Core Path · L2.5.04

Sensitivities

12 minMarket riskSensitivities

Know these first

Math Atoms

By the end you will be able to

  1. 1Use DV01, beta and delta to turn a market move into an estimated profit or loss.
  2. 2Explain a sensitivity as a slope, and why it is a first-order (straight-line) estimate.
  3. 3Estimate a share's beta from return data.
  4. 4Add up sensitivities across a portfolio, and say when the straight-line estimate breaks down.

Why it matters

“If yields rise 10 basis points we lose $24,008. If the index falls 1% we lose $45,966.” Two sentences like these open many morning risk meetings. Each is a sensitivity at work.

Core explanation

One number per risk factor

Volatility and correlation (L2.5.01: Measuring volatility, L2.5.03: Correlation and diversification) describe how markets move. A sensitivity describes how your position responds. Pick a risk factor, move it a little, hold everything else fixed, and see how much the value changes. Multiply that by the size of the move you are worried about, and you have an estimated profit or loss.

Illustrative dataThe Module 2.5 data set

500 trading days of daily returns for an equity index fund, a bond fund and a bank share, generated for this module by a seeded model with a calm stretch, a turbulent spell and a few extreme days. It is not real market history. The same series runs through every lesson in the module, so the numbers line up. The fund we follow is worth $10 million: $6 million in the equity index fund and $4 million in the bond fund.

In this lesson the fund also considers a hedge: 12 put options on the equity index (strike 4,750, three months to expiry, index at 5,000, $100 per index point per contract). The bond fund behaves like a 7-year bond with a 4% annual coupon priced at par.

Three sensitivities cover most first conversations about market risk:

DV01

Bonds and rates

The fund's bond fund: modified duration 6.00 on $4,000,000 gives a DV01 of $2,401. Yields up 1 bp, value down about $2,401.

Beta

Shares and equity funds

The index fund has β = 1 by construction. The bank share in our data has β = 1.29: it tends to move 1.29% for each 1% the index moves.

Delta

Options

Each put has Δ = −0.234. Index down 1 point, each put up about 0.234 points, or $281 for the 12 contracts.

DV01 comes straight from duration (L1.4.08: Duration): value × modified duration × 0.0001. Beta is new here. It measures how strongly a share follows the market, and it is estimated from data as the covariance with the index divided by the index's variance (L2.5.03: Correlation and diversification). Delta is the option sensitivity you met in words in L1.7.09: What drives an option’s price: a put loses value when the index rises, so its delta is negative, which is exactly what makes it a hedge.

Adding them up. Sensitivities to the same factor simply add. The fund's equity exposure is the index fund's $6,000,000 plus the puts' −$1,403,449 of dollar delta: about $4,596,551 net. The puts cut the fund's exposure to a small index move by almost a quarter. Sensitivities to different factors (rates and equities) cannot be added as exposures, but the P&L they imply for a given scenario can.

Where the straight line breaks. A sensitivity is a slope at today's prices: a linear estimate. For small moves it is very good. For large ones the true value curve bends away. For bonds that bend is convexity (L1.4.09: Convexity); for options it is gamma, and it can be large:

Index movePuts: delta estimatePuts: full revaluation
−1%+$14,034+$15,087
−5%+$70,172+$99,252
−10%+$140,345+$265,887
−20%+$280,690+$772,779

For a 1% fall the two agree closely. For a 20% crash the delta estimate captures little more than a third of the gain. That is why risk managers use sensitivities for everyday moves and switch to full revaluation for scenarios and stress tests (L2.5.05: Scenario analysis).

Your callWhich sensitivity would you quote?

0 of 3
  1. A pension fund holds £800 million of long-dated government bonds.
  2. A fund manager holds €50 million of European bank shares and asks how much she loses if the market falls 5%.
  3. A market maker in Tokyo has sold call options on the Nikkei 225 and wants to hedge with futures.

The formal version

Every sensitivity is a derivative: the slope of value VV against a risk factor xx (M-B01: Derivatives). For a small move Δx\Delta x, start from the slope and expand one case at a time:

ΔV≈∂V∂x Δxbonds:ΔV≈− DV01×Δybp,DV01=V×Dmod×0.0001shares:ΔV≈β×V×Rm,β=cov⁡(R,Rm)var⁡(Rm)options:ΔV≈Δ×N×m×ΔS\begin{aligned} \Delta V &\approx \frac{\partial V}{\partial x}\,\Delta x \\[4pt] \text{bonds:}\quad \Delta V &\approx -\,\text{DV01} \times \Delta y_{\text{bp}}, \qquad \text{DV01} = V \times D_{\text{mod}} \times 0.0001 \\[4pt] \text{shares:}\quad \Delta V &\approx \beta \times V \times R_m, \qquad \beta = \frac{\operatorname{cov}(R, R_m)}{\operatorname{var}(R_m)} \\[4pt] \text{options:}\quad \Delta V &\approx \Delta \times N \times m \times \Delta S \end{aligned}

Here Δybp\Delta y_{\text{bp}} is the yield change in basis points, RmR_m the market's return, NN the number of option contracts, mm the contract multiplier and ΔS\Delta S the move in the underlying, in points. Beta can also be written as β=ρ σ/σm\beta = \rho \, \sigma / \sigma_m: correlation times the ratio of the volatilities.

The next term of the expansion adds curvature, 12 ∂2V∂x2 (Δx)2\tfrac{1}{2}\,\frac{\partial^2 V}{\partial x^2}\,(\Delta x)^2: convexity for bonds, gamma for options. It is tiny for small moves and decisive for large ones.

Signs and units

DV01 is quoted as a positive number for a long bond position, and the P&L for a rise in yields is minus DV01 per basis point. A put's delta is between −1 and 0, a call's between 0 and 1. Delta here is per option per index point; multiply by contracts and the multiplier for dollars per point, and by the index level for “dollar delta”.

Try it

The defaults are the worked example: index −2%, yields −10 bp. Switch on full revaluation and compare. Then drag the index move to −20% and watch the puts' two columns part company. Swap the index fund for the bank share to see beta at work.

Try itSensitivity calculator

Market moves

Positions

Equity holding

PositionSensitivityEstimated P&L
Bond fundDV01 $2,401 per bp+$24,008
Equity index fundβ 1.00 × $6,000,000−$120,000
12 index putsΔ −0.234: −$281 per point+$28,069
Total−$67,923
Index at 5,000, $100 a point per contract. Full revaluation reprices the bond and the puts (Black-Scholes); the equity line is linear in both.

Worked example

A bad morning for the fund

The index opens 2% down and government bond yields fall 10 basis points as investors look for safety. Estimate the fund's P&L from its sensitivities, with the put hedge in place. Try each step before revealing it.

From sensitivities to P&L

Step 1 of 6
  1. 1

    Bond fund: DV01

    DV01=$4,000,000×6.0021×0.0001=$2,400.82\text{DV01} = \$4{,}000{,}000 \times 6.0021 \times 0.0001 = \$2{,}400.82. Yields fall 10 bp, so the estimate is −$2,400.82×(−10)=+$24,008-\$2{,}400.82 \times (-10) = +\$24{,}008.

For bigger rate moves, convexity shows up in the bond fund too:

YieldsDV01 estimateFull revaluation
+10 bp−$24,008−$23,920
+50 bp−$120,041−$117,854
+100 bp−$240,082−$231,455
+200 bp−$480,164−$446,591

Regulatory lens

Sensitivities are the building blocks of the Basel market risk standard (FRTB). Its standardised approach, the sensitivities-based method, asks banks to compute delta and vega sensitivities to a prescribed list of risk factors, plus a curvature charge for the bend you saw above. Each sensitivity is multiplied by a regulatory risk weight and the results are combined with prescribed correlations. Inside banks, sensitivities are also the everyday currency of trading limits, such as a cap on a desk's DV01 (L2.3.06: Limits and escalation). The detail is in M4.3: Market risk capital (FRTB) and CVA, part of Level 4: Regulation, Capital & Accounting.

Sparks

SparkNot all Greeks are Greek
Option sensitivities are called “the Greeks”: delta, gamma, theta, rho. The sensitivity to volatility is called vega, which is not a Greek letter at all. The name stuck anyway, perhaps because it starts with a v, like volatility.
SparkOrange County, 1994
Orange County in California ran an investment pool for local governments that borrowed heavily to buy bonds whose value fell as interest rates rose. Its sensitivity to rates was far larger than the size of the pool suggested. When the US Federal Reserve raised rates through 1994, the pool lost about $1.6 billion, and in December 1994 the county filed for bankruptcy.

Key takeaways

  • A sensitivity is the change in value for a small move in one risk factor: a slope, or derivative.
  • DV01 = value × modified duration × 0.0001; P&L ≈ −DV01 × change in yield (bp).
  • Beta = cov(asset, market) ÷ var(market); P&L ≈ β × value × market return.
  • Delta: P&L ≈ Δ × contracts × multiplier × move in the underlying. Puts have negative delta.
  • Sensitivities to the same factor add up. They are straight-line estimates: good for small moves, poor for large ones, where convexity and gamma take over.

Reaction

Check what stuck

Question 1 · objective 1

A bond portfolio has a DV01 of $3,500. Yields rise by 15 basis points. What is the estimated profit or loss, in dollars? (A loss is negative.)

$

Question 2 · objective 1

You hold $5 million of a share with a beta of 1.4. The market index falls 3%. What is the estimated P&L, in dollars?

$

Question 3 · objective 1

You are long 20 call contracts on an index, each worth $100 per index point, with a delta of 0.6. The index rises 50 points. What is the estimated gain, in dollars?

$

Question 4 · objective 3

Over the same days, a share's covariance with the index is 0.00012 and the index's variance is 0.0001 (both from daily returns). What is the share's beta?

Question 5 · objective 4

The index falls 20%. Your delta-based estimate says your put options gain $280,000. Full revaluation says they gain much more. Why?

Glossary

Risk factor
A market variable whose moves change a portfolio's value, such as an equity index, a yield or an exchange rate.
Sensitivity
How much a position's value changes for a small move in one risk factor, all else held fixed.
DV01
The change in value, in money, for a one basis point (0.01%) move in yield. A rise in yields lowers a bond's value.
Beta
How much a share or portfolio tends to move for a 1% move in the market index: cov(asset, market) ÷ var(market).
Delta
How much an option's value changes for a one-point move in its underlying price.
Gamma
How fast delta itself changes as the underlying moves: the curvature of an option's value.
Linear estimate
An estimate that treats value as a straight line in the risk factor. Good for small moves, poor for large ones.

Deep Dives & Origins

Deep Dives

  • J. C. Hull, Options, Futures, and Other Derivatives, Pearson, the chapter on the Greek letters. Intermediate.
  • B. Tuckman and A. Serrat, Fixed Income Securities, Wiley, the chapter on DV01, duration and convexity. Intermediate.

Origins

  • J. C. Hull, Options, Futures, and Other Derivatives, Pearson.
  • Basel Committee on Banking Supervision, Minimum capital requirements for market risk, January 2019 (the sensitivities-based method).
  • P. Jorion, Big Bets Gone Bad: Derivatives and Bankruptcy in Orange County, Academic Press, 1995.

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Correlation and diversification

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Scenario analysis