free rate through the risky portfolio. The steepest such line touches the frontier at the tangency portfolio — the risky mix with the highest Sharpe ratio, $(\mu_P - r_f)/\sigma_P$. That line is the capital market line, and every mean-variance investor holds a point on it: a cautious one lends at the risk-free rate, an aggressive one borrows.

For two risky assets, the tangency weight has a closed form in their excess returns over the risk-free rate, $e_1 = \mu_1 - r_f$ and $e_2 = \mu_2 - r_f$, and their covariance $\sigma_{12} = \rho\sigma_1\sigma_2$:

from math import sqrt

mu_e, sd_e, mu_b, sd_b, rho, rf = 0.08, 0.18, 0.038, 0.045, 0.10, 0.03
cov = rho * sd_e * sd_b

def portfolio(w):
    mean = w * mu_e + (1 - w) * mu_b
    sd = sqrt(w ** 2 * sd_e ** 2 + (1 - w) ** 2 * sd_b ** 2 + 2 * w * (1 - w) * cov)
    return mean, sd

w_min = (sd_b ** 2 - cov) / (sd_e ** 2 + sd_b ** 2 - 2 * cov)
ex_e, ex_b = mu_e - rf, mu_b - rf
w_tan = (ex_e * sd_b ** 2 - ex_b * cov) / (ex_e * sd_b ** 2 + ex_b * sd_e ** 2 - (ex_e + ex_b) * cov)
for name, w in (("minimum variance", w_min), ("tangency", w_tan)):
    m, s = portfolio(w)
    print(f"{name:<16} equities {w:.2%}   return {m:.2%}   volatility {s:.2%}   Sharpe {(m - rf) / s:.4f}")
# Output:
# minimum variance equities 3.70%   return 3.96%   volatility 4.45%   Sharpe 0.2147
# tangency         equities 30.23%   return 5.07%   volatility 6.55%   Sharpe 0.3161

What the framework assumes

Standard deviation is a complete description of risk only if the distribution of returns is fully described by its mean and standard deviation — which is true of the normal distribution — or if investors care about nothing else. Neither holds well for real assets. Standard deviation also treats a large gain as just as "risky" as a large loss, which no risk manager believes. The rest of this chapter is about what replaced it.

1.2 How Real Returns Depart From the Normal

The normal distribution is attractive: it is described by two numbers, sums of normal variables are normal, and its percentiles are tabulated. Daily returns on equities and most risky assets, though, differ from it in three consistent ways.

1. Fat tails (excess kurtosis). Very large moves, in both directions, happen far more often than a normal distribution with the same standard deviation allows. The kurtosis of daily equity returns is typically well above the normal's 3.
2. Negative skew. The largest moves are more often falls than rises: markets tend to drift up and crash down. A normal distribution is symmetric.
3. Volatility clustering. Large moves follow large moves. Volatility is not constant; it rises in bad periods and stays high for a while. Chapter 3 models this directly.

There is also a fourth, quieter difference: daily returns tend to be more peaked than the normal near zero. With the same total variance, the distribution puts more probability in the very centre and in the far tails, and less in the shoulders between.

Same volatility, different tail

The cleanest way to see what fat tails do is to hold the volatility fixed and change only the shape. This course uses a Student-t distribution with 4 degrees of freedom, rescaled so that its standard deviation equals that of the normal. A t with $\nu$ degrees of freedom has variance $\nu/(\nu-2)$, so the scaling factor that gives unit variance is

The t is symmetric, so it captures fat tails but not skew. That is a simplification, and not a conservative one: real equity returns put extra mass in the left tail, which a symmetric t spreads equally over both, so it can understate the left tail it is meant to measure.

from math import sqrt
from scipy.stats import norm, t

nu = 4
scale = sqrt((nu - 2) / nu)          # rescales a t(4) variable to unit variance
for k in (3, 4, 5):
    p_normal = norm.cdf(-k)
    p_t = t.cdf(-k / scale, nu)
    print(f"loss beyond {k} sd: normal once in {1 / p_normal / 252:>9,.1f} years   "
          f"t(4) once in {1 / p_t / 252:5.2f} years")
# Output:
# loss beyond 3 sd: normal once in       2.9 years   t(4) once in  0.60 years
# loss beyond 4 sd: normal once in     125.3 years   t(4) once in  1.65 years
# loss beyond 5 sd: normal once in  13,843.5 years   t(4) once in  3.76 years

1.3 Value at Risk

Standard deviation answers "how much do returns typically vary?" A risk manager, a board or a regulator wants an answer to a different question: how much could we lose? Value at risk was built to answer it in one number.

Value at risk (VaR) The loss over a stated time horizon that will not be exceeded with a stated probability, the confidence level. A one-day 99% VaR of $21 million means that on 99% of days the loss should be no more than $21 million — equivalently, that the loss exceeds $21 million on 1% of days.

A VaR number therefore has two parameters that must always be stated with it:

• the confidence level $c$, typically 95%, 99% or 99.9%;
• the time horizon $T$, typically one day or ten days for trading books and one year for credit and capital.

Formally, if $L$ is the loss over the horizon, VaR at confidence $c$ is the $c$-quantile of the loss distribution: the smallest $\ell$ for which $P(L > \ell) \le 1 - c$. Because VaR is stated as a loss, it is reported as a positive number.

VaR when returns are normal

If returns over the horizon are normal with mean $\mu$ and standard deviation $\sigma$, and the position is worth $V$, the loss is normal too, and

where $z_c$ is the standard normal percentile: 1.6449 for 95%, 2.3263 for 99% and 3.0902 for 99.9%. Over a short horizon such as one day, the expected return is tiny compared with the standard deviation and is normally set to zero, which gives the simplest and most quoted form, $\text{VaR}_c = V z_c \sigma$.

Scaling across horizons — the square-root-of-time rule

Daily volatility and annual volatility are linked by the number of trading days. If daily returns are independent and identically distributed, variances add across days, so standard deviations grow with the square root of time:

With 252 trading days, an annual volatility of 18% is a daily volatility of $0.18/\sqrt{252} = 1.1339\%$. The rule is used everywhere — Basel's ten-day market-risk VaR was for years computed as one-day VaR times $\sqrt{10}$ — but it rests on two assumptions that real returns break:

• Independence. If returns are autocorrelated, the rule is wrong. Positive autocorrelation (trends) makes multi-day losses larger than the rule says; negative autocorrelation (mean reversion), smaller.
• Identical distribution with zero mean. Volatility clustering means tomorrow's volatility depends on today's, so the ten-day distribution is not a scaled-up one-day distribution. And over long horizons the mean is no longer negligible: it grows with $T$ while the standard deviation grows only with $\sqrt{T}$.

What VaR assumes, and what it does not

VaR is a definition, not a model. To compute it, you need the distribution of the position's loss over the horizon — and every choice about that distribution changes the answer:

• The shape of the distribution. Normal, fat-tailed, or the empirical distribution of past returns (Chapter 2's historical simulation).
• The volatility. A long-run average, a recent estimate, or the market's implied volatility (Chapter 3).
• The holding period. The horizon should be the time it would take to close or hedge the position. A liquid equity book can be cut in a day; a book of illiquid loans cannot, and a one-day VaR on it is meaningless.
• The portfolio is fixed over the horizon. VaR asks what happens to today's positions, not to whatever the traders hold next week.

1.4 VaR on the Core Equity Fund, Normal and Fat-Tailed

The Core Equity Fund is worth $800 million with an annual volatility of 18%. Its daily standard deviation is

That is $9.07 million. Under the normal, the one-day 99% VaR is $9.07 \times 2.3263 =$ $21.10 million, and the ten-day 99% VaR by the square-root rule is $66.73 million.

Over one day the mean barely matters. If the fund is expected to return 8% a year, the one-day expected gain is 800 × 0.08 / 252 = $0.25 million, which lowers the VaR from $21.10 million to $20.85 million. Over a year it matters a great deal: the one-year 99% VaR is $334.99 million with a zero mean and $270.99 million after subtracting the $64 million expected gain.

The same fund, fat-tailed

Now keep everything — the $800 million, the 18% volatility — and change only the distribution, to the rescaled t with 4 degrees of freedom. The one-day VaR becomes the daily standard deviation times the t's percentile times 0.7071. The t(4) percentiles, read from a t table, are 2.1318, 3.7469 and 7.1732 at 95%, 99% and 99.9%, against the normal's 1.6449, 2.3263 and 3.0902. So at 99.9% the fat-tailed VaR is 9,071,147 × 7.1732 × 0.7071 = $46.01 million; at 99% it is 9,071,147 × 3.7469 × 0.7071 = $24.03 million; and at 95% it is 9,071,147 × 2.1318 × 0.7071 = $13.67 million.

Two things stand out. First, at 99.9% the fat-tailed VaR is 64% higher. Second, at 95% it is lower. That is the more peaked shape of §1.2 at work: with the same variance, the t puts more probability right at the centre and in the far tail, so less of it falls in the moderate losses that a 95% VaR measures. A fat-tailed distribution is not "riskier everywhere". It moves risk into the far tail, which is exactly where a 95% VaR does not look.

1.5 The Limitations of VaR

VaR became the industry's standard risk measure in the 1990s, through J.P. Morgan's RiskMetrics and then the Basel market-risk rules. Its appeal is real: one number, in currency, comparable across desks and asset classes, and easy to explain to a board. Its limitations are just as real.

1. It says nothing about the size of losses beyond it. A 99% VaR of $21 million tells you that 1% of days are worse than $21 million. Whether those days lose $22 million or $200 million, VaR does not say.
2. It can be gamed. Because VaR ignores everything past its percentile, a trader can lower it by taking on risk that sits just beyond the percentile — for example, by writing deep out-of-the-money options that lose rarely but heavily.
3. It is only as good as its distribution. As §1.4 showed, the normal and fat-tailed VaRs of one fund differ by 64% at 99.9%. Past data are a poor guide to future tails, and a VaR estimated from a calm period understates the risk that arrives in a crisis.
4. It is not subadditive. The VaR of a combined portfolio can exceed the sum of the VaRs of its parts, which means that VaR can penalise diversification. §1.8 shows how.

1.6 Expected Shortfall

The natural repair for limitation 1 is to ask the question VaR refuses to answer: if we are in the tail, how bad is it on average?

Expected shortfall (ES) The expected loss over the horizon, given that the loss is in the worst $1-c$ of outcomes. Also called conditional VaR (CVaR), expected tail loss or tail conditional expectation. For a continuous distribution, it is the expected loss conditional on the loss exceeding VaR.

For a continuous loss distribution,

Expected shortfall is always at least as large as the VaR at the same confidence level, because it averages losses that are all at least VaR.

Expected shortfall for a normal position

When the loss is normal with zero mean and standard deviation $\sigma$, ES has a closed form:

where $\phi(z_c)$ is the standard normal density (not the cumulative probability) at the percentile $z_c$:

At 99%, $\phi(2.3263) = 0.026652$, so the ES multiplier is $0.026652/0.01 = 2.6652$, against 2.3263 for VaR: ES is 15% above VaR. At 97.5%, $\phi(1.9600) = 0.058445$ and the multiplier is $0.058445/0.025 = 2.3378$ — almost exactly the 99% VaR multiplier. That near-equality is why the Basel Committee, when it replaced 99% VaR with expected shortfall for market-risk capital, chose a 97.5% confidence level: under the normal the capital number barely changes, while under a fat tail it rises.

The constant is stored first because the square-root key acts on whatever is displayed: keyed at the end of a chain, it would take the root of the whole running result rather than of 2π.

The keystrokes end at $24,179,243 rather than the Worked Problem's $24,176,421 because they carry $z$ to four places, 2.3263, where the Worked Problem used the full 2.326348. The $2,822 gap is rounding in $z$, and both figures round to $24.18 million.

The fund's ES, normal and fat-tailed

The t distribution's expected shortfall also has a closed form. Nothing on the exam requires it; it is shown here for reference only, so that the fat-tailed column below can be reproduced by hand. With $q_c$ the t percentile and $f_\nu$ the standard t density,

At 99%, $q = 3.7469$, so $q^2 = 14.0393$ and $f_4(q) = 0.375 \times 4.5098^{-2.5} = 0.008682$. The multiplier is $0.7071 \times (0.008682/0.01) \times (18.0393/3) = 3.6916$, and 9,071,147 × 3.6916 is $33.49 million. The same steps at 95% ($q = 2.1318$, $f_4 = 0.056228$, multiplier 2.2648) and at 99.9% ($q = 7.1732$, $f_4 = 0.000524$, multiplier 6.8491) give $20.54 million and $62.13 million.

Notice that at 95%, where the fat-tailed VaR was lower than the normal one, the fat-tailed ES is higher. ES looks past the percentile and into the tail, so it sees the fat tail at every confidence level. At 99.9% it doubles.

from math import sqrt
from scipy.stats import norm, t

fund, vol, nu = 800e6, 0.18, 4
sd_day = fund * vol / sqrt(252)
scale = sqrt((nu - 2) / nu)

for c in (0.95, 0.99, 0.999):
    z = norm.ppf(c)
    q = t.ppf(c, nu)
    var_n, es_n = sd_day * z, sd_day * norm.pdf(z) / (1 - c)
    var_t = sd_day * q * scale
    es_t = sd_day * scale * t.pdf(q, nu) / (1 - c) * (nu + q * q) / (nu - 1)
    print(f"{c:.1%}  normal VaR {var_n / 1e6:6.2f}M  ES {es_n / 1e6:6.2f}M   "
          f"t(4) VaR {var_t / 1e6:6.2f}M  ES {es_t / 1e6:6.2f}M")
print(f"10-day 99% VaR, normal: {sd_day * norm.ppf(0.99) * sqrt(10) / 1e6:.2f}M")
# Output:
# 95.0%  normal VaR  14.92M  ES  18.71M   t(4) VaR  13.67M  ES  20.54M
# 99.0%  normal VaR  21.10M  ES  24.18M   t(4) VaR  24.03M  ES  33.49M
# 99.9%  normal VaR  28.03M  ES  30.54M   t(4) VaR  46.01M  ES  62.13M
# 10-day 99% VaR, normal: 66.73M

Expected shortfall for a discrete distribution

For a discrete distribution, ES is the probability-weighted average of the worst $1-c$ of outcomes, taking a fraction of an outcome when the tail boundary falls inside it. The put writer of §1.5 loses $50 million with probability 0.8%. Its 99% tail is 1% wide: 0.8% of it is the $50 million loss and the remaining 0.2% is a loss of zero. So

The 99% ES is $40 million for a desk whose 99% VaR is zero. ES cannot be gamed by hiding losses just past the percentile, because it counts them.

VaR and ES compared

The last two rows are VaR's remaining advantages. VaR can be checked by counting exceptions — Chapter 2 shows how — while ES depends on the few observations deep in the tail, so it is estimated less precisely and checked less easily.

1.7 Coherent Risk Measures

VaR and ES are two candidates. How should anyone decide which is better, or whether some third measure is better than both? In 1999 Artzner, Delbaen, Eber and Heath answered by asking what properties any sensible risk measure should have. A risk measure $R$ assigns a number — the capital needed — to each portfolio's future value. They proposed four properties, and called a measure that has all four coherent.

Coherent risk measure A risk measure that satisfies monotonicity, translation invariance, positive homogeneity and subadditivity.

1. Monotonicity. If portfolio $X$ has a lower value than portfolio $Y$ in every state of the world, then $X$ is at least as risky: $R(X) \ge R(Y)$. A portfolio that always does worse cannot need less capital.

2. Translation invariance. Adding a riskless amount of cash $K$ to a portfolio reduces its risk measure by exactly $K$:

This is what lets a risk measure be read as capital. If a portfolio's risk measure is $21 million, adding $21 million of cash brings it to zero: the cash covers the loss.

3. Positive homogeneity. Scaling a portfolio by a positive factor $\lambda$ scales its risk by the same factor:

Doubling every position doubles the capital required. This assumes the position can be scaled without changing its character — a strong assumption for a large position in an illiquid market, where selling twice as much costs more than twice as much.

4. Subadditivity. The risk of a combined portfolio is no greater than the sum of the risks of its parts:

This is the mathematical form of diversification. Merging two portfolios should never create extra risk. If it could, a firm could lower its measured risk by splitting itself into pieces, and a regulator could not add up the capital of a firm's desks and know the total was enough.

1.8 Why VaR Is Not Coherent

When returns are normal — or, more generally, elliptical — VaR is subadditive, because it is then a fixed multiple of standard deviation, and standard deviation is subadditive. The failure comes from discrete, lumpy losses, exactly the kind a loan book produces.

# Two independent $10M loans, each defaulting with probability 4% and losing everything
p, size, c = 0.04, 10.0, 0.95

p_both = p * p
p_one = 2 * p * (1 - p)
print(f"P(no default) {(1 - p) ** 2:.4f}   P(one) {p_one:.4f}   P(both) {p_both:.4f}")

# VaR: the smallest loss L with P(loss > L) <= 5%
var_single = 0.0 if p <= 1 - c else size
var_pair = 0.0 if p_one + p_both <= 1 - c else (size if p_both <= 1 - c else 2 * size)
print(f"95% VaR: each loan {var_single:.2f}, sum {2 * var_single:.2f}, both together {var_pair:.2f}")

# ES: the average loss in the worst 5% of outcomes
es_single = size * p / (1 - c)
es_pair = (p_both * 2 * size + ((1 - c) - p_both) * size) / (1 - c)
print(f"95% ES:  each loan {es_single:.2f}, sum {2 * es_single:.2f}, both together {es_pair:.2f}")
# Output:
# P(no default) 0.9216   P(one) 0.0768   P(both) 0.0016
# 95% VaR: each loan 0.00, sum 0.00, both together 10.00
# 95% ES:  each loan 8.00, sum 16.00, both together 10.32

What the example generalises to

The loans are small, but the mechanism is general. A VaR treats any loss rarer than its tail as if it did not exist. Each loan's default, at 4%, is rarer than the 5% tail; combine two loans and the chance of some default, 7.84%, is not. Diversification spread the loss into more frequent, smaller pieces, and VaR saw them for the first time.

Expected shortfall is coherent for every distribution: it satisfies all four properties. Why it is subadditive can be seen without algebra. ES at confidence $c$ is the largest average loss over any set of outcomes with total probability $1-c$: the worst $1-c$ of outcomes give the highest average, and any other set of the same size gives less. Now take the worst $1-c$ of outcomes for the combined portfolio $X + Y$. Its ES is $X$'s average loss over those outcomes plus $Y$'s average loss over the same outcomes. Those outcomes need not be the worst ones for $X$ alone, so $X$'s average over them is at most $\text{ES}(X)$; likewise $Y$'s is at most $\text{ES}(Y)$. Hence $\text{ES}(X+Y) \le \text{ES}(X) + \text{ES}(Y)$. VaR has no such argument, because it is not an average over a set of outcomes but a single cut-off, and the cut-off of a sum can sit above the sum of the cut-offs, as the two loans showed.

That subadditivity, together with its sensitivity to the size of tail losses, is why the Basel Committee moved market-risk capital from VaR to ES. VaR keeps its place where losses are close to normal and where its ease of backtesting matters.

1.9 Spectral Risk Measures and Risk Aversion

VaR and ES can both be written as weighted averages of the loss quantiles. Let $q_p$ be the loss at percentile $p$ (so $q_{0.99}$ is the 99% VaR), and let $\psi(p)$ be a weight on each percentile. A spectral risk measure is

The integral sign here only means a weighted average: multiply each quantile by its weight and add them up across all percentiles, exactly as a weighted average of a list of numbers does.

The two measures of this chapter are special cases:

• VaR at confidence $c$ puts all of its weight on one percentile, $p = c$, and none anywhere else.
• ES at confidence $c$ puts zero weight on percentiles below $c$ and an equal weight of $1/(1-c)$ on every percentile above it.

A spectral risk measure is coherent if and only if its weights are non-negative, sum to one, and never decrease as $p$ rises — that is, if a worse outcome never gets less weight than a better one. This is how the weights should reflect risk aversion: a risk-averse investor cares more about worse outcomes. ES meets the condition, with weights flat across the tail. VaR does not: its weight drops from everything to nothing just past $c$, so the outcomes worse than VaR get less weight than VaR itself.

A common choice is the exponential spectrum, $\psi(p) = k e^{-k(1-p)} / (1 - e^{-k})$, which rises smoothly towards the worst outcomes; the parameter $k$ is the degree of risk aversion. The larger $k$, the more the measure concentrates on the far tail.

from math import sqrt, exp
import numpy as np
from scipy.stats import norm, t

sd_day, nu = 800e6 * 0.18 / sqrt(252), 4
scale = sqrt((nu - 2) / nu)
p = (np.arange(200_000) + 0.5) / 200_000            # midpoints of the probability scale


def spectral(k, quantile):
    weight = k * np.exp(-k * (1 - p)) / (1 - exp(-k))   # rises with p: worse outcomes weigh more
    return np.mean(weight * quantile)


normal_q = sd_day * norm.ppf(p)
t_q = sd_day * scale * t.ppf(p, nu)
for k in (25, 100):
    print(f"risk aversion k = {k:>3}:  normal {spectral(k, normal_q) / 1e6:6.2f}M   "
          f"t(4) {spectral(k, t_q) / 1e6:6.2f}M")
# Output:
# risk aversion k =  25:  normal  17.73M   t(4)  19.57M
# risk aversion k = 100:  normal  22.73M   t(4)  30.28M

For a mildly risk-averse holder ($k = 25$), the fat tail adds 10% to the measure; for a strongly risk-averse one ($k = 100$), it adds 33%. The more a measure cares about the worst outcomes, the more the choice of distribution drives it.

1.10 Putting It All Together: What the Fund Can Lose Tomorrow

Ravensmoor's board asks the chief risk officer a simple question: how much could the Core Equity Fund lose tomorrow? Every input the CRO needs has been fixed since Foundations: $800 million, an 18% volatility, a one-day horizon. The answer still depends on two choices this chapter has made explicit.

The choice of distribution. Volatility pins down the typical day, not the bad ones. With an identical 18% volatility, a fat tail adds 14% to the 99% VaR and doubles the 99.9% ES. The data the CRO has — a few years of daily returns — contains only a handful of days beyond the 99th percentile and perhaps none beyond the 99.9th, so the shape of the far tail is largely an assumption.

The choice of measure. VaR says where the tail begins; ES says how bad it is. At 95%, VaR would report that the fat-tailed fund is safer. ES would not. And of the two tail measures, only ES is guaranteed to reward the diversification that the mean-variance framework of §1.1 was built to capture.

A responsible answer to the board is therefore not one number but a short table and a sentence: on one day in a hundred we expect to lose more than about $21 to $24 million, and on those days about $24 to $33 million on average, depending on how fat the tail is. The width of each range is the honest part of the answer.

Chapter Summary

What's Next

This chapter took the distribution as given — normal or t — and computed from it. Chapter 2 asks how VaR and ES are computed in practice, when the distribution is unknown and the portfolio contains more than a single equity fund: by historical simulation from past returns, by the delta-normal method for linear positions and options, and by Monte Carlo simulation. Applied to one Ravensmoor portfolio, the three methods give three different answers, and each difference has a cause that can be named.

Concept Checks

Exam-Style Practice

Ten standalone questions in the style of the FRM Level I exam. Allow 24 minutes — the exam's pace of 2.4 minutes a question.

Get the full course: FRM Level I: Risk Models