> For the complete documentation index, see [llms.txt](https://laurence-wilse-samson.gitbook.io/textbooks/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://laurence-wilse-samson.gitbook.io/textbooks/financial-economics-claims-prices-holders/part-vi-risk-and-the-real-economy/chapter_26_risk_management.md).

# Chapter 26: Risk Management in Firms and Financial Institutions

*Part VI: Risk and the Real Economy — Financial Economics: Claims, Prices, and Holders*

***

## Opening Episode: A Hedge That Worked and a Firm That Did Not

In December 1993 the supervisory board of Metallgesellschaft AG, then among Germany's largest industrial groups, removed its chief executive and ordered the immediate liquidation of a derivatives position held by an American subsidiary most of the board had barely heard of. The unwind crystallized a loss reported at around $1.3 billion, and the parent survived only on a rescue package from a syndicate of well over a hundred banks, led by Deutsche Bank, reported at some DM 3.4 billion. What makes the episode worth an opening is that thirty years of argument have not settled whether the position that killed the firm was a mistake.

The subsidiary, MG Refining and Marketing, had built an attractive business. It sold fixed-price forward contracts for heating oil, gasoline and diesel to independent retailers and small distributors — customers with no access to the futures market and a real appetite for knowing their input cost for the next decade. By late 1993 MGRM had committed to deliver on the order of 160 million barrels at prices fixed as far as ten years forward. That commitment is a short position in energy, and leaving it unhedged would have been reckless. So MGRM hedged: it bought a **rolling stack** of front-month NYMEX energy futures, supplemented by over-the-counter swaps, roughly barrel for barrel, and rolled the whole stack forward each month as the near contract approached expiry.

Two things then happened at once. Crude fell through 1993, from around nineteen dollars a barrel in the summer to under fifteen in December, producing mark-to-market losses on a long futures position of roughly two-thirds of a billion dollars — paid in cash, daily, through the margin system. And the energy curve, which had spent most of the preceding decade in backwardation, moved into contango, turning the monthly roll from a source of income into a recurring cost: a long stack rolled in contango sells the cheap expiring contract and buys the dearer next one, and at twenty cents a barrel on 150 million barrels that is thirty million dollars a month. Meanwhile the offsetting gain — the customer contracts, now enormously valuable because MGRM was owed a decade of deliveries at prices well above spot — was unrealized, on an illiquid, unmarked, non-financeable asset, and under the German accounting of the day not recognized at all.

The academic fight began at once and is worth teaching as a fight. Christopher Culp and Merton Miller argued that MGRM was running a coherent program of **synthetic storage** — manufacturing a decade of physical inventory out of paper — that the hedge was economically sound, and that the loss was manufactured by the parent's decision to liquidate at the bottom, abetted by an accounting convention that showed one leg and hid the other. Antonio Mello and John Parsons replied that a barrel-for-barrel stack is not the right hedge for a ten-year obligation at all: near-dated futures move more than the present value of distant delivery commitments, so the variance-minimizing hedge ratio is well below one, and MGRM's choice of one loaded the firm with a purely financial cash-flow exposure unrelated to the risk it was removing.

Both sides are right about different questions, and the difference is this chapter's first theme. Culp and Miller describe an **economic hedge**: a position that offsets the exposure in present-value terms. Mello and Parsons describe a **survivable hedge**: one whose intermediate cash flows the firm can finance. The two coincide only where a firm can borrow freely against an unrealized gain, which is precisely the world Chapter 16 §16.5 says does not exist. MGRM's hedge failed not as a hedge but as a funding structure, and the mechanism — margin calls arriving on the liquid leg while the offset sits in an asset no lender will advance against — is §16.5's margin spiral running inside one firm's treasury.

The second theme follows. If the danger is the gap between an offset and a financeable offset, then measuring risk is not bookkeeping. It is a decision about which states the firm has to survive, taken with a model — and the model becomes part of the market it measures.

***

## 26.1 Why Nonfinancial Firms Hedge

Start where Chapter 23 §23.1 starts, because the logic is identical. In a frictionless market, hedging is irrelevant. A fairly priced forward has zero net present value by construction, and adding a zero-value contract to a portfolio of claims cannot change the portfolio's value. Shareholders who dislike the firm's oil exposure can hedge it themselves, exactly as the homemade-leverage argument says of capital structure. Modigliani-Miller applies to the risk-management decision without amendment.

That baseline is a map of the frictions, and Clifford Smith and René Stulz drew it in 1985. Four entries matter.

**Convex taxes.** If the effective tax schedule is convex over the relevant range — a progressive rate structure at low income, loss carryforwards worth less than immediate deductions, credits that expire unused — then expected tax on a volatile pre-tax income exceeds the tax on its certainty equivalent, and hedging raises after-tax value with no change in the pre-tax distribution. The effect is real and, in most estimates, small.

**Costs of financial distress.** Chapter 23's trade-off theory prices leverage against the deadweight costs of distress: legal expense, lost customers and suppliers, forgone investment, key employees leaving. Those costs are convex in a bad outcome, so reducing the probability of reaching the bad region is worth money to the firm's existing claimants. This entry scales with leverage, which is why the empirical literature finds hedging and leverage moving together.

**Costly external finance.** The sharpest version is Kenneth Froot, David Scharfstein and Jeremy Stein's. Suppose the firm has an investment program to fund, internal funds are cheaper than external ones for the standard information reasons, and the marginal product of investment is decreasing. Then a shortfall in internal cash flow is absorbed by cutting investment, at a shadow cost equal to the forgone marginal project. Hedging is then not about smoothing income for its own sake. It is about **matching the supply of internal funds to the demand for them**, so the firm does not arrive at a good project with an empty account. Chapter 25 §25.1's opening fact makes this the operative theory: internal funds cover the large majority of nonfinancial corporate capital expenditure in a typical year, so the marginal dollar of investment is very often a dollar of cash flow. It also carries a testable corollary the other three lack: a firm should hedge more when its investment opportunities are *uncorrelated* with the risk it faces, and less when a fall in oil prices also cuts the value of the projects an oil producer would fund — because then the shortfall in funds and the shortfall in demand for funds arrive together, and the exposure is partly self-hedging.

Figure 26.1 draws the first two entries on one firm. **Managerial risk aversion.** Chapter 24 §24.2's third conflict is about variance rather than mean. An executive holds undiversified human capital and an undiversified equity stake in one firm and can hedge neither. She therefore values a reduction in total firm risk that a diversified shareholder does not, and if she controls hedging policy she will buy it with shareholders' money. This is the one entry that predicts hedging the shareholders would not choose, and compensation structure separates it from the rest: managers holding stock hedge more, managers holding options hedge less, exactly as convexity in the payoff predicts.

![Figure 26.1: Why nonfinancial firms hedge](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-cd5657ff8be47f66627ac6ebd8dacc3ff1b57bb2%2Ffig_26_01_why_firms_hedge.png?alt=media)

**Figure 26.1: Why nonfinancial firms hedge.** Smith and Stulz's first two entries, drawn on one firm with two equally likely pre-tax outcomes, −10 and 210, and therefore an expected cash flow of 100 whether the firm hedges or not. Panel (a): the cost of the frictions. The schedule is convex — a loss carries no immediate deduction, so tax kinks at zero, and the distress cost rises faster than linearly in the shortfall — so the chord joining the two outcomes lies above the curve. The gap at the expected cash flow is 4.72: 1.05 of it from the convex tax and 3.68 from the distress cost, which is roughly the relative weight the section's prose gives them. Panel (b): the same statement on the value scale the firm is actually measured on, where the concavity is real, is under five percent of expected cash flow, and is very nearly invisible. That is why this figure is normally drawn with its curvature exaggerated, and why it is not exaggerated here. The distress parameters are a drawing choice; the tax kink is not. *Source: Author's construction, after Smith and Stulz (1985).*

**What firms actually hedge.** Survey and disclosure evidence across three decades agrees on a rough picture. Something over half of large nonfinancial firms use derivatives; the share falls steeply with firm size, and small firms hedge almost not at all, which the fixed cost of a treasury function explains without any theory of value. Currency exposure is hedged most, then interest rates, then commodities. Horizons are short — most programs run a year or less, a long way from Metallgesellschaft's ten — and hedge ratios are partial, which is consistent with every friction on Smith and Stulz's list being real and none being large. These are the end users on the other side of Chapter 8's pricing theory and of the market structure in *International Finance* Chapters 11 and 13: when Chapter 8 §8.6 says dealers are the counterparty to end-user demand, this section is the end user.

***

## 26.2 Hedging Practice: Delta, Gamma, and the Rolling Stack

Chapter 8's Box 8.2 defined the Greeks and deferred their use. Their use is a hedged book, and a hedged book is a claim about what happens between rebalances.

**Delta hedging, worked.** Take Chapter 8's benchmark option: a one-year call struck at 100 on a stock at 100, with a five percent riskless rate and twenty percent volatility, worth 10.4506 with delta 0.6368. A dealer sells calls on 100,000 shares for $1,045,060 and immediately buys 63,683 shares, financing the difference. The position is now locally insensitive to the stock — a one-cent move changes the option liability and the share holding by the same amount — and it does not stay that way, because delta is a function of price and of time.

**Table 26.1: A weekly delta hedge of a short call on 100,000 shares** ($$K=100$$, $$r = 5$$ percent, $$\sigma\_{\text{imp}} = 20$$ percent, one year to expiry at week 0)

| Week | Stock  | Call value | Delta  | Shares held | Shares traded |
| ---- | ------ | ---------- | ------ | ----------- | ------------- |
| 0    | 100.00 | 10.4506    | 0.6368 | 63,683      | +63,683       |
| 1    | 102.50 | 11.9736    | 0.6813 | 68,128      | +4,445        |
| 2    | 99.00  | 9.5777     | 0.6148 | 61,484      | −6,644        |
| 3    | 103.50 | 12.4072    | 0.6974 | 69,740      | +8,256        |
| 4    | 101.00 | 10.5921    | 0.6510 | 65,101      | −4,639        |
| 5    | 104.00 | 12.4977    | 0.7051 | 70,506      | +5,405        |

*Source: Author's calculation from the Black-Scholes formula at the stated parameters.*

Read the last column. The dealer buys after the stock rises and sells after it falls, every time, because delta is increasing in the stock price. That is what being **short gamma** means operationally, and it is the trading rule Chapter 7 §7.6's portfolio insurance followed into October 1987.

Now the outcome. Carrying the share position and the financing forward, the five weeks leave the dealer down $13,026, about thirteen cents per option, on a book that was delta-neutral at the start of every week. Nothing went wrong. The path's realized volatility, annualized from its five weekly log returns, was 23.4 percent against the 20 percent the option was sold at, and a short-gamma book that sells volatility at twenty and receives twenty-three loses by construction.

**Why static hedges decay.** Over a short interval, a delta-hedged short option earns roughly

$$
\tfrac{1}{2}\Gamma\big\[\sigma\_{\text{imp}}^2 S^2 \Delta t - (\Delta S)^2\big],
$$

which is positive when the realized move is smaller than the implied move and negative when it is larger. At the benchmark parameters the weekly break-even move is $$\sigma\_{\text{imp}} S\sqrt{\Delta t} = 2.77$$ dollars: a move of exactly that size leaves the hedged position within half a cent of flat, a two-dollar move earns about four cents, a six-dollar move loses about twenty-four. A delta hedge does not remove risk. It converts a directional bet into a bet on realized versus implied volatility, and gamma is the exchange rate.

**The rolling stack, resolved.** MGRM's hedge is the same problem in a different market. A ten-year fixed-price delivery obligation is, in present-value terms, a short position whose sensitivity to spot is far less than one barrel per barrel owed, because a distant delivery is discounted and because distant forward prices move less than one-for-one with the front month. Hedging it with front-month futures at a ratio of one is therefore an over-hedge in variance terms — Mello and Parsons's point — and an over-hedge in exactly the direction that maximizes the cash-flow consequence, since the front month is the contract that margins daily. Add the roll: its cost is the slope of the curve, and the sign of that slope is not a hedging decision but a market outcome, so MGRM's program was long the front month and short the curve's shape, an exposure no one had underwritten. Chapter 8 §8.2's cost-of-carry relation says what determines that shape and why the theory of storage makes it unstable. The lesson generalizes past oil: **a hedge whose maturity does not match its exposure has replaced price risk with basis risk and funding risk**, and the second pair is what closes firms.

***

## 26.3 Measuring Risk: Value at Risk and Expected Shortfall

**The definition.** Let $$\mathcal{L}$$ be the loss on a portfolio over a stated horizon — one day for a trading book, ten days for regulatory capital, a year for an insurer. The **value at risk** at confidence level $$\alpha$$ is the smallest loss threshold not exceeded with probability $$\alpha$$:

$$
\mathrm{VaR}\_\alpha = \inf\lbrace x : \Pr(\mathcal{L} \le x) \ge \alpha \rbrace.
$$

Three parameters must be stated for the number to mean anything — horizon, confidence level, distribution — and stripping any one makes it unreadable.

**A worked parametric case.** Take an equity portfolio of 100 million dollars with a daily return standard deviation of one percent and mean zero, and assume normality. Then $$\mathrm{VaR}\_\alpha = z \_\alpha \sigma$$ times the position, with $$z \_{0.95} = 1.645$$ and $$z \_{0.99} = 2.326$$: a one-day 99 percent VaR of $2.33 million and a 95 percent VaR of $1.64 million. Scaling to ten days under the square-root-of-time rule multiplies by 3.162, giving $7.36 million — a rule that assumes independent daily returns and is therefore wrong in exactly the episodes the number is for. Note what "one percent daily volatility" is: an annual volatility of about sixteen percent, an unremarkable equity market. The number is not conservative.

**Three routes to the number.** *Parametric (variance-covariance)* imposes a distribution — usually the normal — estimates a covariance matrix, and reads the quantile off a formula: fast, able to handle thousands of positions, and wrong about tails and about options, whose payoffs are not linear in the risk factors. *Historical simulation* revalues today's portfolio at every set of factor moves observed in some past window and takes the empirical quantile. It imposes no distribution and captures whatever fat tails the window contained, which is also its weakness: the window is the model, a quiet window gives a small number, and an event absent from it has no probability at all. *Monte Carlo* simulates factor paths from an assumed process and revalues on each — the only route that handles path-dependent and heavily optioned books, at the cost of putting the assumed process back in and a computational bill that forces approximations of the portfolio being measured.

**What VaR is not.** It is not a worst case; it is the *best* of the bad cases, the threshold at which the tail begins, and it says nothing about the region beyond. Two portfolios with identical VaR can have expected losses in the worst one percent differing by a factor of ten. It also inherits the tail assumption entirely: at the parameters above, a Student-$$t$$ with four degrees of freedom scaled to the same variance raises the 99 percent VaR only from $2.33 million to $2.65 million — a 14 percent difference that flatters the normal assumption, and conceals that the same change raises the *average* loss in the worst one percent by 38 percent.

**Expected shortfall.** The fix is to average over the tail rather than to point at its edge; Figure 26.2 draws both measures on this section's own numbers. **Expected shortfall** at level $$\alpha$$ is

$$
\mathrm{ES}\_\alpha = E\big\[\mathcal{L} \big| \mathcal{L} \ge \mathrm{VaR} \_\alpha\big],
$$

the mean loss conditional on being in the worst $$1-\alpha$$ of outcomes. Under normality $$\mathrm{ES}\_\alpha = \sigma\phi(z \_\alpha)/(1-\alpha)$$: at 99 percent the multiplier is 2.665 against VaR's 2.326, so the portfolio above has an expected shortfall of $2.67 million. A coincidence explains a regulatory choice: expected shortfall at 97.5 percent has a normal multiplier of 2.338, almost exactly the 99 percent VaR multiplier, which is why the Basel Committee's revised market-risk framework could switch measures without recalibrating the capital level.

![Figure 26.2: VaR and expected shortfall](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-9ae223d5457949ce1c6ba527a81198ab6efcae36%2Ffig_26_02_var_and_expected_shortfall.png?alt=media)

**Figure 26.2: VaR and expected shortfall.** This section's worked case — a $100 million equity book with a one percent daily return standard deviation — under the normal assumption and under a Student-t with four degrees of freedom scaled to the *same* variance, so the two differ in shape and not in volatility. Panel (a): at the scale a risk report is read, the two are almost the same picture. Panel (b): the same tail, magnified. The 99 percent VaR moves from $2.33 million to $2.65 million, a 14 percent difference that flatters the normal; the average loss beyond it moves from $2.67 million to $3.69 million, which is 38.5 percent — the figure the section rounds to 38. The shaded regions are the worst one percent of days — the region VaR points at and says nothing about, and the region expected shortfall averages over. Two portfolios can share the left panel and not the right one, and that is the whole argument for the measure the Basel Committee switched to. *Source: Author's calculation from the worked parametric case of §26.3.*

**Why the switch was made: coherence.** Philippe Artzner, Freddy Delbaen, Jean-Marc Eber and David Heath asked in 1999 what properties a risk measure ought to satisfy and named four: monotonicity, translation invariance, positive homogeneity, and **subadditivity** — the requirement that the risk of a combined portfolio never exceed the sum of its parts' risks. Subadditivity is the one with institutional teeth: it is what keeps diversification from ever looking harmful, and what lets a firm set desk limits that add up to a firm limit. VaR fails it.

Here is the failure in four lines. Two corporate bonds, independent, each with face value 100, each defaulting with probability 0.04 and recovering nothing. For one bond $$\Pr(\mathcal{L} = 0) = 0.96 \ge 0.95$$, so the 95 percent VaR is zero. Hold both, and $$\Pr(\mathcal{L} = 0) = 0.96^2 = 0.9216$$ falls below 0.95, so the quantile moves up to the next outcome: $$\mathrm{VaR}\_{0.95} = 100$$, against a sum of parts of zero. Diversifying into a second independent credit made measured risk appear from nothing. Expected shortfall gives the sensible answer on the same numbers: 80 for each bond, 103.2 for the pair, below the sum of 160. The corollary is that a firm running VaR limits desk by desk cannot add them up and does not know what it owns.

**Backtesting.** A risk number that is never checked is a claim, not a measurement. The check is mechanical: count the days on which realized loss exceeded that day's VaR, and compare the count with the model's own prediction. Over 250 trading days a correct 99 percent model produces 2.5 exceptions on average, and the Basel traffic-light zones follow from the binomial distribution — four or fewer, which a good model delivers 89 percent of the time; five to nine, a yellow zone that raises the capital multiplier; ten or more, a probability of about one in four thousand, and the model is rejected. Two cautions belong with the arithmetic. The test has very little power at 99 percent, because the events counted are rare by construction, which argues for backtesting at 95 percent even when capital is set at 99. And exceptions cluster: the count may be right while every exception falls in one fortnight, which is the pattern that matters and which a count cannot see.

Figure 26.3 runs that backtest on the simplest model a desk actually uses.

![Figure 26.3: When VaR failed](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-1df029ea88b0b57d704aab6eeba8d3c502662479%2Ffig_26_03_when_var_failed.png?alt=media)

**Figure 26.3: When VaR failed.** A rolling 250-day historical-simulation 99 percent value at risk on the US market return, and the number of days each year on which the realized loss exceeded it. The dashed line is what a correct model predicts, 2.5 a year, and the bands are Basel's traffic-light zones. The model is not merely imprecise; it is biased. Most years land in the yellow zone rather than the green one, which means a correct model would be a better model than the one every risk system in the world runs — historical simulation assumes the next 250 days resemble the last 250, and they do not. Worse, the failures are not spread out. The two red years are 2008 and 2022, and the near-misses are 2007 and 2020: the exceptions cluster in exactly the episodes the number exists to warn about, and cluster because volatility does. A count of fifteen in 2008 is not fifteen independent surprises; it is one regime the model had never seen, arriving on fifteen consecutive-ish days. That is the caution §26.3 states and the one the count cannot express. *Source: Kenneth R. French data library, daily market excess return; rolling 250-day historical-simulation VaR at 99 percent and the Basel traffic-light thresholds, computed by the author.*

***

## 26.4 Stress Testing as Regulation and Practice

A stress test abandons the distribution: it asks not what loss corresponds to a probability but what loss corresponds to a *scenario*. The scenario is written down — unemployment reaching ten percent, equity prices halving, a sovereign defaulting, a counterparty failing — and the portfolio revalued in it. The gain is that no probability need be estimated for an event that has happened once or never; the cost is that the scenario carries no probability at all, so a stress loss cannot be compared with a capital requirement without the judgment the statistical model was supposed to supply.

The institutional form in the United States is the pair of annual supervisory exercises the Federal Reserve has run since the 2009 crisis-era assessment: the Dodd-Frank Act Stress Test, which projects losses and capital ratios for large bank holding companies under supervisor-specified baseline and severely adverse scenarios, and the Comprehensive Capital Analysis and Review, which converts the results into a constraint on dividends and buybacks. That regulatory architecture belongs to *International Finance* Chapter 25; what belongs here is scenario design.

**Reverse stress testing** inverts the exercise and is the more useful half. Rather than asking what a scenario does to the firm, it asks what scenario would exhaust the firm's capital, and how implausible that scenario really is. The output is a description of the binding vulnerability rather than a number, and it is the technique that would have caught MGRM: nobody at Metallgesellschaft appears to have asked what price path would generate margin calls larger than the group's borrowing capacity, and the answer — a moderate fall in oil plus a flip in the curve's slope — was not a tail event.

The standing critique is that scenarios are models with the uncertainty hidden rather than removed. Someone chooses the variables, the paths, and the correlation between them; a scenario is a joint distribution collapsed to a single draw, with all the judgment that implies and none of it visible. Supervisory scenarios are also published, which makes them optimizable: a portfolio can be arranged to perform well in the announced scenario at the cost of performing badly in a neighboring one. And they are written by people looking at the last crisis. The 2020 pandemic scenario resembled nothing in the 2019 exercise, and the 2023 failures of banks holding long-duration securities into rising rates were not the credit-loss scenario a decade of testing had rehearsed.

***

## 26.5 Model Risk

Every number in the last two sections came out of a model, and a model is a map. **Model risk** is the risk of loss from using the map as the territory, and it has three forms: the model is wrong for the claim; the model is right but its parameters are estimated on data lacking the relevant states; the model is right and well estimated but used outside the range it was fitted for.

The parameter failure has the highest body count, and correlation is where it lives. Correlations estimated in calm periods understate those that obtain in crises, for a reason Chapter 19 §19.5 makes structural: when the marginal holder's constraint binds, claims sharing nothing but their holder move together. Chapter 10 §10.8's tranche arithmetic is the clean demonstration — a senior tranche's expected loss rising from about one percent to five and a half as the default correlation parameter moves from a plausible estimate to a crisis value, with nothing about the underlying loans changing — and the copula carrying that parameter became structured credit's standard device precisely because it produced a whole loss distribution from one unobservable number. What the models did in 2007-2009, and the epistemics of the ratings built on them, is the companion volume's subject.

Governance is the response, and it is unglamorous: independent **model validation**, staffed and reporting separately from the desk that uses the model; **benchmarking** against an alternative built on different assumptions, with the gap reported rather than reconciled away; backtesting where an outcome exists to test against; an inventory of models with owners and documented approved input ranges. The recurring finding of supervisory reviews is that failures are organizational rather than mathematical — a model approved for one product used for another, a limit exceeded without escalation, a validation function reporting to the business it validates. None of that is solved by a better model.

> **Box 26.1 — VaR as convention**
>
> Chapter 7 §7.6's argument was that a model in wide use stops being a description of the market and becomes part of it. Value at risk is the cleanest instance in this book, because its path from private tool to public infrastructure is documented.
>
> It began inside one firm as a reporting device: a single number, delivered to the chief executive at the end of each trading day, summarizing what the whole institution could lose. The appeal was never analytical. It was that one number could be read by someone who could not read a position sheet, and compared across desks trading unrelated instruments. In 1994 the methodology and a set of covariance estimates were published rather than sold; by 1996 the Basel market-risk amendment let banks set trading-book capital from their own VaR models. Within a decade the number was in annual reports, board packs, desk limits, and counterparty credit terms.
>
> That is what infrastructure means: not that everyone believes the number, but that everyone must produce it and decisions are keyed to it. A shared measure then has a consequence a private one does not. When many institutions run the same measure on similar portfolios, a volatility spike tightens all their constraints at once; each firm's sell decision is individually correct and jointly destabilizing, and the selling raises measured volatility, tightening the constraint further. Chapter 16 §16.5 supplies the balance-sheet version of that loop; §26.6 supplies its pricing consequence. The homogenization is not a defect of the mathematics but a property of any measure adopted widely enough to coordinate, and it would follow just as forcefully from a universally adopted expected shortfall.

***

## 26.6 Risk Constraints as a Pricing Mechanism

The chapter so far has treated risk measurement as something a firm does to itself. This section is the chapter's argument, and it reverses the direction: **the risk measure is a price-setting mechanism**, because it determines how much of a claim the levered sector can hold.

Start with the arithmetic. Suppose a dealer holds equity equal to the value at risk of its assets — a good approximation of what a VaR-based capital rule, an internal economic-capital regime, and a prime broker's margin schedule all do. Write $$v$$ for VaR per dollar of assets. Then $$E = vA$$, and leverage is

$$
L = \frac{A}{E} = \frac{1}{v}.
$$

Leverage is the reciprocal of measured risk. Put numbers on it: at a ten-day, 99 percent horizon, a daily volatility of one percent gives $$v = 7.36$$ percent and permits leverage of 13.6 times. Let measured volatility fall to six-tenths of a percent, an ordinary bull market, and $$v$$ falls to 4.41 percent while permitted leverage rises to 22.7 times. Nothing about the assets changed. Capacity to hold them rose by two-thirds because a rolling window of returns got quieter.

Tobias Adrian and Hyun Song Shin's contribution was to observe that this is not a curiosity but the operating logic of the dealer sector, and that it runs as a loop. Measured risk falls; the constraint loosens; balance sheets expand; the expansion is spent buying claims; prices rise; the rise, on a volatility measure estimated from realized returns, *reduces* measured risk further. Leverage and balance-sheet size move together and both move procyclically — the opposite of a passive investor with a fixed portfolio, whose leverage mechanically falls as asset values rise. It runs equally hard in reverse, and in reverse it is Chapter 16 §16.5's margin spiral. That section is this book's canonical statement of the mechanism; what this chapter adds is its **regulatory face**. A haircut set by a lender and a capital charge set by a supervisor are the same object seen from different sides of the desk, and both are calibrated to a risk measure that is low precisely when risk is building. Section 16.5 noted that any requirement calibrated to current measured risk is mechanically procyclical, and that making it countercyclical means making it discretionary. This is the measurement statement of that problem: the procyclicality is not a flaw in the calibration, it is the calibration.

![Figure 26.4: Risk constraints as a pricing mechanism](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-b8dba1881e536aa945391768c99fbadd08ff0a5b%2Ffig_26_04_risk_constraints_as_a_pricing_mechanism.png?alt=media)

**Figure 26.4: Risk constraints as a pricing mechanism.** Panel (a) is Adrian and Shin's own picture, drawn from the Financial Accounts' broker-dealer table: quarterly growth in the sector's balance sheet against quarterly growth in its leverage, 225 quarters since 1970. The relation is positive — a slope of 0.78 and a correlation of 0.86 — and the sign is the whole finding. A passive investor holding a fixed portfolio produces the opposite: the assets appreciate, equity rises one for one with them, and leverage falls. A sector whose leverage rises as its balance sheet expands is a sector managing to a constraint rather than to a portfolio, and the 2007-2009 quarters, marked in red, are the extreme of both directions. Panel (b) is the section's arithmetic in the same units. The orange line is the reciprocal of a ten-day, 99 percent value at risk computed on the market portfolio at each quarter's realized volatility — the leverage a VaR-based rule would permit, drawn as a rolling year over the quarterly series behind it. It is capacity, not prediction, and the panel is honest about the difference: quarter to quarter the two changes correlate at about −0.07, and through the decade to 2007 they move in opposite directions, actual leverage climbing to 47 times while the permitted line fell. Two reasons, both worth stating. A VaR model is estimated on a window of days or weeks, not on a quarter of realized volatility. And the Z.1's reporting perimeter moved over the same decades as dealers shifted activity into holding companies, the caveat Figure 19.6 carries. What the constraint sets is a shadow price on balance-sheet space, and a shadow price is not a level of leverage — it is the thing that makes $$\Lambda\_t$$ move in Chapter 19 §19.5's intermediary discount factor. *Source: Financial Accounts of the United States (Z.1), table L.130, through the FRED mirror; realized market volatility computed from Kenneth French's daily research factors. NBER recession dates. Author's calculations.*

The unlevered version of the same loop is **volatility targeting**. A volatility-target fund, a risk-parity portfolio, or a managed-futures program sizes each position inversely to its recent realized volatility, so that forecast portfolio volatility stays fixed. That is a mandate rather than a lender's constraint, but the trading rule is identical: buy as measured risk falls, sell as it rises, without reference to price. Estimates of the assets run to such rules reach into the hundreds of billions of dollars, and two episodes illustrate the consequence without settling its size. On February 5, 2018, a single-session doubling of the VIX destroyed the inverse-volatility exchange-traded complex — one prominent note lost nearly all its value and was wound up within days — and hedging demand from products referencing volatility fed back into the volatility they referenced, which is Chapter 8 §8.6's dealer gamma with a large sign. In March 2020 the same rules produced synchronized deleveraging across risk-parity and volatility-target mandates in the fortnight dealers were withdrawing from the Treasury market, which is Chapter 19's opening episode. These flows are real, mechanical, and one amplifier among several; attributing a given day's move to them is not something the data support.

The pricing consequence connects this chapter to Part IV, and the connection is exact. Chapter 19 §19.5 writes the intermediary's stochastic discount factor as $$m\_{t+1} = \delta \Lambda\_{t+1}/\Lambda\_t$$, where $$\Lambda\_t$$ is the marginal value of a dollar of the sector's equity capital, and states that risk premia are decreasing and convex in the sector's capital ratio $$\eta\_t$$. This section supplies the object that makes $$\Lambda\_t$$ move. **The risk constraint is the source of that shadow value.** When measured volatility rises, $$v$$ rises, the constraint binds harder, and every claim the sector must hold requires a higher expected return — not because anything was learned about its cash flows, but because the shadow price of the balance-sheet space it occupies went up. The intermediary SDF is the pricing shadow of the risk measure, and He-Krishnamurthy's nonlinearity is the nonlinearity of a constraint that is irrelevant when slack and dominant when tight.

Chapter 20 completes the translation. Demand-system asset pricing writes each holder's demand as a function of price and characteristics, with the coefficient vector $$\theta\_h$$ distinguishing an index fund's demand curve from an insurer's, and derives prices from market clearing. A risk limit is a statement about the *slope* of such a curve: an institution whose VaR budget is exhausted has a demand for additional risk that is perfectly inelastic in price, so a fall in price that raises expected return produces no buying from it at all. Aggregate elasticity $$\zeta$$ falls as more holders reach their limits, and the price multiplier $$\mathcal{M} = 1/\zeta$$ rises accordingly — the same dollar of selling moves prices further in exactly the states where selling is happening. Risk management, viewed from Chapter 20, is not a control system layered on top of the market. It is one of the things that determines the shape of the demand curves the market clears against, and therefore one of the things that determines prices.

***

## 26.7 Close: Between the Firm's Claims and the System's Stability

Risk management sits between the two halves of this book's title. Part V's firms design and issue claims, and the hedging decision is part of that design: a fixed-price forward sold to a customer is a claim, the futures position offsetting it is a claim, and the treasury's problem is that the two are financed on different terms even when they offset perfectly in present value. Part IV's holders are constrained, and their constraints are risk measures — a value at risk, a haircut, a capital charge, a volatility target — so how risk is measured is not separable from what prices are.

Three claims survive. Hedging matters because Modigliani-Miller's assumptions fail, and the failure that matters most is that a firm's investment depends on its own cash flow. Measurement is a modeling choice with a coherence property attached, and the measure the industry standardized on lacks it. And a risk constraint is not a passive observer of the market it constrains: it makes risk-bearing capacity a function of measured risk, which makes the system's capacity procyclical and turns a private prudential rule into a public pricing mechanism.

That last point is where the book has been heading. Chapter 27 asks what an economy gets from a financial system built this way, and what the prices such a system produces can and cannot tell us about the real economy underneath them.

***

## Elsewhere in the Series

* **Regulatory architecture** — *International Finance*, **Chapter 25**: Basel's evolution, the supervisory stress-testing regime as an institution, macroprudential policy and its politics, and the international coordination problem. Section 26.4 keeps the measurement question and the scenario-design critique; the architecture is there.
* **Derivatives markets as markets** — *International Finance*, **Chapter 13** (rates, credit, equity, commodity and structured products; central clearing, initial and variation margin, and the February 2018 short-volatility unwind) and **Chapter 11** (FX). Section 26.2 owns the practice of running a hedged book; the market structure in which it is run is there.
* **What the models did in the crisis** — the companion volume on 2008, which owns the narrative of model failure under stress, the epistemics of the ratings built on those models, and the institutional history. Section 26.5 keeps only the general statement of model risk and its governance.
* Within this book: **Chapter 8** owns option pricing, the Greeks (Box 8.2), the cost of carry, and demand-based option pricing; **Chapter 10 §10.8** owns the tranche loss distribution and the correlation parameter §26.5 cites; **Chapter 16 §16.5** is the canonical statement of constrained capital, fire sales and margin spirals, of which §26.6 is the regulatory face; **Chapter 19 §19.5** owns the intermediary stochastic discount factor whose shadow §26.6 measures; **Chapter 20 §§20.2-20.3** owns the demand system and the price multiplier in which a risk limit is a demand-curve slope; **Chapter 23 §23.1** owns the irrelevance baseline §26.1 starts from; **Chapter 24 §24.2** owns managerial risk aversion; **Chapter 25 §25.1** owns the internal-funds fact that makes Froot-Scharfstein-Stein the operative theory; **Chapter 7 §7.6** owns performativity, of which Box 26.1 is an instance.

***

## Summary

1. **Metallgesellschaft is the standing case for the difference between an economic hedge and a survivable one.** MGRM offset roughly 160 million barrels of fixed-price delivery commitments with a rolling stack of front-month futures; falling prices and a shift into contango produced cash margin calls against an unrealized, unfinanceable gain on the delivery contracts, and the parent liquidated at a loss reported at around $1.3 billion. Culp and Miller argued the hedge was sound and the liquidation was the error; Mello and Parsons argued the hedge ratio was wrong and the funding requirement was unsustainable. The funding-liquidity mechanism is Chapter 16 §16.5's.
2. **Hedging is irrelevant under Modigliani-Miller's assumptions**, for the same reason capital structure is: a fairly priced forward has zero net present value, and shareholders can hedge on their own account.
3. **Smith and Stulz's four frictions are why firms hedge anyway** — convex taxes, deadweight costs of distress, costly external finance, and managerial risk aversion. Froot, Scharfstein and Stein make the third the operative one: hedging matches the supply of internal funds to the demand for them, which matters because internal funds finance most investment (Chapter 25).
4. **What firms actually do is partial and short-horizon.** Over half of large nonfinancial firms use derivatives and small firms almost none; currency exposure is hedged most, then rates, then commodities; programs typically run a year or less; hedge ratios are fractions rather than one.
5. **A delta hedge converts a directional bet into a volatility bet.** Table 26.1's weekly rebalancing of a short call buys after rises and sells after falls, and loses 13,026 dollars over five weeks because the path's realized volatility was 23.4 percent against 20 percent implied. The governing approximation is $$\tfrac{1}{2}\Gamma\[\sigma\_{\text{imp}}^2S^2\Delta t - (\Delta S)^2]$$, with a weekly break-even move of $2.77 at the benchmark parameters.
6. **A maturity-mismatched hedge replaces price risk with basis risk and funding risk.** A one-for-one front-month stack against a ten-year obligation over-hedges in variance terms and concentrates the entire cash-flow consequence in the contract that margins daily, while making the roll's cost a bet on the slope of the curve.
7. **Value at risk is the quantile of the loss distribution and needs three parameters to be readable**: horizon, confidence level, distribution. A $100 million portfolio at one percent daily volatility has a one-day 99 percent normal VaR of $2.33 million and a ten-day VaR of $7.36 million under square-root-of-time scaling, a rule that fails in exactly the episodes the number is for.
8. **VaR is silent about the tail beyond it and is not coherent.** Two independent bonds, each defaulting with probability 0.04, each have a 95 percent VaR of zero; the pair has a 95 percent VaR of 100, because $$0.96^2 = 0.9216$$ falls below the confidence level. Expected shortfall — 80 each, 103.2 for the pair against a sum of 160 — is subadditive and gives the sensible answer.
9. **Expected shortfall at 97.5 percent has almost the same normal multiplier as VaR at 99 percent** (2.338 against 2.326), which is why the measure could be switched without recalibrating capital — and it is far more sensitive to tail shape, rising 38 percent under a Student-$$t$$ with four degrees of freedom where VaR rises 14 percent.
10. **Backtesting is arithmetic and has low power.** A correct 99 percent model produces 2.5 exceptions in 250 days; four or fewer occurs 89 percent of the time and ten or more about once in twenty thousand. Clustering of exceptions is the pattern that matters and a count cannot see it.
11. **Stress tests trade a probability for a scenario**, and the scenario is a model with its judgment hidden. Reverse stress testing — asking what would exhaust the firm's capital — is the more informative direction and is the exercise that would have caught MGRM.
12. **Model risk is mostly parameter risk, and mostly correlation.** Chapter 10 §10.8's senior tranche moves from about one percent expected loss to five and a half on a change in one unobservable correlation parameter. The governance response — independent validation, benchmarking, documented approved ranges — addresses failures that are organizational rather than mathematical.
13. **A shared risk measure is market infrastructure** (Box 26.1). VaR moved from one firm's internal report to a published methodology to a capital rule to a common language in about a decade, and a measure everyone runs tightens for everyone at once. The homogenization would follow from any widely adopted measure, expected shortfall included.
14. **Risk constraints price claims.** If equity equals value at risk, leverage is the reciprocal of measured risk per dollar of assets: 13.6 times at one percent daily volatility, 22.7 times at six-tenths of a percent. Adrian and Shin's loop — measured risk falls, balance sheets expand, prices rise, measured risk falls further — is Chapter 16 §16.5's margin spiral seen from the regulator's side, and volatility-targeting mandates run the same rule without leverage.
15. **The risk constraint is what makes the intermediary's shadow value of capital move.** It supplies the mechanism behind Chapter 19 §19.5's $$\Lambda\_t$$ and, in Chapter 20's language, is a statement about the slope of a demand curve: a holder at its limit is perfectly inelastic, aggregate elasticity $$\zeta$$ falls as limits bind, and the price multiplier $$\mathcal{M} = 1/\zeta$$ rises in exactly the states where selling is happening.

***

## Key Terms

* **Hedge**: A position taken to offset an existing exposure. An *economic* hedge offsets in present value; a *survivable* hedge also has intermediate cash flows the firm can finance, and the two differ whenever the offsetting gain cannot be borrowed against
* **Delta hedge**: Holding $$\Delta$$ units of the underlying against an option position so that the combination is locally insensitive to the underlying's price; requires rebalancing because $$\Delta$$ moves with the price and with time
* **Rolling stack**: Hedging a long-dated obligation with near-dated contracts and rolling the whole position forward at each expiry; introduces exposure to the slope of the forward curve and concentrates margin flows in the most liquid, most volatile contract
* **Basis risk**: The residual risk that the hedging instrument and the exposure do not move together — because of maturity, location, grade, or contract specification
* **Value at risk (VaR)**: The loss threshold not exceeded with a stated probability over a stated horizon; a quantile of the loss distribution, not a worst case, and silent about the tail beyond it
* **Expected shortfall (ES)**: The mean loss conditional on exceeding the VaR threshold; the coherent alternative, and far more sensitive to tail shape
* **Subadditivity**: The requirement that the risk of a combined portfolio never exceed the sum of its parts' risks, so that diversification never appears harmful and desk limits can be aggregated. Expected shortfall satisfies it; VaR does not
* **Coherent risk measure**: A measure satisfying monotonicity, translation invariance, positive homogeneity, and subadditivity (Artzner, Delbaen, Eber and Heath)
* **Backtesting**: Counting realized losses that exceeded the model's own VaR and comparing the count with the model's prediction; low-powered at high confidence levels, and blind to clustering
* **Stress test**: Revaluation of a portfolio under a specified scenario rather than a specified probability; *reverse* stress testing instead solves for the scenario that would exhaust the firm's capital
* **Model risk**: Loss arising from using a model outside what it can support — wrong model, parameters estimated on data lacking the relevant states, or use outside the fitted range
* **Model validation**: Independent review of a model's conceptual soundness, implementation, and approved input range, staffed and reporting separately from the desk that uses it
* **Procyclicality**: The property of a constraint calibrated to current measured risk that it loosens in booms and tightens in busts, expanding risk-bearing capacity when risk is building and contracting it when risk materializes
* **Volatility targeting**: A mandate that sizes positions inversely to recent realized volatility so that forecast portfolio volatility stays fixed; produces the same buy-high, sell-low trading rule as a leverage constraint, without leverage

***

## Readings

### Required

* Smith, C. W. and R. M. Stulz (1985). "The Determinants of Firms' Hedging Policies." *Journal of Financial and Quantitative Analysis* 20(4): 391-405. *The paper that made corporate risk management a question in the Modigliani-Miller tradition rather than a treasury practice. Read it for the structure of the argument — irrelevance first, then the list of frictions that break it — and notice that the tax argument, which comes first in the paper, is the smallest of the four in every subsequent measurement.*
* Froot, K. A., D. S. Scharfstein and J. C. Stein (1993). "Risk Management: Coordinating Corporate Investment and Financing Policies." *Journal of Finance* 48(5): 1629-1658. *The version of the theory with the sharpest empirical content. The object being smoothed is not income but the match between internal funds and investment opportunities, which yields the prediction that the optimal hedge depends on the correlation between the risk and the value of the firm's projects — and can be far from full coverage.*

### Recommended

* Artzner, P., F. Delbaen, J.-M. Eber and D. Heath (1999). "Coherent Measures of Risk." *Mathematical Finance* 9(3): 203-228. *The axiomatic paper behind §26.3. It is short, and the subadditivity axiom is worth reading in the authors' own framing as a statement about whether a firm can decentralize risk limits at all.*
* Adrian, T. and H. S. Shin (2014). "Procyclical Leverage and Value-at-Risk." *Review of Financial Studies* 27(2): 373-403. *The paper §26.6 is built on. The empirical fact — leverage and balance-sheet size move together for dealers, and in the opposite direction for passive investors — is visible in a scatter plot, and the model behind it is the VaR constraint of this chapter written as an equilibrium.*
* Shin, H. S. (2003). "Disclosures and Asset Returns." *Econometrica* 71(1): 105-133. *The earlier statement of §26.6's mechanism by one half of the Adrian-Shin pair: what a firm is required to report changes what it holds, and therefore changes the return on what it holds.*
* Morris, S. and H. S. Shin (2008). "Financial Regulation in a System Context." *Brookings Papers on Economic Activity*, Fall 2008. *Risk models as a source of procyclicality rather than a measurement of it — the system-level reading of §26.6, written while the mechanism was operating.*
* Grossman, S. J. and M. H. Miller (1988). "Liquidity and Market Structure." *Journal of Finance* 43(3): 617-633. *The supply side of the spiral in §26.6. Liquidity is produced by market makers who must be paid to carry inventory, which is why a simultaneous tightening of everyone's risk limits raises the price of immediacy exactly when it is most demanded; connects to Chapters 11 and 16.*
* Hanson, S. G., A. K. Kashyap and J. C. Stein (2011). "A Macroprudential Approach to Financial Regulation." *Journal of Economic Perspectives* 25(1): 3-28. *The survey statement of why capital and risk rules calibrated firm by firm can be destabilizing in aggregate; the policy face of the pricing mechanism §26.6 describes.*
* Culp, C. L. and M. H. Miller (1995). "Metallgesellschaft and the Economics of Synthetic Storage." *Journal of Applied Corporate Finance* 7(4): 62-76. *The defense of MGRM's program, and the more instructive half of the debate to read first because it forces the reader to see what the hedge was actually doing.*
* Mello, A. S. and J. E. Parsons (1995). "Maturity Structure of a Hedge Matters: Lessons from the Metallgesellschaft Debacle." *Journal of Applied Corporate Finance* 8(1): 106-120. *The reply, and the source of §26.2's point about hedge ratios below one. Read the two together; the disagreement is more useful than either verdict.*
* Miller, M. H. (1996). "The Social Costs of Some Recent Derivatives Disasters." *Pacific-Basin Finance Journal* 4(2-3): 113-127. *Miller generalizing a year after the Metallgesellschaft defense of §26.2: which of the celebrated derivatives losses imposed costs on anyone other than the losing firm's own claimants, and which merely transferred wealth.*
* MacKenzie, D. (2010). "Models as Coordination Devices." In M. Akrich, Y. Barthe, F. Muniesa and P. Mustar (eds.), *Débordements: Mélanges offerts à Michel Callon*. Paris: Presses des Mines, 299-302. *The argument behind Box 26.1, in general form: a model in wide use is valuable as a shared language and not only as a representation, which is why one known to be inaccurate can be retained rather than replaced. Extends the performativity account of Chapter 7 §7.6 that the box is an instance of, and is the one sociology-of-finance entry this chapter's reader needs.*
* Bernstein, P. L. (1996). *Against the Gods: The Remarkable Story of Risk*. Wiley. *The long history of measuring risk, from which Box 26.1's claim borrows its force: quantification arrived late, it arrived as a set of conventions, and each convention reorganized the practice that adopted it.*
* Jorion, P. *Value at Risk: The New Benchmark for Managing Financial Risk*, 3rd edition. McGraw-Hill. *The technical reference for everything in §26.3 that this chapter states without deriving — mapping positions to risk factors, the delta-gamma approximation for optioned books, and the full apparatus of backtesting tests beyond the exception count.*

***

## Discussion Questions

1. **Did Metallgesellschaft die of a bad hedge or of a funding constraint?** State each side's strongest version. Then answer a harder question: if you accept Culp and Miller's claim that the position was economically sound and the liquidation was the mistake, does that make the risk-management failure smaller or larger? A hedge that is correct in present value and unfinanceable in cash is a design choice, and someone made it.
2. **Do shared risk models stabilize or synchronize?** A common measure lets supervisors compare institutions, lets counterparties price collateral, and lets boards read a number they could not otherwise read. It also makes every institution's constraint tighten on the same day. Is there a risk measure that could be adopted universally without producing correlated selling, or is the homogenization inherent in the coordination? If the latter, what follows for the design of capital rules?
3. **Should a mining company hedge its output?** Its shareholders may hold it *for* the commodity exposure, and can hedge on their own account. Apply the four Smith-Stulz frictions and the Froot-Scharfstein-Stein correlation condition, and reach a recommendation. Then consider whether your answer changes if the firm is highly levered, and say which friction did the work.
4. **Expected shortfall is coherent and VaR is not, so why did it take twenty years to switch?** Consider backtesting (an exception count has an obvious test; a conditional mean does not), the elicitability literature, the cost of rebuilding systems, and the fact that a measure everyone already uses has coordination value that a better measure does not yet have. Which of these is a good reason?
5. **If risk constraints price claims, is a capital requirement a monetary policy instrument?** Section 26.6 argues that a haircut and a capital charge are the same object and that both change how much of a claim the levered sector can hold. Trace what a countercyclical capital buffer does to Chapter 20's aggregate demand elasticity, and say whether a supervisor setting that buffer is doing something different in kind from a central bank buying the same claims.

***

## Problems

**Problem 1 — VaR and expected shortfall, parametric.** A trading book is worth 400 million dollars with a daily return standard deviation of 1.4 percent and mean zero. (a) Compute the one-day 95, 97.5 and 99 percent normal VaR. (b) Compute the corresponding expected shortfalls using $$\mathrm{ES}\_\alpha = \sigma\phi(z \_\alpha)/(1-\alpha)$$. (c) Verify that the 97.5 percent expected shortfall is close to the 99 percent VaR, and state in one sentence what regulatory convenience that coincidence buys. (d) Scale the 99 percent number to a ten-day horizon and state the assumption you used and one reason it fails.

**Problem 2 — The subadditivity counterexample, generalized.** Take $$n$$ independent defaultable bonds, each with face value 100, zero recovery, and default probability $$p$$. (a) For $$n = 2$$ and $$p = 0.04$$, reproduce §26.3's result that the 95 percent VaR of each bond is zero and of the pair is 100. (b) Find the largest $$p$$ for which the 95 percent VaR of a single bond is still zero, and the smallest $$n$$ at which the 95 percent VaR of the equally weighted portfolio *per bond* falls back below 100. (c) Compute the 95 percent expected shortfall for $$n = 1, 2, 5$$ and confirm that it is subadditive at each step. (d) Explain in two sentences why the failure is not a small-sample artifact but a property of quantiles of discontinuous loss distributions.

**Problem 3 — Delta-hedge rebalancing.** Using the benchmark parameters of Table 26.1 ($$K = 100$$, $$r = 5$$ percent, $$\sigma\_{\text{imp}} = 20$$ percent, one year to expiry at week 0) and a short position in calls on 100,000 shares: (a) recompute the table for the alternative path 100.00, 100.80, 101.40, 100.60, 101.20, 100.90, and report the total shares traded over the five weeks against Table 26.1's. (b) Compute the path's annualized realized volatility and predict, before computing it, whether the hedged position gains or loses. (c) Compute the hedged profit and loss and check your prediction. (d) Using the approximation $$\tfrac{1}{2}\Gamma\[\sigma\_{\text{imp}}^2S^2\Delta t - (\Delta S)^2]$$ with the week-0 gamma held fixed, produce an estimate of the total and explain the two sources of the discrepancy with your exact answer.

***

## Selected Solutions

*Solutions to Problems 1 and 2 follow. Solutions to the remainder are in the instructor materials.*

**Problem 1.**

(a) The book's daily standard deviation in currency terms is $$0.014 \times 400 = 5.6$$ million. Parametric VaR is the normal quantile times that figure, and the three quantiles are 1.645, 1.960 and 2.326, so the one-day VaR is **9.21 million**, **10.98 million** and **13.03 million** dollars at 95, 97.5 and 99 percent. Nothing about the portfolio changed across the three numbers. Only the question did, which is why §26.3 insists that a VaR quoted without its confidence level is unreadable.

(b) Expected shortfall replaces the quantile $$z$$ with the multiplier $$\phi(z)/(1-\alpha)$$: $$0.1031/0.05 = 2.062$$ at 95 percent, $$0.0584/0.025 = 2.338$$ at 97.5 percent, and $$0.0267/0.01 = 2.665$$ at 99 percent. Multiplying by 5.6 million gives expected shortfalls of **11.55 million**, **13.09 million** and **14.93 million** dollars. Each exceeds the VaR at the same level, as it must: the mean of a tail cannot be smaller than the point at which the tail begins.

(c) The 97.5 percent expected shortfall of 13.09 million sits within half a percent of the 99 percent VaR of 13.03 million, because under normality the multipliers 2.338 and 2.326 differ by that much. The convenience is that a supervisor can replace a 99 percent VaR requirement with a 97.5 percent expected shortfall requirement and leave the *level* of required capital essentially untouched, which separates the argument about which measure is right from the argument about how much capital is enough. That is the Basel Committee's calibration, and §26.3 notes that the coincidence is what made the switch politically feasible.

(d) Under the square-root-of-time rule the ten-day figure is $$13.03 \times 3.162 = 41.20$$ million dollars. The assumption is that daily returns are independent and identically distributed, so that variance is additive across days. It fails because volatility is persistent, and it is persistent in exactly the ten-day windows a regulatory capital number is meant to cover, so the scaled figure understates the loss in a stressed fortnight. A second and quieter failure is that the rule holds the position fixed for ten days, which is false for a trading book and false in the worst direction for one that is being forced to sell into the same move.

**Problem 2.**

(a) One bond loses 100 with probability 0.04 and nothing otherwise, so $$\Pr(\mathcal{L} \le 0) = 0.96 \ge 0.95$$ and the 95 percent VaR is **zero**. For the pair, $$\Pr(\mathcal{L} = 0) = 0.96^2 = 0.9216$$ falls below 0.95, so the quantile moves up to the next atom: one default costs 100 and $$\Pr(\mathcal{L} \le 100) = 0.9984$$, so the 95 percent VaR of the pair is **100** against a sum of the parts of zero.

(b) The single-bond VaR stays at zero as long as $$\Pr(\mathcal{L} \le 0) = 1 - p \ge 0.95$$, so the largest such default probability is **0.05** exactly — and the boundary is a cliff rather than a slope, since at $$p = 0.0501$$ the same bond's VaR is 100. For the portfolio, let $$D$$ be the number of defaults, binomial in $$n$$ and $$p$$, and let $$q\_n$$ be the smallest integer with $$\Pr(D \le q\_n) \ge 0.95$$. The portfolio's VaR is $$100q\_n$$ and the figure per bond is $$100q\_n/n$$. At $$p = 0.04$$ that sequence runs 0, 50, 33.3, 25, 20 for $$n = 1$$ through 5. It never reaches 100 at all: the pair's 100 is a portfolio figure, and per bond it is already only 50, so the answer as posed is $$n = 2$$ — the very step that produces the violation is also the step at which diversification starts working per unit of exposure. From there the per-bond figure declines monotonically toward $$100p = 4$$, the expected loss per bond, as the law of large numbers takes hold.

(c) Expected shortfall at 95 percent is the average of the worst five percent of outcomes, taking a fraction of an atom where an atom straddles the cutoff. For one bond the worst five percent is the four percent that defaults plus one percent of the surviving mass, $$(0.04 \times 100 + 0.01 \times 0)/0.05 = 80$$. For the pair it is all 0.0016 of the double default and 0.0484 of the single, $$(0.0016 \times 200 + 0.0484 \times 100)/0.05 = 103.2$$. The same construction at $$n = 5$$ gives 130.75, or 26.15 per bond.

**Table 26.2: VaR and expected shortfall on independent bonds in Problem 2** (face 100, zero recovery, $$p = 0.04$$)

| Bonds | 95 percent VaR | Sum of the parts | 95 percent ES | Sum of the parts |
| ----- | -------------- | ---------------- | ------------- | ---------------- |
| 1     | 0              | 0                | 80.00         | 80               |
| 2     | 100            | 0                | 103.20        | 160              |
| 5     | 100            | 0                | 130.75        | 400              |

*Source: Author's calculation from the problem's assumptions.*

VaR exceeds the sum of its parts at every portfolio above one bond, which is the subadditivity failure. Expected shortfall falls short of the sum by a widening margin — 80 against 80, 103.20 against 160, 130.75 against 400 — which is diversification appearing in the measure as it should.

(d) The failure is a property of the quantile, not of the sample. A quantile of a discontinuous loss distribution is the location of an atom, and adding an independent position redistributes probability mass across the confidence level, which can push the quantile to a strictly higher atom while every moment of the loss distribution behaves impeccably. Nothing here is noisy: the figures above are exact, not estimated. And the construction can be reproduced at any $$n$$ by choosing $$p$$ so that the tail atoms straddle the confidence level, which is generic for portfolios of skewed, lumpy exposures — credit, insurance, catastrophe covers, digital options, anything that pays a fixed amount in a rare state. That is why §26.3's corollary is organizational rather than statistical: a firm running VaR limits desk by desk cannot add them up, and no quantity of data repairs it.

***

## Data Exercise: Does the VaR Model Hold Up?

**Part A — A rolling parametric VaR and its exceedances (free data: FRED, or Kenneth French's data library).** Download the daily S\&P 500 series `SP500` from FRED, or — for a much longer sample, which this exercise wants — construct the daily market return from Kenneth French's daily **Fama/French 3 Factors** as `Mkt-RF` + `RF`.

1. On each day $$t$$, estimate $$\sigma\_t$$ as the standard deviation of the previous 250 daily returns, and compute a one-day 99 percent parametric VaR as $$2.326\sigma\_t$$ applied to a notional of $100 million. Plot the VaR series from 1990 to the present alongside realized daily losses.
2. Count **exceedances**: days on which the realized loss exceeded that day's VaR. Report the total count, the count as a fraction of days, and the expected fraction. Then report the count year by year and compare each year with the Basel traffic-light zones for 250 observations (green 0-4, yellow 5-9, red 10 or more).
3. The count is not the interesting statistic. For each exceedance, record the number of trading days since the previous one, and plot the histogram of those gaps against the geometric distribution a correct and independent model implies. State what the clustering does to a firm that sized its positions on this model.
4. Repeat step 1 with an exponentially weighted volatility estimate (decay factor 0.94, the published RiskMetrics value) in place of the equally weighted 250-day window. Report both exceedance counts and both clustering pictures, and say which failure each specification fixes and which it does not.
5. Replace the normal quantile with the empirical 1 percent quantile of the trailing 250 returns — historical simulation — and repeat. Report all three exceedance counts in one table and write one paragraph on what the comparison establishes about where the parametric model's error actually lies.

**Part B — The volatility risk premium (reuses Chapter 8's data).** Using the VIX series `VIXCLS` from FRED and the daily market return from Part A:

6. Compute, for each month, the average VIX and the realized volatility of the *following* month. Regress the exceedance indicator from Part A step 2 on the level of VIX at the start of each month and report whether VaR breaches are predictable from a forward-looking volatility measure the model does not use. Interpret a significant coefficient as a statement about the trailing-window specification rather than about the market.

**Part C ★ — Bank VaR disclosures (free data: SEC EDGAR).** Large bank holding companies disclose trading-book VaR in the market-risk section of their 10-K filings, usually as an average, high, low and period-end figure by risk category, together with a count of backtesting exceptions for the year.

7. For four large dealers, hand-collect average daily 99 percent trading VaR and the disclosed exception count for each year available. Tabulate VaR scaled by common equity and plot it against the year.
8. Compare the four series through 2008-2009 and 2020. Do the disclosed numbers rise before, during, or after the episodes? Section 26.6 predicts that a measure estimated from trailing returns rises *after* risk has materialized. State whether the disclosures support that, and name three reasons — position changes, model changes, and disclosure discretion — why this test is weaker than it looks.