MR 5 – VaR Mapping

A bank that reports a single value-at-risk number is aggregating everything it holds: bonds, equities, currencies, commodities and the derivatives written on all of them. J.P. Morgan Chase described its own VaR calculation in its 2004 annual report as covering more than 2.1 million positions and 240,000 pricing series. Modelling each of those positions as a separate source of randomness would require an enormous covariance matrix, and the calculation would become too slow to be useful for daily risk control.

VaR mapping is the shortcut. Each position is marked to market and its value is then re-expressed as exposures on a limited set of primitive risk factors, such as a spot exchange rate, a handful of zero-coupon yields or an equity index. The risk engine only has to model the joint behaviour of those factors, and the positions enter the calculation through their exposures.

Consider a currency desk with thousands of outstanding dollar/euro forward contracts, each with its own delivery date and contracted price. Pricing these contracts correctly needs every detail, but measuring their risk does not. All of them move mainly with the dollar/euro spot rate, so the desk can be summarised by one aggregate exposure to that rate without losing meaningful risk information. This is why a risk system is allowed to look different from a pricing system: aggregation that would be unacceptable for valuation is perfectly acceptable for risk measurement.

Instruments whose risk profile changes over time

Mapping is also the only workable approach when the instrument itself keeps changing. A bond that had seven years to run last year now has six, so its own price history describes a different risk profile from the one it carries today. The bond has to be mapped onto the yields that match its current maturity structure. Options change even faster, because their sensitivity to the underlying moves with every change in price and time, so they too are mapped onto their main risk drivers.

Mapping as a fix for missing or stale data

Sometimes the history simply does not exist. A fund that buys shares at their initial public offering (IPO) holds stocks with no past returns at all, yet those positions cannot be left out of the risk system. The risk manager replaces each one with exposures on comparable factors that the system already tracks, such as an industry or market index.

A subtler data problem is timing. The Tokyo market closes at 1:00 A.M. EST (3:00 P.M. in Japan), while New York closes at 4:00 P.M., so as much as 15 hours separate the two closing prices. A Tokyo closing price observed alongside a New York close ignores everything that happened in between, and is said to be stale. Because daily returns from the two markets are not measured over the same window, their measured daily correlation is too low, which understates the risk of a portfolio holding both.

Stale prices also created a trading abuse. In September 2003 the New York Attorney General, Eliot Spitzer, accused several fund companies of allowing market timing, a short-term strategy of rapidly buying and selling the same funds. When U.S. stocks rallied on good news, timers bought funds holding Japanese stocks at the stale Tokyo-based price, expecting Japan to follow the next day, and sold a day later. The trading costs fell on the other investors, and the funds involved eventually settled by paying more than USD 2 billion. Funds responded with short-term redemption fees and earlier cut-off times for orders.

Mapping offers a remedy for the risk measurement side. The risk manager can regress Japanese returns on U.S. returns and use the fitted value, conditional on the latest U.S. move, as an estimate of where Japanese prices would stand at the U.S. close. Alternatively, correlations can be estimated from weekly rather than daily returns, which reduces the effect of non-synchronous closes. The broader point is that the data collection process has to be designed so that the resulting risk estimates are meaningful.

The process starts by marking every position to market in a single reference currency, usually the bank’s reporting currency. Each market value is then split across the risk factors the system tracks. Figure 1 shows six positions mapped onto three factors: some positions load on a single factor, others are spread across two or three.

The bookkeeping can be laid out as a grid. Position i has market value V_i, and its allocation to factor j is written x_ij. After every row has been filled in, each column is summed to give the portfolio exposure on that factor.

If a position’s value is not fully allocated across the factors, the unallocated remainder is treated as cash. Cash is not a risk factor because its value does not fluctuate, so that part of the position carries no market risk in the calculation.

Exact mapping and estimated mapping

There are two ways of arriving at the x_ij. For many derivatives the price is an exact mathematical function of the risk factors, so the exposures follow analytically from the partial derivatives of the pricing formula. Forwards, FRAs, swaps and options, covered later in this lesson, all work this way, and the mapping leaves nothing out.

For other positions the exposure has to be estimated statistically. A single stock replaced by a position in the market index is the standard case: the size of the index position comes from the slope coefficient of a regression of the stock’s returns on the index returns. Because the regression never fits perfectly, part of the stock’s risk is left outside the mapped exposure, which leads directly to the split between general and specific risk.

Choosing the set of primitive risk factors is a trade-off. More factors give a tighter approximation of the portfolio’s behaviour, but every extra factor adds modelling effort and computing time. The choice also determines how much risk ends up classified as specific. Specific risk is the risk due to issuer-specific price moves that remain after the general market factors have been accounted for, so its size depends entirely on how general market risk has been defined. The Basel rules impose a separate capital charge for specific risk; typically it is 4 percent of the position value for equities and unrated debt when a bank’s internal model does not capture specific risk itself.

An equity portfolio mapped on one index

Take a portfolio of N stocks, each mapped onto the stock market index. The return on stock i is regressed on the market return:

The residuals are assumed to be uncorrelated with each other and with the market. With portfolio weights w_i, the portfolio return splits into a market part and a residual part, and the market exposures add up to a portfolio beta:

If the portfolio is worth W, its mapped position on the index is x = Wβ_p. Because the residuals are independent, the portfolio variance separates cleanly into two pieces:

For a fixed total variance, adding more general factors (industry indices, for example) explains more of each stock’s movement, so the residual specific term becomes smaller.

A corporate bond portfolio

Bonds describe a stream of cash flows spread over time and across issuers of different credit quality, which in principle creates a continuum of risk factors from overnight to very long maturities and across every rating. In practice the set has to be cut down: a single factor may suffice for some portfolios, while others need as many as 15 maturities, and portfolios with options also need factors for implied volatility.

A natural design uses J government yields z_j and K credit spreads s_k sorted by rating. Each bond’s yield change dy_i is mapped onto the government yield at the nearest maturity plus the spread for its rating, and whatever is left is the residual ε_i. With DVBP denoting the dollar value of a basis point, summed across all bonds that map to the same factor, the change in portfolio value is

More general factors leave a smaller residual, but the residual still has to be sized. Individual bonds rarely have enough clean price history for that, so in practice all issuers in the same risk class are often assumed to share the same specific risk.

Once the yield-curve factors have been chosen, a fixed-income portfolio can be mapped onto them in three ways. Principal mapping places the whole portfolio on one vertex at the average maturity of the principal repayments. Duration mapping places it on one zero-coupon bond whose maturity equals the portfolio duration. Cash-flow mapping discounts each cash flow at the zero-coupon rate for its date and places the present value on the corresponding maturity vertex. All three preserve the portfolio’s market value; the better methods also preserve its market risk.

The running example is a USD 200 million portfolio made of two par bonds: USD 100 million of a 5-year 6 percent issue and USD 100 million of a 1-year 4 percent issue. Its average maturity is 3 years and its duration is 2.733 years, which is the simple average of the 1-year bond’s duration of 1 and the 5-year bond’s duration of about 4.465 at its own 6 percent yield.

The monthly 95 percent VaR of zero-coupon bonds at each vertex (as a percentage of value) is 0.4696 at 1 year, 0.9868 at 2 years, 1.4841 at 3 years, 1.9714 at 4 years and 2.4261 at 5 years. Rounded, the 2-year and 3-year figures are 0.987 and 1.484.

Principal mapping has simplicity in its favour and little else. By looking only at when principal is repaid, it ignores the coupons that arrive earlier, so the portfolio looks longer than it is and its risk is overstated. Duration mapping corrects for the timing of all payments and so gives the lower, more accurate number, but it still compresses the whole portfolio into a single point on the curve.

Cash-flow mapping groups every payment onto the term-structure vertices for which volatilities are supplied, each represented by its present value at the matching zero rate. Writing x for the vector of mapped present values, V = ασ for the vector of vertex VaRs (in percent) and R for the correlation matrix of zero returns, the diversified VaR is

If every vertex were perfectly correlated with every other, the portfolio VaR would simply be the sum of the individual VaRs:

The correlation matrix is symmetric, so it is shown in full. Individual and component VaRs are in USD millions.

Why cash-flow mapping comes in lower than duration mapping

The gap between the duration VaR of USD 2.70 million and the diversified cash-flow VaR of USD 2.57 million is USD 130,000, and it has two separate causes. First, vertex risk does not rise in a straight line with maturity, so spreading cash flows across the actual vertices gives a different answer from one interpolated point: that effect is the USD 70,000 between the duration VaR and the undiversified cash-flow VaR (USD 2.70 million less USD 2.63 million). Second, correlations between vertices are below one, which removes a further USD 60,000 (USD 2.63 million less USD 2.57 million). Cash-flow mapping is the only one of the three methods that captures both effects.

Once a portfolio has been mapped onto zero-coupon vertices, its VaR can also be reached by repricing rather than by matrix algebra. Suppose every zero is shocked downward by its own VaR at the same time, which is what perfect correlation across the curve would imply. The shocked discount factors are then applied to the original cash flows, and the fall in portfolio value is the loss under that scenario.

The final column is derived from the two PV columns.

The two routes agree because both assume all vertices move together. The repricing route has the advantage of being transparent: it shows a concrete set of shocked discount factors, which is the essence of a stress test, and it extends naturally to any scenario a risk manager wants to impose, such as a steepening or flattening of the curve applied factor by factor. Matrix multiplication is more direct, however, and it is the better tool for VaR itself because it allows correlations below one across different parts of the yield curve.

Many fixed-income managers are judged against an index rather than in absolute terms, so the risk that matters is the risk of deviating from that index. The same vertex machinery handles this directly. The benchmark here is the J.P. Morgan U.S. bond index, mapped onto 14 vertices from 1 month to 30 years, with a duration of 4.62 years. For a USD 100 million portfolio, the monthly 95 percent VaR of the index itself is USD 1.99 million, roughly the risk of a 4-year note.

If x is the vector of portfolio positions and x₀ that of the index, the VaR of the deviation is

Five two-bond tracking portfolios are tested, each with its duration matched to the index’s 4.62 years. Portfolio 1 uses the two vertices that bracket the duration most tightly, 4 and 5 years; the others widen the bracket step by step until portfolio 5 combines 1-month bills with 30-year zeros.

What the five portfolios show

Moving to portfolio 2 (3 and 7 years) cuts the TE-VaR to USD 0.29 million, and portfolio 3 (2 and 9 years) cuts it further to USD 0.16 million, the lowest of the five. Portfolio 3 resembles the index most closely because the index carries its heaviest weight at the 2-year vertex. Spreading further out hurts: portfolio 4 (1 and 10 years) has a TE-VaR of USD 0.20 million, and the barbell of 1-month bills and 30-year zeros in portfolio 5 jumps to USD 0.81 million.

Absolute risk tells a different story. Portfolio 5 has the lowest absolute VaR, USD 1.71 million, because correlations fall as maturities move apart, so a duration-matched barbell of cash and a 30-year zero diversifies the most. Minimising absolute risk and minimising risk relative to a benchmark are different goals, and duration matching is only a first approximation for either. A manager trying to track an index needs a fine decomposition of the index across maturities.

Forwards and futures are linear in the underlying, so they are the easiest derivatives to break into building blocks. For a forward on a foreign currency, let S_t be the spot price of one unit of the asset, K the contracted forward price, r the domestic risk-free rate, y the income yield on the asset (for a currency, the foreign risk-free rate r*) and τ the time to maturity, with continuous compounding.

Two strategies end with one unit of the foreign currency at maturity. One buys e^−yτ units today and lets the income build them up to one unit. The other enters the forward contract, worth f_t, and sets aside Ke^−rτ in cash to pay the delivery price. Since both end in the same place, they must cost the same today, which gives the value of an outstanding forward. Setting f_t = 0 identifies the current forward rate F_t, and substituting it back expresses the value as the discounted gap between today’s forward rate and the locked-in price.

The last line makes intuitive sense: a holder of the long forward position could close it by selling forward at F_t, locking in F_t − K at maturity, worth that amount discounted to today.

The three building blocks

Differentiating the value with respect to the spot rate and the two interest rates, and then expressing rate changes as changes in bill prices P = e^−rτ and P* = e^−r*τ, gives

So a long currency forward is a long spot position in the foreign currency, plus a long position in a foreign currency bill, plus a short position in a U.S. dollar bill, each with the present value shown in brackets.

The spot leg dominates. Its 4.54 percent monthly VaR (about 1.65 standard deviations) dwarfs the 0.14 percent on the euro bill and the 0.21 percent on the dollar bill, so the forward’s VaR of USD 5.735 million is barely above the USD 5.713 million of the spot leg alone. The positive 0.13 correlation between EUR spot and the euro bill means that a stronger euro tends to come with lower euro rates, adding a little risk, while the short dollar bill slightly offsets it.

The same decomposition handles long-dated currency swaps. A 10-year swap paying dollars and receiving euros is a string of 10 forward contracts, so it maps into a currency component plus a series of USD and EUR fixed-income positions, and its VaR is again driven mainly by the currency component.

Currencies, bonds and stock indices pay a well-defined income stream (a foreign interest rate, a coupon or a dividend yield), which is what makes the three-block decomposition possible. Physical commodities such as metals, energy and agricultural products do not pay cash. Holding them yields an implied benefit from having the physical good available, which, net of storage cost, is called the convenience yield. That yield is not tied to any traded financial variable and is itself highly variable, so it cannot be hedged out into a separate building block.

The practical solution is to use the forward-price form of the valuation directly and treat the forward price for each delivery date as the risk factor. Differentiating gives

Precious metals, cash contracts: gold 6.18, silver 14.97, platinum 7.70.

Two patterns stand out. Commodity risk is high: near-month natural gas reaches about 29 percent a month, against roughly 6 percent for typical currency and equity markets. And volatility falls with maturity, sharply for energy products that are costly to store, less for base metals, and imperceptibly for precious metals, which have low storage cost and essentially no convenience yield. For financial assets volatility is driven by the spot price, so it stays roughly constant across contract maturities.

Contract dates rarely line up exactly with the vertices for which volatilities are available, so in practice the present value of each contract has to be apportioned between the neighbouring maturities.

A forward rate agreement fixes an interest rate for a future period. The buyer locks in a borrowing rate and the seller a lending rate, so the buyer, who is long the FRA, receives a payment if the spot rate at settlement is above the agreed forward rate. With the near date at τ₁ and the far date at τ₂, and using simple (linear) interest, the forward rate is the rate that makes rolling over a τ₁ investment equal to investing for τ₂:

Consider a party that has sold a 6 × 12 FRA on USD 100 million. Economically this is the same as borrowing USD 100 million for 6 months and investing the proceeds for 12 months. If the 6-month rate at expiry turns out above the forward rate, the seller pays the difference on the principal, which exactly offsets the higher rate the reinvested funds would otherwise earn, so the investor ends up with the forward rate. So, an FRA splits into two zero-coupon building blocks; from the buyer’s side, a long 6 × 12 FRA equals a long 6-month bill plus a short 12-month bill, and the seller holds the reverse.

The component VaR of the 180-day leg is negative: holding the short leg reduces the risk of the 360-day leg, which is exactly what makes an FRA a spread position on two nearby points of the curve rather than an outright rate bet.

An interest rate swap exchanges fixed payments for floating ones, so it can be split into a fixed leg, priced like a coupon bond, and a floating leg, which behaves like a floating-rate note (FRN). Just before a reset, the FRN is about to be repriced at the prevailing market rate, so its value is at par and it carries no market risk: it can be mapped onto cash. Right after the reset the next coupon is known, and the FRN behaves like a bill that matures on the next reset date.

Take a USD 100 million 5-year swap in which the bank pays 6.195 percent annually and receives LIBOR, analysed just before a reset. Assuming an at-the-market swap, the short fixed-rate bond is worth USD 100 million and so is the long FRN, so the swap is worth zero. The fixed flows are discounted at zero rates of 5.813, 5.929, 6.034, 6.130 and 6.217 percent for years 1 to 5, and the vertex risks and correlations are those of the bond example earlier.

The individual VaR column sums to the undiversified VaR; the component column sums to the diversified VaR.

The same swap as a strip of forwards

The swap can equally be viewed as five forward contracts. The first pays USD 100 million plus the 6.195 percent coupon in 1 year, worth −USD 100.36 million at the 5.813 percent spot rate, against USD 100 million received now, which carries no risk. The 1 × 2 contract pays USD 106.195 million in 2 years, worth −USD 94.64 million, against USD 100 million in 1 year, worth USD 94.50 million. The pattern continues to the 4 × 5 contract.

Contract VaRs as published. Recomputing each contract from the vertex risks and correlations of the bond example reproduces 0.471 and 0.571, but gives 0.466, 0.435 and 0.368 for the last three contracts, so the undiversified total comes to about 2.311 rather than 2.401.

Adding the five printed contract VaRs gives an undiversified USD 2.401 million (about USD 2.31 million on recomputation), higher than before because the netting of flows at each date now happens across contracts rather than inside them. The diversified VaR is USD 2.152 million, identical to the two-bond view, which confirms that the two decompositions describe the same risk.

After the first floating payment has been set, the FRN becomes a 1-year bond valued at par, so USD 100 million moves to the year-1 vertex and the year-1 position changes from −USD 5.855 million to USD 94.145 million. The VaR falls from USD 2.152 million to USD 1.763 million, because the long 1-year FRN now offsets part of the short fixed leg. In general, a swap’s VaR shrinks toward zero as it approaches maturity, dipping each time a coupon is set.

Options are nonlinear in their underlying, which creates difficulties for the delta-normal approach, since that approach is fundamentally linear. The Black-Scholes model, extended by Merton (1973) to an asset with a continuous yield r*, gives the value of a European call when the spot price follows geometric Brownian motion with constant volatility σ:

A Taylor expansion approximates the change in option value through its partial derivatives, all of which have closed forms under Black-Scholes:

Values as published. An independent Black-Scholes calculation reproduces every entry to within 0.001 (for example, rho* at K = 100 recomputes to −0.134).

Delta is not constant

The first sensitivity, delta, matters most. For a European call it is

Linear VaR methods hold delta fixed over the risk horizon, and the quality of that shortcut depends on the parameters. With a 1-day horizon, the worst down move is αSσ√(1/252) = 1.645 × 100 × 0.20 × √(1/252), about USD 2.07 (quoted as USD 2.08, which matches a multiplier of 1.65), for a worst price of USD 97.92. For the 90-day option, delta only moves from 0.536 to 0.452, so the linear effect dominates and a linear approximation is acceptable for longer options over short horizons. A 10-day option near the money is a very different story.

Options as forwards with N(d) weights

Keeping only the linear terms in the spot price and the two rates gives

This is the currency forward decomposition with weights attached: the spot and foreign bill positions are scaled by N(d₁), and the dollar bill by N(d₂). Deep in the money, both N(d₁) and N(d₂) approach one and the call collapses into a forward, c = Se^−r*τ − Ke^−rτ. Since the dollar bill position Ke^−rτN(d₂) equals ΔS − c, a long call is equivalent to a long position of Δ in the asset financed by a short bill of (ΔS − c).

Across a portfolio, all options on the same underlying are converted to delta equivalents and summed. The approach extends to net vega (Λ) where implied volatility is a risk factor, and to net gamma (Γ) for second-order effects, and these exposures can be combined with simulated factor moves to produce a full risk distribution. Derivatives like these are mapped exhaustively, with no specific risk left over, whereas individual stocks and corporate bonds keep some specific risk, which washes out only in large, well-diversified portfolios.