---
title: "Limits and Capital Allocation"
book: "The Desk and the Firm"
subject: quant
language: en
chapter: 7
exercises: 8
source: https://one-course.com/books/quant/16/en/chapter/7-limits-and-capital-allocation
---

# Chapter 7 — Limits and Capital Allocation

A firm holds $213 million of risk capital against six strategies, the expected shortfall of its annual P&L in its worst one year in a hundred. Measured one by one, the strategies would need $394 million, and three of them would earn less than the firm’s 15% cost of capital. Measured by what each contributes to the firm’s worst years, two still do; a third clears it almost three times over; and the trend-following book, which alone would need $69 million, lowers the firm’s capital by $21 million, because it makes money in the years the others lose. Who is charged for the tail decides which strategies the firm keeps.

## 7.1 What a limit framework is for

A risk limit (One Quant Book 6, chapter 29) caps one measure of one exposure. A firm has thousands of them; what holds them together is a framework.

**Definition 7.1 (Limit framework).**

A firm’s *limit framework* is the set of its risk limits and the rules that relate them: the tree of levels (firm, business, desk, strategy, trader) at which limits are set, the measures limited at each level, how a level’s limits constrain those below it, who may set, change and temporarily raise each limit, and what happens when one is breached.

A framework allocates the firm’s [risk appetite](https://one-course.com/books/quant/16/en/chapter/6-the-head-of-desks-job#def-fm-the-head-of-desks-job-appetite) (chapter 6) down the risk hierarchy; its limits are the [risk appetite statement](https://one-course.com/books/quant/16/en/chapter/6-the-head-of-desks-job#def-fm-the-head-of-desks-job-appetite) in numbers. It has three jobs: to keep the firm inside its appetite, to make every desk’s risk visible against an agreed number, and to force a conversation before a limit is crossed, not after. A framework that only records breaches does the second job; one whose limits are raised whenever they are reached does none (chapter 12).

![A limit framework as a tree: the firm’s appetite is cascaded to desks and strategies, each level limiting the measures that fit its risks. Model: firm.limitalloc.Node, check.](https://one-course.com/images/onecourse/chapters/quant-16/fm-limits-and-capital-allocation/fig-3c9de9669109.svg)

***Figure 7.1.** A [limit framework](#def-fm-limits-and-capital-allocation-framework) as a tree: the firm’s appetite is cascaded to desks and strategies, each level limiting the measures that fit its risks. Model: `firm.limitalloc.Node`, `check`.*

## 7.2 Measures and levels: notional, sensitivities, VaR, stress, loss

No single measure is enough, because each misses what another sees. Notional and position limits are simple and cannot be gamed by a model, but say nothing about risk; sensitivity limits (delta, vega, DV01, credit spread sensitivity) are precise for small moves and blind to large ones; value at risk and expected shortfall (One Quant Book 6, chapter 21) aggregate across risks but rest on a model and a history; stress losses (chapter 22 of the same book) see scenarios the history has not; loss limits (One Quant Book 11, chapter 27) act only after the fact. A desk carries a few of each, chosen for its risks.

**Definition 7.2 (Concentration limit).**

A *concentration limit* caps an exposure to one name, issuer, sector, counterparty or instrument, or a position as a share of a market’s volume or open interest, whatever the diversification elsewhere: it limits what could not be sold or hedged quickly.

**Method 7.3 (Choosing a desk’s limits).**

1. List the desk’s risks and the moves that would hurt it most; choose one measure that sees each.
2. Set a VaR or ES limit for the aggregate, stress limits for the scenarios the firm’s appetite names, sensitivity limits for the large risks, a loss limit, and [concentration limits](#def-fm-limits-and-capital-allocation-conc) for what cannot be sold in days (chapter 28’s cases show why).
3. Size each limit from the desk’s share of the firm’s appetite (chapter 6), and check that the desk can run its expected business at 60–80% of each.
4. Record who may change each limit and how temporary increases expire.

## 7.3 Utilisation, breaches and temporary increases

**Definition 7.4 (Hard limit, soft limit, limit utilisation, limit breach).**

A *hard limit* is a limit that must not be exceeded: an excess must be cured at once, by reducing the exposure, or escalated for an approved increase. A *soft limit* is a lower threshold whose crossing triggers a warning and a review, not a mandatory action. *Limit utilisation* is the exposure as a share of the limit. A *limit breach* is an exposure above a hard limit, recorded with its size, duration and resolution.

The chapter’s vol-selling strategy has a daily VaR limit of 8 with a soft threshold at 6. After a volatility spike its VaR reaches 7.6: a soft breach and 95% utilisation of the [hard limit](#def-fm-limits-and-capital-allocation-breach). The desk asks for a temporary increase to 12 for sixteen days; utilisation falls to 63% and the position stays ([Figure 7.2](#fig-fm-limits-and-capital-allocation-var)). A temporary increase is a decision to take more risk for a stated time; one that is renewed at each expiry is a permanent increase that nobody approved as one.

![The vol-selling strategy’s daily VaR through a volatility spike, against its soft limit and its hard limit, raised temporarily from 8 to 12 for days 25 to 40. Illustrative path; model fm_limits.var_path.](https://one-course.com/images/onecourse/chapters/quant-16/fm-limits-and-capital-allocation/fig-ef8d895dd36a.svg)

***Figure 7.2.** The vol-selling strategy’s daily VaR through a volatility spike, against its [soft limit](#def-fm-limits-and-capital-allocation-breach) and its [hard limit](#def-fm-limits-and-capital-allocation-breach), raised temporarily from 8 to 12 for days 25 to 40. Illustrative path; model `fm_limits.var_path`.*

The same check runs at every level. At the chapter’s firm on the spike’s first day, when vol selling’s VaR has doubled to 6.4, the six strategies’ daily VaRs add up to $21.5 million against a firm limit of $30 million (72% utilisation), while vol selling alone is in soft breach: the firm is well inside its appetite and one desk is not. A framework that checked only the firm would see nothing; one that checked only desks would never see the day when six desks at 70% add up to a firm breach.

## 7.4 Allocating risk capital

**Definition 7.5 (Economic capital).**

A firm’s *economic capital* is the capital it holds, by its own measure, against unexpected losses: typically the expected shortfall or value at risk of its P&L over a year at a high confidence level, computed on its own model of its risks.

The chapter’s firm (illustrative, $ millions a year) runs six strategies: stat arb, trend following, credit carry, vol selling, macro and market making, with expected P&L of 11, 12, 8, 12, 6 and 10 and volatilities of 25, 30, 20, 22, 20 and 8, correlated at 0.2 except trend. In one year in twenty a crash adds losses of 40, 60 and 110 to stat arb, credit carry and vol selling and a gain of 30 to trend. Its [economic capital](#def-fm-limits-and-capital-allocation-ec), the 99% expected shortfall of annual P&L over 200 000 simulated years, is $212.8 million for an expected P&L of $59.1 million.

Three allocations divide it ([Figure 7.3](#fig-fm-limits-and-capital-allocation-capital)). *Stand-alone*: each strategy’s own expected shortfall; they add up to $393.9 million, far more than the firm’s, because the firm is diversified. *Euler* (One Quant Book 6, chapter 20): each strategy’s expected P&L in the firm’s tail years, with the sign flipped; these add up exactly to the firm’s. *Incremental*: the firm’s expected shortfall minus what it would be without the strategy.

**Proposition 7.6 (Euler contributions of expected shortfall).**

Let the firm’s P&L be $X=\sum_iX_i$ and $\mathrm{ES}_\alpha(X)=-\E[X\mid X\le q_{1-\alpha}(X)]$. For a continuous distribution, the Euler contribution of strategy $i$, $w_i\,\partial\mathrm{ES}_\alpha(\sum_jw_jX_j)/\partial w_i$ at $w=\mathbf 1$, equals $-\E[X_i\mid X\le q_{1-\alpha}(X)]$, and the contributions add up to $\mathrm{ES}_\alpha(X)$.

**Partial proof.** Expected shortfall is positively homogeneous of degree one in the scales $w$, so Euler’s theorem gives $\sum_iw_i\,\partial_i\mathrm{ES}=\mathrm{ES}$. The derivative of the tail-conditional mean with respect to $w_i$ is the tail-conditional mean of $X_i$, because the quantile’s own derivative contributes nothing at the boundary (the density of $X$ at the quantile times a zero-length change of the tail set); the full argument, with the regularity conditions, is Tasche’s (One Quant Book 6, chapter 20). On scenarios, the contribution is the mean of $-X_i$ over the firm’s worst $(1-\alpha)n$ scenarios, and the sum is exactly the firm’s scenario ES. ∎

![Economic capital (99% expected shortfall of annual P&L) allocated to six strategies three ways. Stand-alone allocations add up to $394 million, the Euler contributions to the firm’s $213 million; trend following’s contribution is negative because it gains in the firm’s worst years. Illustrative firm, 200 000 simulated years. Data: fm_limits.allocation.](https://one-course.com/images/onecourse/chapters/quant-16/fm-limits-and-capital-allocation/fig-69e2c8c9c78c.svg)

***Figure 7.3.** [Economic capital](#def-fm-limits-and-capital-allocation-ec) (99% expected shortfall of annual P&L) allocated to six strategies three ways. Stand-alone allocations add up to $394 million, the Euler contributions to the firm’s $213 million; trend following’s contribution is negative because it gains in the firm’s worst years. Illustrative firm, 200 000 simulated years. Data: `fm_limits.allocation`.*

```python
def allocate_capital(scen: np.ndarray, alpha: float = 0.99) -> dict:
    """scen: (n scenarios, m strategies) annual P&L. Stand-alone, Euler and incremental ES allocations."""
    firm = scen.sum(1)
    n = len(firm)
    k = max(1, int(round((1 - alpha) * n)))
    tail = np.argsort(firm)[:k]
    euler = -scen[tail].mean(0)
    stand = np.array([es(scen[:, i], alpha) for i in range(scen.shape[1])])
    total = es(firm, alpha)
    incr = np.array([total - es(firm - scen[:, i], alpha) for i in range(scen.shape[1])])
    return {"standalone": stand, "euler": euler, "incremental": incr, "firm": total}


def raroc(mu, capital):
    return np.asarray(mu, float) / np.asarray(capital, float)


def capital_charge(capital, h_k: float):
    return h_k * np.asarray(capital, float)

```

***Listing 7.1.** Stand-alone, Euler and incremental allocations of expected shortfall on scenarios, and the return on and charge for capital. code/firm/limitalloc/firm_limitalloc.py*

## 7.5 Charging for capital

**Definition 7.7 (Risk-adjusted return on capital, capital charge).**

A strategy’s *risk-adjusted return on capital* (RAROC) is its expected P&L divided by the [economic capital](#def-fm-limits-and-capital-allocation-ec) allocated to it. Its *capital charge* is the hurdle rate (One Quant Book 6, chapter 19) times that capital, deducted from its P&L when its performance is measured and its pay pool sized.

A strategy adds value when its RAROC exceeds the hurdle, that is when its expected P&L exceeds its [capital charge](#def-fm-limits-and-capital-allocation-raroc) ([Figure 7.4](#fig-fm-limits-and-capital-allocation-va)). With a 15% hurdle:

|  | expected | RAROC | value added |
| --- | --- | --- | --- |
| strategy | P&L | stand-alone | Euler | stand-alone | Euler |
| stat arb | 10.9 | 16.7% | 22.8% | 1.12 | 3.74 |
| trend | 12.0 | 17.4% | (negative capital) | 1.66 | 15.24 |
| credit carry | 8.0 | 10.5% | 12.6% | $-3.50$ | $-1.55$ |
| vol selling | 12.1 | 9.8% | 11.1% | $-6.34$ | $-4.20$ |
| macro | 6.0 | 12.6% | 43.3% | $-1.17$ | 3.93 |
| market making | 10.0 | 84.9% | (near zero capital) | 8.24 | 10.02 |

![Expected P&L less a 15% charge on allocated capital. On stand-alone capital three strategies destroy value; on Euler capital two do, credit carry and vol selling, the two that lose most in the firm’s worst years; macro and trend change sign or grow. Data: fm_limits.value_added.](https://one-course.com/images/onecourse/chapters/quant-16/fm-limits-and-capital-allocation/fig-790b1628d22c.svg)

***Figure 7.4.** Expected P&L less a 15% charge on allocated capital. On stand-alone capital three strategies destroy value; on Euler capital two do, credit carry and vol selling, the two that lose most in the firm’s worst years; macro and trend change sign or grow. Data: `fm_limits.value_added`.*

Stand-alone charging penalises diversifiers and flatters strategies whose losses come together; Euler charging is consistent with the firm’s actual capital and rewards what reduces the firm’s tail. It has its own trap: a strategy whose contribution is near zero or negative has an undefined or negative RAROC and a [capital charge](#def-fm-limits-and-capital-allocation-raroc) that is a credit, and a desk paid on it will want more of it than the firm’s tail can use (the contribution changes as the book changes). Firms therefore cap credits, charge a floor, or use the incremental allocation for decisions about entering or leaving a business and Euler for the running charge.

**Method 7.8 (Setting a capital charge).**

1. Compute the firm’s [economic capital](#def-fm-limits-and-capital-allocation-ec) on scenarios that include its crash years, not only its recent history.
2. Allocate it by Euler contributions for the monthly charge; recompute the contributions when positions change materially.
3. Use incremental capital for decisions to add or close a strategy.
4. Charge capital at the hurdle rate in each desk’s P&L before its pay pool is sized (chapter 10), with a floor for strategies whose contribution is near zero.

A firm that re-sizes its strategies to maximise expected P&L less a 15% charge on its whole expected shortfall, each scaled between zero and twice today’s size (`firm.limitalloc.best_scales`), grows four of them to the cap, cuts credit carry to 1.57 times and vol selling to 0.57 times, and raises the firm’s value added from $27.2 million to $59.0 million a year. The cap does most of that work; the useful output is the direction, the two tail-heavy books shrinking while the rest grow.

## 7.6 Tutorial: six strategies and one tail

**Goal.** Check a [limit framework](#def-fm-limits-and-capital-allocation-framework) on one day, allocate a firm’s [economic capital](#def-fm-limits-and-capital-allocation-ec) three ways, and charge for it. **End state:** the table of RAROCs and [Figure 7.4](#fig-fm-limits-and-capital-allocation-va).

1. **Limits.** `fm_limits.framework()` builds the tree; `firm.limitalloc.check(framework(), daily_var_exposures(), 1)` reports the soft breach and the firm’s 72% utilisation; `framework(temp=…)` adds a temporary increase of 4 on vol selling’s VaR limit until day 10.
2. **Scenarios.** `fm_limits.scenarios()` simulates 200 000 years of the six strategies with the crash.
3. **Allocation.** `allocation()` computes the three allocations and the RAROCs ( [Listing 7.1](#lst-fm-limits-and-capital-allocation-alloc) ); check that the Euler contributions add up to the firm’s ES ( [Proposition 7.6](#prop-fm-limits-and-capital-allocation-euler) ).
4. **Charge and scale.** `value_added()` and `scaled()` .

**What to change next.** Remove the crash and see the Euler allocation of trend turn positive; halve vol selling and recompute every strategy’s contribution.

## 7.7 Build: limits and capital

**Purpose.** The firm’s [limit framework](#def-fm-limits-and-capital-allocation-framework) as data and its risk capital as a price: which desk is inside its limits today, and what each strategy costs in capital.

**Interface.** `firm.limitalloc`: `Limit(measure, soft, hard)`, `Node(name, limits, children, temp)`; `check(node, exposures, day)`; `es(pnl, alpha)`, `allocate_capital(scen, alpha)`; `raroc`, `capital_charge`; `best_scales(scen, alpha, h_k, lo, hi)`; `to_riskctl(node)`, the pre-trade part in the shape of One Quant Book 11’s `firm.riskctl` snapshot.

**Rules.** A temporary increase applies only until its last day; a parent’s exposure is the sum of its children’s unless measured directly; Euler contributions add up to the firm’s ES exactly on the same scenarios.

**Acceptance tests.** `code/firm/limitalloc/tests/`: soft and hard statuses, a parent breach, an expiring increase; Euler contributions add up and stand-alone ones exceed the firm’s ES; incremental allocations do not exceed stand-alone ones; the optimiser prefers the better strategy; the export’s shape.

**Stretch.** Non-additive parent measures (a parent’s VaR computed from the joint P&L); an optimiser with the firm’s limits as constraints; a charge floor.

Sources and further reading

- E. Zaik, J. Walter, G. Kelling and C. James, “RAROC at Bank of America: from theory to practice”, *Journal of Applied Corporate Finance* 9(2), 1996.
- Financial Stability Board, *Principles for an Effective Risk Appetite Framework* , 2013.
- One Quant Book 6, chapters 19–21 and 29 (hurdle rate, Euler allocation, VaR and ES, the risk engine’s limits).

## 7.8 Exercises

**Exercise 7.1 ★.**

A strategy’s daily VaR is 7.6 against a [soft limit](#def-fm-limits-and-capital-allocation-breach) of 6 and a [hard limit](#def-fm-limits-and-capital-allocation-breach) of 8. What is its utilisation, and what is its status? What if the [hard limit](#def-fm-limits-and-capital-allocation-breach) is raised temporarily to 12?

**Solution of Exercise 7.1.**

$7.6/8=95\%$ of the [hard limit](#def-fm-limits-and-capital-allocation-breach), above the soft threshold of 6: a soft breach. With the [hard limit](#def-fm-limits-and-capital-allocation-breach) at 12: 63%, still above the soft threshold.

**Exercise 7.2 ★.**

Compute stat arb’s RAROC on stand-alone and on Euler capital, and its value added at a 15% hurdle on each.

**Solution of Exercise 7.2.**

Stand-alone: $10.9/65.5=16.7\%$, value added $10.9-0.15\times65.5=1.12$. Euler: $10.9/48.0=22.8\%$, value added 3.74.

**Exercise 7.3 ★.**

The six strategies’ stand-alone ES add up to $393.9 million and the firm’s is $212.8 million. What is the diversification benefit, as a share of the stand-alone sum?

**Solution of Exercise 7.3.**

$393.9-212.8=181.1$, 46.0% of the stand-alone sum.

**Exercise 7.4 ★★.**

At what hurdle rate would vol selling break even on its Euler capital? And credit carry?

**Solution of Exercise 7.4.**

At its Euler RAROC: 11.1% for vol selling, 12.6% for credit carry.

**Exercise 7.5 ★★.**

Explain why trend following’s Euler contribution is negative, and why a desk charged on it might take more risk than the firm wants.

**Solution of Exercise 7.5.**

It gains in the crash years that make the firm’s tail, so its expected P&L there is positive and its contribution negative. A desk charged a negative amount earns a credit for adding risk; but the contribution changes as it grows (a large trend book dominates the tail itself), so the credit must be capped.

**Exercise 7.6 ★★.**

Six desks each use 70% of a VaR limit of 8 and the firm’s limit is 30. Is the firm in breach if the desks’ VaRs add? If they are correlated at 0.2 and each VaR is a multiple of its volatility?

**Solution of Exercise 7.6.**

Adding: $6\times0.7\times8=33.6>30$, a breach. Correlated at 0.2 with VaR proportional to volatility: $5.6\sqrt{6+30\times0.2}=19.4$, no breach. The firm’s limit must be checked on the measure it is set in.

**Exercise 7.7 ★★★.**

*Coding.* Rerun `fm_limits.allocation` with no crash (`P_CRASH = 0`). What are trend following’s and vol selling’s Euler contributions now?

**Solution of Exercise 7.7.**

Trend following 24.7 and vol selling 20.3 (firm ES 114.7): without the crash, trend is an ordinary volatile strategy and vol selling looks cheap. The allocation is only as good as the scenarios’ tails.

**Exercise 7.8 ★★★.**

*Find the flaw.* “Every desk is inside its limits, so the firm is inside its [risk appetite](https://one-course.com/books/quant/16/en/chapter/6-the-head-of-desks-job#def-fm-the-head-of-desks-job-appetite).”

**Solution of Exercise 7.8.**

Desk limits do not add up to the firm’s appetite unless the framework makes them: desks at 70% can sum to a firm breach, and risks that no desk measures (concentration across desks, a shared crash) appear only at the firm.

## 7.9 Problem: Who Pays for the Tail

**Problem 7.1.**

Weekend problem — who pays for the tail

The firm’s head of risk must decide how to charge six strategies for $213 million of [economic capital](#def-fm-limits-and-capital-allocation-ec), and the vol-selling desk has just asked for a temporary limit increase.

**Part I — The framework.**

1. Define a [limit framework](#def-fm-limits-and-capital-allocation-framework) , and hard and [soft limits](#def-fm-limits-and-capital-allocation-breach) .
2. Give the vol-selling strategy’s utilisation and status at the peak, before and after a temporary increase to 12.
3. Give the firm’s VaR utilisation on the day of the spike.
4. Why must a framework check both the firm and each desk?
5. What turns a temporary increase into an unapproved permanent one?

**Part II — Capital.**

6. Define [economic capital](#def-fm-limits-and-capital-allocation-ec) and give the firm’s, with its expected P&L.
7. Give each strategy’s stand-alone, Euler and incremental capital.
8. State and prove [Proposition 7.6](#prop-fm-limits-and-capital-allocation-euler) ; check that the contributions add up.
9. Why does the stand-alone sum exceed the firm’s capital?
10. Which strategy’s capital changes sign between the stand-alone and Euler allocations, and why?

**Part III — The charge.**

11. Define RAROC and a [capital charge](#def-fm-limits-and-capital-allocation-raroc) .
12. Which strategies destroy value at a 15% hurdle under each allocation?
13. Give each strategy’s value added under both.
14. What goes wrong with a near-zero or negative Euler contribution, and what do firms do about it?
15. Which allocation should decide whether to close a strategy?

**Part IV — The decision.**

16. Give the best scales in $[0,2]$ and the value added before and after.
17. Which part of that result would you trust, and which not?
18. Should the vol-selling desk get its temporary increase?
19. State the *named result* : the hurdle rates at which credit carry and vol selling break even on stand-alone and on Euler capital, and trend’s capital under both.
20. In two sentences, write the head of risk’s recommendation.

**Solution of Problem 7.1.**

1. See Definitions [7.1](#def-fm-limits-and-capital-allocation-framework) and [7.4](#def-fm-limits-and-capital-allocation-breach) .
2. 95% of the [hard limit](#def-fm-limits-and-capital-allocation-breach) , a soft breach; 63% after the increase, still soft.
3. 72% ($21.5 million against $30 million).
4. Desks can each be inside while their sum breaches, and one desk can breach while the firm is well inside.
5. Renewing it at each expiry without the approval a permanent change needs.
6. The 99% ES of annual P&L: $212.8 million; expected P&L $59.1 million.
7. Stand-alone 65.5, 69.0, 76.9, 122.8, 47.9, 11.8; Euler 48.0, $-21.5$ , 64.0, 108.5, 13.9, $-0.1$ ; incremental 42.5, $-31.2$ , 59.6, 93.6, 10.0, $-0.7$ ($ million).
8. See [Proposition 7.6](#prop-fm-limits-and-capital-allocation-euler) ; the Euler column adds up to 212.8.
9. Expected shortfall is subadditive: diversification means the tails do not all arrive together.
10. Trend following (and market making, near zero): it gains in the firm’s worst years.
11. See [Definition 7.7](#def-fm-limits-and-capital-allocation-raroc) .
12. Stand-alone: credit carry, vol selling, macro. Euler: credit carry and vol selling.
13. Stand-alone 1.12, 1.66, $-3.50$ , $-6.34$ , $-1.17$ , 8.24; Euler 3.74, 15.24, $-1.55$ , $-4.20$ , 3.93, 10.02.
14. Undefined or negative RAROC and a credit that rewards unlimited growth; firms cap credits or charge a floor.
15. The incremental allocation, which measures the firm’s capital with and without the strategy.
16. 2, 2, 1.57, 0.57, 2, 2; value added $27.2 million to $59.0 million a year.
17. The direction (shrink the tail-heavy books, grow the others), not the size, which the cap at 2 sets.
18. Yes if it expires, the firm’s utilisation allows it (72%), and the desk’s contribution to the firm’s tail at the larger size is charged; its Euler RAROC (11.1%) is already below the hurdle, so a larger book destroys more value.
19. Credit carry: 10.5% stand-alone and 12.6% Euler; vol selling: 9.8% and 11.1%; trend: $69.0 million stand-alone, $-\$21.5$ million Euler.
20. Charge capital by Euler contributions with a floor, and shrink vol selling and credit carry, which do not earn their share of the tail; approve the temporary increase only with an expiry and the charge applied.

## 7.10 Interview questions

**Interview question 7.1 ★ risk.**

What is the difference between a soft and a [hard limit](#def-fm-limits-and-capital-allocation-breach)? Give an example of each at a trading desk.

**Solution of Interview question 7.1.**

A [soft limit](#def-fm-limits-and-capital-allocation-breach) warns and triggers a review (a VaR at 75% of its limit); a [hard limit](#def-fm-limits-and-capital-allocation-breach) must be cured or escalated at once (a loss limit that stops trading).

*What the interviewer is looking for: the difference in required action.*

**Interview question 7.2 ★ trader, risk.**

Why does a desk carry several kinds of limit instead of one VaR limit?

**Solution of Interview question 7.2.**

Each measure misses something: VaR depends on its model and history, sensitivities on small moves, notionals on nothing; stress, loss and [concentration limits](#def-fm-limits-and-capital-allocation-conc) cover what they miss.

*What the interviewer is looking for: complementary blind spots.*

**Interview question 7.3 ★★ risk, researcher.**

Three strategies each have a stand-alone ES of 50 and the firm’s ES is 90. How would you allocate the 90, and why not 50 each?

**Solution of Interview question 7.3.**

By Euler contributions, each strategy’s expected loss in the firm’s tail; they add up to 90. Fifty each ignores diversification and double counts.

*What the interviewer is looking for: additivity to the firm’s measure.*

**Interview question 7.4 ★★ bank, risk.**

A strategy earns 12 a year on Euler capital of 108. At a 15% hurdle, does it add value? What would change your answer?

**Solution of Interview question 7.4.**

$12/108=11.1\%<15\%$: it destroys 4.2 a year. A different tail model, or a franchise value outside its P&L, could change it.

*What the interviewer is looking for: RAROC against hurdle, and the model behind the capital.*

**Interview question 7.5 ★★ researcher.**

Why can a strategy’s Euler contribution to expected shortfall be negative? Is that a reason to grow it without limit?

**Solution of Interview question 7.5.**

It makes money in the firm’s worst scenarios. No: its contribution is marginal at today’s size and turns positive once it dominates the book.

*What the interviewer is looking for: marginal, not average.*

**Interview question 7.6 ★★★ risk.**

How would you estimate each strategy’s contribution to a firm’s tail when the tail is driven by rare crashes that the last five years do not contain?

**Solution of Interview question 7.6.**

Add crash scenarios, historical and hypothetical, to the simulation (One Quant Book 6, chapter 22), with each strategy’s loss in them estimated from its positions, and allocate on the combined scenario set.

*What the interviewer is looking for: scenarios beyond the sample.*