Mapping Positions to Risk Factors

A mid-sized bank's trading book might hold forty thousand distinct instruments: bonds of every maturity, loans, swaps, futures, thousands of individual equities, options on many of them. Now try to compute the risk of that book. You would need a covariance matrix across forty thousand things, which means estimating eight hundred million numbers from price histories that, in many cases, do not exist, because half the instruments were issued in the last two years.

That approach is hopeless, and nobody attempts it. What every risk system does instead is a translation step. Before measuring anything, it rewrites each position not as itself but as a bundle of exposures to a small set of shared risk factors: a handful of points on each yield curve, a few equity indices, currency rates, credit spreads, volatility levels. Fifty thousand instruments become perhaps two hundred factors. Only then does the arithmetic become possible.

That translation step is risk factor mapping, and it is where most of the judgement in a risk system lives.

An analogy before any symbols

Think about how nutrition labels work. A restaurant might serve four hundred dishes, and you cannot possibly model each one. But every dish, however elaborate, can be described by a handful of standard quantities: calories, protein, fat, carbohydrate, sodium. Once each dish carries those five numbers, you can add up a whole day's eating, compare two menus, or ask what happens if you cut fat by a third. You have not captured everything about the food, you have thrown away flavour and texture entirely, but you have kept the part that answers your question.

Risk factor mapping is the same move. A callable bond, a convertible, and an interest-rate swap are wildly different contracts, but all three can be described by "how much do you lose if five-year rates rise one basis point, and how much if credit spreads widen one basis point." Once every position speaks that common language, they can be added together.

The mapping equation

Let $f = (f_1, \ldots, f_K)$ be the risk factors, the small set of market quantities you have chosen to model. For a position worth $V$ today, a small move in the factors changes its value by

$\Delta V \;\approx\; \sum_{k=1}^{K} \frac{\partial V}{\partial f_k}\,\Delta f_k \;=\; \delta^\top \Delta f.$

In words: the change in the position's value is the sum, over every factor, of how sensitive the position is to that factor times how much that factor moved. The vector $\delta$ of sensitivities is the position's mapping, its nutrition label. This is just a first-order Taylor expansion, which is worth saying out loud because it is also the source of every problem later on.

Add up the whole book by summing the $\delta$ vectors, and the portfolio's profit-and-loss variance becomes

$\text{Var}(\Delta V_{\text{book}}) = \delta_{\text{book}}^\top\, \Sigma_f\, \delta_{\text{book}},$

where $\Sigma_f$ is the covariance matrix of the factors, not the instruments. In words: risk equals your exposures, squared and weighted by how the factors move together. This is a $K \times K$ matrix rather than a forty-thousand-square one, and every factor in it has decades of clean history.

How the main asset classes get mapped

• Equity. A single stock is mapped to one or more index factors through its Beta (β), plus a specific (idiosyncratic) term for the part no factor explains. A $5m holding with beta 1.3 becomes $6.5m of index exposure plus $5m of name-specific risk.
• Interest rates. Every fixed cash flow is assigned to standard maturity vertices, typically 1m, 3m, 1y, 2y, 5y, 10y, 30y. A bond becomes a set of DV01s, one per vertex. This is cash-flow mapping, worked through below.
• FX. Each position's value is decomposed into currency legs; a euro-denominated bond held by a dollar book carries a euro-dollar exposure equal to its full market value on top of its rate exposure.
• Credit. Spread duration per rating bucket and sector, so a corporate bond carries both a rates DV01 and a spread DV01.
• Options. Delta to the underlying, vega to the volatility factor, plus gamma and other second-order terms because the first-order approximation is not enough. This is the hard case, and it is why The Option Greeks exist.

Worked example: cash-flow mapping a bond

You hold a single cash flow of $1,000,000 arriving in exactly 4 years. Your risk system has vertices at 3 and 5 years but not 4, so the flow has to be split.

Step 1, present value. At a 4% yield, the flow is worth $1{,}000{,}000/1.04^4 = 854{,}804$ today.

Step 2, sensitivity. Modified duration is $4/1.04 = 3.846$, so the DV01 (the loss from a one basis point rise in yield) is $854{,}804 \times 3.846 \times 0.0001 = 328.8$ dollars per basis point.

Step 3, split it. Four years sits exactly halfway between the 3-year and 5-year vertices, so allocate half the present value to each: $427,402 apiece. Check that duration is preserved: $0.5 \times 3 + 0.5 \times 5 = 4$ years. It is.

Step 4, the two vertex sensitivities.

• 3-year vertex: $427{,}402 \times (3/1.04) \times 0.0001 = 123.3$ per basis point.
• 5-year vertex: $427{,}402 \times (5/1.04) \times 0.0001 = 205.5$ per basis point.
• They sum to $328.8$, matching Step 2 exactly. Mapping has changed the description, not the total risk.

Step 5, risk. Suppose the daily standard deviation of the 3-year yield is 6.0 basis points, the 5-year is 5.5, and they correlate at 0.97. The daily profit-and-loss volatility is

$\sqrt{(123.3 \times 6.0)^2 + (205.5 \times 5.5)^2 + 2(0.97)(123.3 \times 6.0)(205.5 \times 5.5)}$

$= \sqrt{547{,}157 + 1{,}277{,}126 + 1{,}621{,}724} = \sqrt{3{,}446{,}007} = 1{,}856.$

So about $1,856 a day of profit-and-loss volatility, and a 99% one-day VaR of roughly $2.33 \times 1856$, about $4,300. The correlation of 0.97 does nearly all the work here, which is the point of the next explorer.

That 0.97 is not a modelling convenience, it is what neighbouring points on a yield curve actually do. Drag the correlation slider below down toward zero and watch the cloud go round: at $\rho$ near 1 the two vertices are effectively one factor, which is why curve risk collapses onto a few directions (see PCA (Principal Component Analysis)):

Worked example: an equity book and what hedging really removes

Position 1: $5m long a stock with beta 1.3. Index daily volatility 1.0%, the stock's specific daily volatility 1.4%.

• Systematic exposure: 5m times a beta of 1.3 is 6.5m of index-equivalent, and 6.5m times the index's 1.0% daily move gives $65,000 of daily volatility.
• Specific: 5m times 1.4% gives $70,000.
• Total, since specific risk is uncorrelated with the index, add in quadrature: $\sqrt{65^2 + 70^2} = 95.5$ (in thousands), so about $95,500 a day.

Position 2, added: $3m short a different stock with beta 0.9 and specific volatility 1.8%.

• Net systematic exposure: 6.5m long minus 2.7m short leaves 3.8m, so $38,000 of daily volatility. The market exposure fell by 42%.
• Specific risks add in quadrature because the two names are unrelated: $\sqrt{70^2 + 54^2} = 88.4$, so $88,400.
• Total: $\sqrt{38^2 + 88.4^2} = 96.2$, about $96,200 a day.

Read that again. Hedging cut the market exposure from $65,000 to $38,000, and total risk went slightly up, from $95,500 to $96,200. The hedge removed factor risk and added specific risk, and specific risk does not net. This is the single most useful thing mapping tells you, and it is invisible until you separate the two.

Where mapping breaks

The mapping equation is a straight-line approximation, and options are not straight lines. A long put has a delta that changes as the underlying moves, so a single delta number describes it only for very small moves. Drag the strike and premium below, notice that the payoff is a bend, and ask yourself what one slope could possibly summarise it:

The standard fixes are to add the second-order term ($\Delta V \approx \delta\,\Delta f + \tfrac12\gamma\,\Delta f^2$, delta-gamma mapping) or to abandon approximation entirely and fully reprice every position under each scenario. Full revaluation is exact and expensive; delta-gamma is fast and wrong in the tails. Most desks use full revaluation for the options book and linear mapping for everything else.

Practice

1. Map a $1,000,000 flow at 7 years onto 5-year and 10-year vertices. What weights preserve duration, and what are the two DV01s?
2. In the equity example, at what specific volatility for the second stock would adding the short actually reduce total risk?
3. Your book is delta-neutral in an option position but holds large gamma. Sketch what a linear-mapped VaR would report versus what a full revaluation would report after a 5% index move.
4. You add a new factor to the model and total book VaR falls. Give two different reasons this could happen, only one of which is good news.