VRM 1: Measures of Financial Risk

Combining investments is where the framework earns its keep. Put a proportion w1 into the first investment and the rest, w2, into the second: the combined mean is a weighted average.

The standard deviation refuses to behave so simply, because it turns on how the two returns move relative to one another, which enters through their coefficient of correlation.

With n investments, where investment i carries mean mu i, standard deviation sigma i and weight w i, both results generalise.

Take two investments with mu1 = 5% and mu2 = 8%, standard deviations of 12% and 16%, and a correlation of 0.25. Sliding the weight across traces risk-return pairs.

Source: the chapter’s two-investment illustration, listed from the concentrated end. The weight w2 is one minus w1, and the ratio column is added here.

The last four rows, holding weights of 0.6, 0.4, 0.2 and 0.0 in the first investment, are dominated by nothing on offer, and form the efficient frontier for this pair.

A third investment opens fresh combinations with each of those four, a fourth opens more again, and repeating over every risky investment in existence produces the efficient frontier of all risky investments. The region below and right of that curve can be built; anything above and left of it cannot.

Adding borrowing and lending at the risk-free rate

Introduce an asset paying a risk-free return, plotted as point F on the vertical axis because its return standard deviation is zero. Draw the line from F that is tangent to the frontier and call the tangency point M. Splitting money between F and M places the investor on that line: half in each lands at the midpoint, and three quarters in the risk-free asset lands a quarter of the way along. A zero standard deviation on investment 1 kills the correlation term, leaving portfolio standard deviation proportional to the fraction held in M. Borrowing at that rate extends the line past M, since once the interest is deducted the coordinates trace the same line further along.

The trade-off therefore turns linear, and a striking conclusion follows. Every investor should hold the same risky portfolio, M, and express risk appetite purely through how much is lent or borrowed at the risk-free rate. The cautious sit near F, and risk takers at M or beyond.

The market portfolio and the assumptions holding it up

M is the market portfolio, holding every investment in the same proportion that it represents of all available investments. Any other composition unravels, since an under-represented investment would see demand fall short of supply and its price drop until the shortfall closed.

All of this rests on assumptions that are approximately true at best: investors care only about the mean and standard deviation of portfolio return, agree on every mean, standard deviation and correlation, and borrow freely at the risk-free rate. Frontiers of this kind underpin the capital asset pricing model.

Describing a portfolio by two numbers invites the assumption that returns are normally distributed, since the normal or Gaussian distribution is itself fixed by two parameters.

Height of the density at x measures how likely values near x are, and the probability of landing between a and b is the area beneath it over that interval, which a cumulative distribution supplies. Zero mean and unit standard deviation give the standard normal.

Between minus infinity and 1.0 it accumulates 0.8413. Take a normal variable centred on 2 whose standard deviation is 5. The chance of a value below 1 equals the chance a standard normal falls below -0.2, tabulated as 0.4207. NORM.DIST(4, 2, 5, TRUE) returns 0.6554, so a value between 1 and 4 has probability 0.2347.

What equity returns actually do

Convenience is the main argument for the assumption, and daily equity data undercuts it. Over the 20 years running from January 1, 2000, to December 31, 2019, day-to-day moves in the S&P 500 Index carried a standard deviation of 1.187%. Their mean of 0.024%, almost 50 times smaller, is treated as zero here.

Source: the chapter’s S&P 500 comparison, ordered from the largest move down.

Take the three standard deviation row: 82 days saw a move beyond 3 x 1.187% = 3.56%, which is 1.64% of the sample, over six times the 0.27% a normal model allows across both tails. Days at four, five and six standard deviations should be all but impossible, yet 0.64% of days clear four, 0.30% clear five and 0.12% clear six.

The distribution is both more peaked and fatter in the tails than a normal one of matching standard deviation, with the middle thinned to pay for both. Moves under one standard deviation take 78.95% of days against a predicted 68.27%, and moves of one to three standard deviations take 21.05% – 1.64% = 19.41% against 31.46%.

A standard deviation says how spread out a distribution is and nothing about the shape of its tail. Value at risk, written VaR, points at the tail itself.

Two parameters have to be fixed first: the time horizon over which the loss is measured, and the confidence level. VaR is then the loss not expected to be breached over that horizon at that level. From here the object of study is the distribution of losses in currency rather than of returns, with gains carried through as negative losses.

Horizon and the scaling of the loss distribution

Lengthening the horizon changes both parameters of a normal loss distribution, and not at the same rate. Where daily changes are independent, the mean grows with the number of days and the standard deviation with its square root.

Suppose a portfolio changes each day around a mean of 0.5, with a daily standard deviation of 2. Over ten days the mean becomes 0.5 x 10 = 5, and the standard deviation becomes 2 times the root of 10, or 6.32.

Continuous distributions hide a wrinkle that discrete ones expose. Where only a handful of outcomes are possible the cumulative loss distribution moves in jumps, so the percentile a confidence level names can sit inside one.

Consider a one-year project with three outcomes: a loss of USD 2 million with probability 88%, a loss of USD 5 million with probability 10%, and one of USD 8 million with probability 2%. Accumulating from the smallest loss upward gives each outcome a band.

Source: the chapter’s discrete loss example, ordered from the largest loss down.

Finding VaR now means locating the confidence level among those bands. At 99% the answer is USD 8 million, because 99% falls in the 98% to 100% band belonging to the largest loss. Drop the level to 97% and the answer becomes USD 5 million, since 97% falls in the 88% to 98% band.

A confidence level of exactly 98% lands on a boundary and is genuinely ambiguous. One reading places it in the 98% to 100% band for USD 8 million, the other in the 88% to 98% band for USD 5 million. Averaging them, for a VaR of USD 6.5 million, is one workable convention.

VaR reports one percentile of the loss distribution and stops. Whatever happens past that percentile leaves no trace in the number, and that silence is the central weakness.

Picture two books each reporting a VaR of USD 20 million at 99% confidence. In the first, a loss beyond USD 20 million could not conceivably run past USD 30 million. In the second, once the loss passes USD 20 million it is most likely to land near USD 100 million. VaR grades them identically, though the second is far more dangerous. Put another way, if a loss level V has a 1% chance of being exceeded, investment A may never lose more than 1.1V while B can lose 10V, and both report a VaR of V.

Positions of the second kind are not exotic. A trader who has sold a credit default swap has promised a payout if a named counterparty defaults. Default may carry a probability of only 0.9%, outside a 99% cut-off entirely, while the payout runs to USD 100 million.

Assumptions that travel with the number

Other limitations come from how the figure is produced rather than from its definition. A VaR estimate inherits whatever distributional assumption built the loss distribution, so assuming normality imports thin tails and understates the outcomes VaR exists to describe. Scaling a one-day figure by the root of time assumes successive daily changes are independent, and confidence level and horizon are choices rather than facts, so firms choosing differently produce figures that cannot be compared.

Expected shortfall repairs the defect directly, reporting not where the tail begins but the average loss across the whole tail, conditional on being in it. It also travels as conditional VaR, or C-VaR, and as tail loss.

It separates the two books at once, since the one whose losses cluster near USD 100 million averages far higher than the one whose losses stop at USD 30 million. Expected shortfall always exceeds VaR at the same confidence level, every value it averages lying beyond that cut-off. For normal losses the tail average has a closed form.

Where regulators have landed

Capital requirements built on internal models relied on VaR for a long time. Market risk rules introduced in 1996 set capital from a VaR over a ten-day horizon at a 99% confidence level, and Basel II set credit risk capital from a VaR over one year at 99.9%. The framework known as FRTB replaces VaR with expected shortfall and lowers the confidence level to 97.5%.

Choosing between the two measures for something as consequential as capital requirements calls for a standard that is not a matter of taste. Artzner, Delbaen, Eber and Heath supplied one, writing in Mathematical Finance (volume 9, 1999, pages 203 to 228) under the title “Coherent Measures of Risk” and setting out four properties a risk measure ought to have. A measure meeting all four is coherent.

Monotonicity

Suppose one portfolio finishes worse than another in every state of the world. Monotonicity requires it to carry the larger risk measure, since something that always does worse is riskier and needs more capital.

Translation invariance

Add an amount of cash K to a portfolio and its risk measure must fall by exactly K. Cash inside the portfolio cushions losses and substitutes for capital, so it should cut required capital one for one.

Homogeneity

Scale every component of a portfolio by a factor and the risk measure scales by that same factor, so halving a book halves the number attached to it. This axiom is the most open to challenge, since a position multiplied a hundredfold can be more than a hundred times as dangerous once the difficulty of unwinding it is counted.

Subadditivity

Merge two portfolios A and B. Subadditivity requires the merged risk measure to come out no higher than the two separate measures added together, which is the benefit of diversification written as an axiom. Perfect correlation gives a merged figure equal to the sum, and anything short of it should push the figure below, never above.

Expected shortfall satisfies all four and is coherent. VaR satisfies the first three, and subadditivity is the one it can fail.

One counterexample settles the question, and nothing about it is contrived: it needs only two positions whose damaging outcomes are rare enough to hide behind the confidence level. Take two insurance contracts, each losing USD 20 million with probability 0.8% and USD 1 million with probability 99.2%, treated as independent.

The published figure for that merged expected shortfall is USD 21.1236 million. Recomputing the weighted average from the probabilities above gives USD 21.1216 million, a difference in the last two digits that leaves every comparison intact.

Combining the two contracts therefore raises total VaR by USD 19 million, when diversification ought to have reduced it or at worst left it alone. Each separate measurement had placed the dangerous outcome just outside the 1% tail, and merging pulled it inside. A measure reading one percentile and no further is fooled in exactly this way.

A general way of building risk measures explains the previous result in a line. Start from a probability distribution of outcomes, then define a measure by attaching a weight to each percentile of the loss distribution. Both measures already met are special cases. VaR at a confidence level of X% is the Xth percentile, so all the weight sits there and every other percentile gets zero. Expected shortfall spreads it: percentiles below the VaR level get nothing, and every percentile above gets the same weight.

A measure defined this way is coherent if and only if the weight is a non-decreasing function of the percentile.

VaR breaks the condition immediately. At 99% confidence all the weight goes to the 99th percentile, so setting p2 at 99% and p1 at 99.5% gives w(p2) = 1 alongside w(p1) = 0, a higher percentile with a lower weight. Expected shortfall complies, its weights stepping up from zero to a constant and never falling back.

Spectral risk measures

Expected shortfall is not the only coherent measure available. A spectral risk measure is one whose weights rise across the percentiles in a pattern expressing risk aversion, so worse outcomes count for progressively more rather than equally.

Lowering gamma concentrates weight closer to the extreme, representing a more risk-averse user, so weights computed with gamma at 0.05 climb far more steeply near the top of the distribution than those computed with gamma at 0.15.