Equity risk · RiskAtoms Academy

By the end you will be able to

1Measure an equity book's index exposure, netting shares, futures and option deltas, and turn it into a daily standard deviation.
2Split a share's risk into a market part (beta) and a stock-specific part, and hedge the first with index futures.
3Explain the dividend and repo risk inside index futures and options, and estimate it.
4Recognise when single-name or concentration risk needs more than an index factor.

Why it matters

RiskAtoms Bank's Equities book holds £60m of UK index futures, £30m of US shares and 1,000 puts. On a typical day it gains or loses about £660,385 from equity moves alone. Then a client asks the desk to buy a £10m block of a single company's shares. How much risk does that add, how much can index futures take away, and what is left that no hedge will touch?

Core explanation

Three layers of equity risk

Illustrative dataRiskAtoms Bank plc's Markets desk

One made-up trading book on one made-up valuation day, in pounds: eight positions in four books, eight risk factors and 750 days of simulated factor moves. The levels (UK index 8,000, GBP/USD 1.30, Brent $80 and so on) are round illustrative numbers, not market data. The same book runs through every lesson in Module 3.2, so the figures agree from lesson to lesson.

Share X and the dividend and repo figures below are extras for this lesson, not part of the book: the book's totals do not change.

Position	Size	Value today	Risk factors
Equities
UK index futures (long £60m)	750 contracts × £10 × 8,000 = £60m	£0.00m	UK index
US equity basket ($39m)	$39m at 1.30 = £30m	£30.00m	US index, GBP/USD
UK index puts (1,000 contracts)	1,000 puts, strike 7,600, 3 months	£0.95m	UK index, UK vol

1. Index exposure

The first layer is how much the book gains if the market rises: its index exposure. Shares count at value × beta, futures at their notional, options at their delta (L3.2.02: Risk factors and position mapping). On the UK index the futures give £60.00m and the puts −£19.27m, a net £40.73m: a 1% rise makes £407,320. On the US index the basket gives £30.00m.

With daily volatilities of 1% (UK) and 1.1% (US), a one-standard-deviation day is worth £407,320 and £330,000. The two indices have a correlation of 0.6, so together, as in L2.5.03: Correlation and diversification, the book's equity standard deviation is £660,385: less than the £737,320 sum.

Desks often quote one number: the exposure in “UK-index equivalents”. The US index's beta on the UK index is 0.6 × 1.1% / 1% = 0.66 (L3.1.09: CAPM, beta and the security market line), so the basket counts as £19.80m and the book as £60.53m. Handy, but it drops the part of the US index the UK index does not explain: a single-index map gives £605,320 a day, not £660,385.

2. Single-name risk

A share moves for two reasons: the market moved (beta times the index), and something happened to the company. The second part is stock-specific risk. An index factor cannot see it, index futures cannot hedge it, and only holding many names makes it small, because independent surprises partly cancel (L3.1.11: Factor models and Fama-French showed the same in factor models).

Take Share X: £10m, beta 1.2, stock-specific volatility 1.6% a day. Its total volatility is 2.0%, and 64% of its variance is stock-specific. Selling beta-weighted index futures (£12m, 150 contracts) cuts the daily standard deviation from £200,000 to £160,000: a fifth less, not all of it.

Concentration

Stock-specific risk is also where the worst single days come from: a profit warning, a failed takeover, a fraud. Share prices can gap by 30% or more overnight, far beyond what a normal distribution with 1.6% daily volatility allows. Desks therefore set concentration limits per name, sector and country (L3.2.16: Market risk limits) as well as VaR limits.

3. Dividends and repo

Index futures and options look like pure index bets, but they carry two quieter risks. Whoever holds the shares receives the dividends and pays to finance them; whoever holds the future does neither. So the futures price is the index adjusted for both (L1.7.02: Forwards): the fair value is $F = S\,e^{(r - q)T}$ , with $r$ the repo (funding) rate and $q$ the dividend yield.

With the index at 8,000, $r$ = 4.0%, $q$ = 3.5% and three months to expiry, $F$ = 8,010.0, 10.0 points above the index. A long future is therefore short dividends: if companies cut dividends, $F$ rises with the index unchanged. It is long the repo rate. Puts are long dividends, because higher dividends pull the future (and so the expected index) down.

Sort itWhich way does it go?

For each position, does it gain or lose if expected dividends rise?

Long UK index futures
Long UK index puts
Long the shares themselves, financed in repo
Long index call options

The formal version

Index exposure and its risk. With UK and US exposures $E_u, E_s$ (£ per 1.00 log return), daily volatilities $\sigma_u, \sigma_s$ and correlation $\rho$ :

\begin{aligned} \Delta V &\approx E_u\,x_u + E_s\,x_s \\ \operatorname{sd}(\Delta V) &= \sqrt{E_u^2\sigma_u^2 + E_s^2\sigma_s^2 + 2\rho\,E_uE_s\,\sigma_u\sigma_s} \end{aligned}

Beta mapping and what it drops. Regress the US move on the UK move: $x_s = \beta\,x_u + e$ , with $\beta = \operatorname{cov}(x_s, x_u)/\sigma_u^2 = \rho\,\sigma_s/\sigma_u$ and $e$ uncorrelated with $x_u$ . Then

\begin{aligned} \Delta V &\approx (E_u + \beta E_s)\,x_u + E_s\,e \\ \operatorname{var}(\Delta V) &= (E_u + \beta E_s)^2\sigma_u^2 + E_s^2\,\sigma_e^2, \qquad \sigma_e^2 = \sigma_s^2 - \beta^2\sigma_u^2 = \sigma_s^2(1 - \rho^2) \end{aligned}

Expanding the first term and adding the second gives back the two-index formula exactly. Keeping only the first term is the single-index shortcut; the second is the residual it misses.

A single share and its hedge. The single-index model $r = \beta\,r_m + \varepsilon$ on a holding of value $V$ , hedged by selling $h$ pounds of index futures:

\begin{aligned} \Delta P &= V(\beta\,r_m + \varepsilon) - h\,r_m = (\beta V - h)\,r_m + V\varepsilon \\ \operatorname{var}(\Delta P) &= (\beta V - h)^2\sigma_m^2 + V^2\sigma_\varepsilon^2 \\ \frac{d\operatorname{var}}{dh} &= -2(\beta V - h)\sigma_m^2 = 0 \;\Longrightarrow\; h^* = \beta V, \qquad \operatorname{var}_{\min} = V^2\sigma_\varepsilon^2 \end{aligned}

Spread $V$ equally over $n$ names with independent residuals of the same size and the residual variance is $n\,(V/n)^2\sigma_\varepsilon^2 = V^2\sigma_\varepsilon^2/n$ : the standard deviation falls with $\sqrt{n}$ .

Dividend and repo sensitivity of a future. Differentiate the fair value:

\begin{aligned} F &= S\,e^{(r - q)T} \\ \frac{\partial F}{\partial q} &= -T\,F, \qquad \frac{\partial F}{\partial r} = +T\,F \\ \text{P\&L per 1bp rise in } q &= -T \times F \times N \times m \times 0.0001 \end{aligned}

for $N$ contracts with multiplier $m$ . The repo sensitivity has the same size and the opposite sign.

Try it

Start with Share X. Raise the hedge to 100% and watch the market bar vanish while the stock-specific bar stays. Then spread the money over more names.

Try itHedge a block of shares with index futures

£10m of shares. The UK index moves 1% a day (one standard deviation).

Beta to the UK index1.2

Stock-specific volatility (daily)1.6%

Hedge: share of the beta sold in futures0%

Spread over this many names1

Market (after hedge)£120,000

Stock-specific£160,000

Total£200,000

Standard deviations of daily P&L. They add as squares: total² = market² + stock-specific².

Futures sold£0.00m

Daily standard deviation£200,000

99% one-day VaR£465,270

Share of variance that is stock-specific64%

Past 100% the hedge over-shoots and the market risk comes back with the opposite sign. No amount of index futures removes the stock-specific part; only spreading the money over more names does.

Worked example

The Equities book and a client's block

Exposure, risk, a new block, and dividends

Step 1 of 5

1
Net index exposure
UK: $\text{£}60.00\text{m} - \text{£}19.27\text{m} = \text{£}40.73\text{m}$ per 1.00, so £407,320 per 1%. US: £30.00m per 1.00 (beta 1).

Regulatory lens · looking ahead

The Basel standardised approach for market risk treats equity spot prices and equity repo rates as separate risk factors for each name, with risk weights by market capitalisation, economy (advanced or emerging) and sector, plus a separate charge for the risk of default of the issuers of shares held. Large single-name positions also fall under the large exposures rules. Both are in M4.3: Market risk capital (FRTB) and CVA and M4.4: Counterparty, operational and large exposures. Models of equity volatility and fat tails come in M6.2: Market risk modelling.

Spark Atoms

Spark AtomVolkswagen, October 2008

On 26 October 2008 Porsche disclosed that it controlled about 74% of Volkswagen's ordinary shares, directly or through options. With the state of Lower Saxony holding about 20% more, few shares were left to buy. Investors who had sold them short scrambled to buy them back, and for a short time Volkswagen was the most valuable company in the world. No index factor saw that coming.

Spark AtomWhen dividends disappear

In spring 2020 many companies cut or cancelled their dividends, and on 31 March 2020 the Bank of England's Prudential Regulation Authority asked the large UK banks to cancel theirs. Dividend futures fell sharply, and several European banks with structured-product desks reported trading losses for that quarter that they partly put down to the cancelled dividends.

Key takeaways

Index exposure nets shares (value × beta), futures (notional) and options (delta). The book's daily equity risk combines the exposures with the indices' volatilities and correlation.
Beta mapping onto one index is a useful summary but drops the residual; with correlation 0.6 the residual is most of the US index's own risk.
Stock-specific risk is untouched by index hedges and falls only with diversification, roughly as 1/√n; concentration limits cover the jumps.
Index futures carry dividend and repo risk: F = S e^{(r − q)T}, so a long future is short dividends and long the repo rate; puts are long dividends.

Reaction

Check what stuck

Question 1 · objective 1

A desk holds £5m of a share with a beta of 1.5 to the index. How much index futures should it sell to hedge the market part, in £?

£

Question 2 · objective 2

A share has a beta of 0.8; the index's daily volatility is 1.5% and the share's stock-specific daily volatility is 0.9%. What is its total daily volatility, in %?

%

Question 3 · objective 2

£20m is spread equally over 25 shares, each with stock-specific daily volatility of 2%, independent of each other. What is the stock-specific daily standard deviation of the whole, in £?

£

Question 4 · objective 3

A desk is long index futures. Companies in the index unexpectedly cut their dividends. What happens to the desk?

AIt loses: lower dividends lower the futures priceBIt gains: the futures price rises because holders of the future do not receive dividendsCNothing: futures have no dividend exposureDIt gains only if it also holds the shares

Question 5 · objective 1

Why does mapping the US basket onto the UK index with a beta understate its risk?

ABecause the beta is too highBBecause it leaves out the part of the US index that does not move with the UK indexCBecause it ignores the putsDIt does not: beta mapping is exact

Glossary

Index exposure: The money a position gains for a 1.00 (100%) rise in an index, so that a 1% move gains one hundredth of it. For shares it is value × beta; for options, delta × index level × contracts × multiplier.
Beta: The slope of a share's (or index's) return on the market's return: cov(r, rm) / var(rm). It says how much of a market move the position follows on average.
Stock-specific risk: The part of a share's return that the market does not explain: company news, earnings, takeovers. Also called specific or idiosyncratic risk.
Beta hedge: Selling index futures worth beta × the value of a share position, so the position no longer gains or loses with the market on average.
Concentration risk: A large position in one name, sector or country, whose risk a portfolio-wide model can understate.
Dividend risk: The risk that the dividends a position is exposed to turn out different from what was priced in.
Repo rate: The rate at which a desk can borrow cash against shares (or lend shares for cash). It sets the cost of holding shares and so enters the futures price.
Fair value (of a future): The futures price that leaves no arbitrage between holding the shares and holding the future: F = S e^{(r − q)T}.

Deep Dives & Origins

Deep Dives

J. C. Hull, Options, Futures, and Other Derivatives, 11th ed., Pearson, 2021, the chapters on futures prices and stock index futures. Intermediate.
C. Bennett, Trading Volatility: Trading Volatility, Correlation, Term Structure and Skew, CreateSpace, 2014, on dividend and repo risk in equity derivatives. Advanced.

Origins

W. F. Sharpe, “A Simplified Model for Portfolio Analysis”, Management Science 9(2), 1963.
Basel Committee on Banking Supervision, Basel Framework, MAR21: Standardised approach: sensitivities-based method (equity risk factors), Bank for International Settlements.
Prudential Regulation Authority, statement on deposit takers' approach to dividend payments, share buybacks and cash bonuses in response to Covid-19, Bank of England, 31 March 2020.

Watch this lesson

Video version

Prefer to watch? The same lesson as a 12-minute video will sit here, made from this page's script.