While working with Value at Risk, we are often looking at calculating the P/L of individual positions, and then calculating the portfolio P/L based on the individual positions. However, there are a few problems in doing so.

First, it is not possible to model each and every position in the portfolio, as there could be many positions with different complexities or lacking historical data to support the data model. Second, the portfolios of these positions can become quite complex with many dependencies. For example, the covariance matrix of risk factors in all the instruments (say n instruments) can become extremely large. The number of volatilities (n), and data on correlations (n(n-1)/2) grows as n grows. This also poses challenge in terms of the computational power required to process the data.

The process can be significantly simplified by mapping these positions to a small set of risk factors. Instead of trying to calculate VaR for each instrument, what we need to do is to decompose these instruments into building blocks, or primitive instruments, which are further mapped to a small set of risk factors. The four common building blocks are: equity positions, zero-coupon bonds, futures/forwards, and spot foreign exchange.

The generic VaR mapping process is as follows:
<ol>
	<li>Choose a set of elementary risk factors and estimate their probability distributions</li>
	<li>Mapping: Decompose financial instruments into exposures on these risk factors. Each position is marked to market. Market value of each instrument is allocated to risk factors.</li>
	<li>Aggregate the exposure for all positions and build the distribution of P/L on portfolio. Arrive at the portfolio VaR.</li>
</ol>

VaR: Mapping Positions to Risk Factors

In the previous article, we learned the need for <a title="VaR: Mapping Positions to Risk Factors" href="https://financetrain.com/var-mapping-positions-to-risk-factors/">VaR Mapping</a>, and how financial instruments can be broken down into simpler building blocks/primitive instruments, which are mapped to a small set of risk factors.

One such building block instrument is spot FX positions, where you hold a foreign currency instrument whose value is fixed in terms of the foreign currency.

The mapping system of the financial institution will have a set of core currencies for which various historical and estimated data such as volatilities and correlations will already be available in the system.

If our FX position is based on one of these core currencies then our work is even easier. However, if our position is on a currency not from this core currency list, then we will have derive a proxy of our position in terms of one of the core currencies.

Assuming the exchange rates follow a normal distribution, the VaR of an FX spot position will be calculated as follows:

VaR = -Zα * σx * P * X

Where,
<ul>
	<li>P is the value of our position if foreign currency units</li>
	<li>X is the exchange rate</li>
</ul>
Let’s say that you have an FX spot position of EUR10,000,000 and the domestic currency is USD. Exchange rate is 1 EUR = 1.23 USD.

The estimated volatility is 20%.

As 99% confidence level, the daily VaR for this position will be:

VaR = -2.32635 * 0.20/Sqrt(250) * 12,300,000

= USD 361,943

This method can be applied to other spot positions also, such as commodities.

Mapping Spot FX Positions

Equity positions are a bit more tricky compared to the <a title="Mapping Spot FX Positions" href="https://financetrain.com/mapping-spot-fx-positions/">FX positions</a>. The main problem arises while dealing with the correlations between the different equity positions. As the number of positions in the equity portfolio increases, the correlation matrix can become substantially large and almost impossible to handle.

In case of equity positions, the number of risk factors are reduced by modeling the individual stock returns using a factor model. The simplest way to do so is to use the CAPM model under which the returns from a security can be represented using the equity market returns.

ri = α + βrm + ε

Where,

α represents the firm-specific constant

ε is a zero-mean specific constant

β is the stock’s beta

rm is the market return.

If we assume that the equity portfolio of a bank is sufficiently diversified, then the α of different stocks will offset each other and the returns of the portfolio can be simply written as

rp = βprm + ε

Where:

βp is the portfolio beta

rm is the market return

Let’s take a simple example to understand how we can use this to calculate the portfolio <a href="https://financetrain.com/series/all-about-value-at-risk/">VaR</a>.

Let’s say we have a portfolio of $100 million invested in three stocks, namely A, B, and C. Also assume that the related stock market index is the S&amp;P500 index.

The details are provided below:
<table border="1" cellspacing="0" cellpadding="0">
<tbody>
<tr>
<td valign="top" width="106"></td>
<td valign="top" width="106">Stock A</td>
<td valign="top" width="106">Stock B</td>
<td valign="top" width="106">Stock C</td>
</tr>
<tr>
<td valign="top" width="106">Position</td>
<td valign="top" width="106">$30 million</td>
<td valign="top" width="106">$30 million</td>
<td valign="top" width="106">$40 million</td>
</tr>
<tr>
<td valign="top" width="106">Beta</td>
<td valign="top" width="106">0.70</td>
<td valign="top" width="106">1.20</td>
<td valign="top" width="106">1.30</td>
</tr>
<tr>
<td valign="top" width="106">Equivalent S&amp;P500 position</td>
<td valign="top" width="106">$21 million</td>
<td valign="top" width="106">$36 million</td>
<td valign="top" width="106">$52 million</td>
</tr>
<tr>
<td valign="top" width="106">Total</td>
<td valign="top" width="106">$109</td>
<td valign="top" width="106"></td>
<td valign="top" width="106"></td>
</tr>
</tbody>
</table>
The equivalent S&amp;P500 position is calculated as Position Size * Beta. This means that $100 million invested in three stocks is equivalent to $109 million invested in S&amp;P500, considering the systematic risk.

Using this information, the VaR of the portfolio can be calculated in terms of the index value. Assume that the daily volatility of S&amp;P500 index is 0.90%. The 99% daily VaR of the three stock portfolio will be:

VaR99% = $109 * 2.326 * 0.9% =  $2.28 million

An alternative way to calculate the VaR of the same portfolio would be by directly using the stock’s weighted average beta:

VaR99% = $100 * 2.326 * 0.9% * (0.70*3/10+1.20*3/10 +1.30*4/10)

= $2.28 million

As you can see, Mapping can significantly simplify the VaR calculation.

If our portfolio had 1,000 stocks, and referenced 10 different markets, then our VaR calculation could be simplified to 10 different positions on these 10 markets, rather than dealing with the complexity of 1000 different stocks as individual exposures.

Mapping Equity Positions

We earlier learned that VaR calculation of a complex portfolio can be simplified by decomposing the instruments into building blocks, or primitive instruments, which are further <a title="VaR: Mapping Positions to Risk Factors" href="https://financetrain.com/var-mapping-positions-to-risk-factors/">mapped to a small set of risk factors</a>. Zero-coupon bonds are one such primitive instrument apart from <a title="Mapping Spot FX Positions" href="https://financetrain.com/mapping-spot-fx-positions/">spot FX positions</a>, <a title="Mapping Equity Positions" href="https://financetrain.com/mapping-equity-positions/">equity positions</a>, and futures/forwards. In this article we will learn about mapping cash flows for a zero-coupon bond.

Consider a 5-year Treasury bond with semi-annual payments. This bond has 10 cash flows that are sensitive to different parts of the term structure. The first coupon is sensitive to the 6-month interest rate, the next coupon is sensitive to the one-year interest rate, and the last (10th) payment will be sensitive to the 5-year zero-coupon interest rate.

For the purpose of mapping each cash flow, the risk manager will need to identify a set of zero-coupon bonds at different maturities. J.P. Morgan suggests 14 different maturities, namely, 1 month, 3 month, 6 month, and 1, 2, 3, 4, 5, 7, 9, 10, 15, 20, and 30 years.

For our 5-year Treasury bond, 6 out of 10 cash flows will directly map to the benchmark maturities, while the others will be converted into an equivalent cash flows suiting one of the benchmark maturities. For example, the cash flow at 2.5 years will be mapped into two equivalent cash flows, one with a maturity of one year and another with two years. The risk sensitivity of cash flow at 2.5 years will be equivalent to the two cash flows, one at 2 years, and the other at 3 years. In other words, the original cash flow at 2.5 years is mapped to two cash flows (at 2 and 3 years).

A simple way to do this is to perform a linear interpolation. However, in reality the method for such transformation to equivalent cash flow can be complex. Even interest rate futures are used to calibrate the curve to the market.

One way to do is to transform the original cash flow into two cash flows with the same value and modified duration. Another way is to use two weights &alpha; and (1-&alpha;) and divide the cash flow such that the return volatility of the portfolio of two equivalent cash flows is the same as the return volatility of the original cash flow.

Mapping Zero-coupon Bonds to Risk Factors

We have learned about how various complex positions can be broken down into elementary blocks which can be further <a href="https://financetrain.com/var-mapping-positions-to-risk-factors/">mapped to risk factors</a>. The four such elementary blocks/instruments are <a href="https://financetrain.com/mapping-spot-fx-positions/">Spot FX positions</a>, <a href="https://financetrain.com/mapping-equity-positions/">Equity Positions</a>, <a href="https://financetrain.com/mapping-zero-coupon-bonds-to-risk-factors/">Zero-coupon bonds</a>, and Future/Forward positions.

In this article we will look at mapping <a title="Introduction to Futures Contracts" href="https://financetrain.com/introduction-to-futures-contracts/">futures/forward positions</a>.

A forward contract is an agreement to buy a specific asset, a commodity or a financial asset, at a fixed price on a fixed future date. While a forward contract is an OTC instrument and is customized for every transaction, a futures contract is a standardized contract and is traded on exchanges. For the purpose of understanding the mapping, let’s assume that they both are the same.

Similar to spot positions, the futures/forward positions also provide daily returns based on the price movement. Let’s say that we have n futures contracts each valuing V in our portfolio. Assume that the futures returns follow a normal distribution with a standard deviation (σ) and a mean of 0. As a certain confidence level (Zα), the VaR of the portfolio can be calculated as follows:
<h2>VaR = -ZασnV</h2>
At 99% confidence level, a portfolio with 10 futures contracts each valuing $10,000, and a standard deviation of 20%. will have a VaR of.

VaR = -2.33*0.20 * 10 * 10000 = -$46,600

The calculation is quite straightforward; the only practical issue is to estimate the standard deviation. Typically we will have pre-estimates for some time horizons, and for the inbetween time horizons we will interpolate the volatilities.

Mapping Futures/Forward Positions to Risk Factors

We have now learned about how VaR can be calculated for primitive positions including spot FX positions, equity positions, zero-coupon bonds, and futures/forward positions. We will now look at how complex positions can be decomposed into these primitive instruments which are mapped to risk factors.

In this article we will take a look at some of these complex positions.

Coupon Paying Bonds: A coupon paying bond can be looked at as a portfolio of zero-coupon instruments (Think each coupon payment as a zero-coupon bond). The VaR of the coupon bond will be calculated based on the VaR of the mapped equivalents of zero-coupon cash flows at benchmark maturities. See <a href="https://financetrain.com/mapping-zero-coupon-bonds-to-risk-factors/">Mapping Zero-coupon Bonds</a>.

Forward Rate Agreement: <a href="https://financetrain.com/forward-rate-agreements-and-calculating-fra-payments/">Forward Rate Agreements</a> (FRA’s) are similar to forward contracts where one party agrees to borrow or lend a certain amount of money at a fixed rate on a pre-specified future date. It’s an exchange of a single floating amount for a single fixed amount at a given future date. The payoff from the floating leg of the FRA can be regarded as a portfolio of two zero-coupon bonds. One long position in the zero-coupon bond maturing at the end of FRA  and a short position in a zero-coupon bond maturing at beginning of FRA period, both bonds having same face value and notional principal. The VaR of the FRA will then be the VaR of the portfolio of these two zero-coupon bonds. See <a href="https://financetrain.com/mapping-zero-coupon-bonds-to-risk-factors/">Mapping Zero-coupon Bonds</a>.

Floating Rate Notes: Floating rate notes (FRNs) have variable coupons and the coupon is reset at the beginning of the coupon period. A floating rate note has the same sensitivity as a zero-coupon bond with maturity equal to the reset interval, and is often regarded as essentially immune to interest rate risk. Therefore, the VaR of an FRN can be calculated as if it’s a zero-coupon bond that matures on the next coupon date. See <a href="https://financetrain.com/mapping-zero-coupon-bonds-to-risk-factors/">Mapping Zero-coupon Bonds</a>.

Vanilla Interest rate Swaps: An interest rate swap can be looked at as a portfolio of a long position in a fixed-coupon bond and a short position in a floating rate bond, which are mapped to zero-coupon bonds as seen above.

Structured Notes: A structured note can be treated as a portfolio of interest-rate swaps and floating rate notes.

FX Forwards: This is a combination of a long position in foreign currency zero-coupon bond and a short position in domestic zero-coupon bond.

Similarly any complex position can be broken down into a combination of the four primitive instruments.

Mapping Complex Financial Positions

In the previous article we learned about how to <a title="Mapping Complex Financial Positions" href="https://financetrain.com/mapping-complex-financial-positions/">map various complex positions</a> such as coupon paying bonds, FRAs, and swaps. All these complex positions had linear payoffs and were fairly straightforward to map. However, there are other positions such as options where the payoff is non-linear. In such a case the mapping process becomes a bit more complex.

For portfolios and positions with non-linear payoff, the mapping is done with the help of first-order and second-order Taylor approximation, i.e., the Delta and Delta-Gamma approach.

Let’s look at these two approaches.

Delta Approach

Let’s say that we have a call option with a value of c. The value of this option will depend on many variables such as the underlying stock price (S), the strike price, and the volatility. Ignoring other factors and considering the underlying stock price as the only factor, we can use the first order approximation, in which case:

Δc = δΔS

Where, δ is the option’s delta, which is readily available for actively traded options.

With this information, we can calculate the option’s VaR as follows:

VaR = δZασS

This approach however assumes that we have a short holding period and the delta remains constant. Also, it may not be very reliable for positions with high optionality or additional nonlinear features.

Delta-Gamma Approach

Our approximation can be made more accurate by using the second order Taylor approximation (Delta-Gamma), in which case the call value will be represented as follows:

Δc = δΔS +g/2(ΔS)2

Note that for a long call option, both Delta and Gamma will be positive, and for a short position both will be negative. A positive gamma favourably impacts the option value, and vice verse.

The VaR of the position can be calculated as follows:

VaR = δZασΔS – γ/2(ZασS)2

Note the minus sign in the formula. This is because positive gamma reduces VaR and vice verse.

Example:

Let’s say we have the following information about a European call option:

S =  100

r = 6%

Option maturity = 3 months

S = 25% (Annual)

D = 0.6

G = 2.2

A 10-day VaR at 99% confidence interval can be calculated as follows:

VaR = 0.6*2.33*Sqrt(10/250)*0.25 –(2.2)/2*(2.33* Sqrt(10/250)*0.25)^2

VaR = 0.0549

Note that VaR without incorporating gamma would have been higher.

VaR Mapping for Options Positions

Mapping Positions to Risk Factors

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Mapping Positions to Risk Factors

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