Mapping a Position to a Risk factor
So far we have assumed that each position in a portfolio is matched directly to an underlying risk factor (see Analytical VaR post). However in practice this may not be feasible or even desirable. The market practice is to map each position to what is known as a risk factor, the process of risk factor mapping.
The main reasons for risk factor mapping are as follows:
– We don’t have enough historical data for certain positions. This may be true of new issues or securities with very little liquidity and hence market price. In these cases it would be advisable to map these securities to a risk factor.
– The covariance matrix may get too large if we treat every asset as a risk factor and leads to large unworkable data files. The idea of mapping is particularly appealing in this scenario. This is why it is best to map each asset to a market risk factor , so for example you might choose to map blue chip UK stocks to the FTSE100 Index.
– The final reason is that of course as the covariance matrix gets too large we will experience very large computational times. Reducing this by using risk mapping greatly speeds up the process of calculating VaR particularly when dealing with simulations.
The key to effective mapping is to recognize that most instruments can be decomposed into a small amount of basic asset classes / risk factors. So we would try and effectively get a portfolio of constituent building blocks. The four main types of basic building blocks are :
– Spot FX
– Zero-Coupon Bonds / Interest Rates
– Futures / Forwards
Mapping Spot Positions
This is probably the more simple of all the building blocks. There is generally a finite number of currencies in the world. So most systems should have the data for them and any volatility and correlations required.
Let us look at the example of a simple position, that has a value of A in the foreign currency, the FX rate is X. SO the position has a value of AFX. If we assume there is no other risk here then the only risk we have is the uncertainty around the exchange rate X.
This gives us a nice and easy way of calculating the VaR analytically using our formula :
Imagine that the position AX was worth $1 million USD. You want to convert this to EUR the post rate is USD 1.25 to the Euro. This means the position is worth EUR 800,000 (our base currency).
The USDEUR volatility is 10% p.a. On a daily basis this is : 0.63%.
Let us now calculate VaR :
VAR(1,0.05) = 1.64485 * 0.63% * 800,000 = EUR 8,290
VAR(1,0.01) = 2.32634 * 0.63% * 800,000 = EUR 11,724
Mapping Equity Positions
This gets somewhat more complicated as the equity universe is quite large. We turn to the CAPM theory for a solution to this problem. If you recall the CAPM says that the return of an equity is related to the market by the following equation :
Rk – Return on Stock
Rm- Return on market
αk- Equity specific
βk- Market Return
εk- Equity spefic error uncorrelated
The variance of the firms return then becomes :
σ2k – Equity Variance
σ2m – Market Variance
σ2k,t – Error term Variance
The VaR of equity would then be:
Xk : Value of Firm
σk : Volatility of Firm
Z : Confidence Interval
When we have a portfolio the diversification effect kicks in, the main contributor in this scenario will be the market risk component of Beta and the market volatility. As the specific risk of each equity is uncorrelated to the market, the share of total risk from each specific risk term will fall as the portfolio size increases. This will even approach zero as the portfolio tends towards the same composition as the market. This in effect is the benefit of diversification.
Now remember that if used a full correlation for 10,000 stocks we would have a correlation matrix of 50 million individual elements. Reduce this to the mapping factor approach and we only need the market volatility , 10,000 volatilities and 10,000 betas. In effect this only 0.04% of the number the full correlation approach would need.
The Systematic VaR of the portfolio under this approach is :
we can modify the above equation to get the net beta of the portfolio as follows:
We have 4 stocks A, B , C, D in a portfolio worth $1million, with equal weightings of each. The stats are as follows :
Market Volatility Daily: 0.015
The Betas are : A : 0.8 , B : 0.9 , C : 0.3 , D, 0.5
All the equities are mapped to the same index so there is no covariance to deal with.
Based on the above equation we would have VaR at 95% as follows :
VaR = 1.64485 * 1000,000 * 0.015 * 0.25 * [0.8+0.9+0.3+0.5] = $15,420
The one point to note is that this approach in an undiversified portfolio will significantly underestimate risk. A portfolio with a lot of concentration risk in a single sector or industry will also lead to underestimation of risk as the firm specific risk component will be quite large.
Mapping a Zero-Coupon Bond
The next simple mapping is to a yield curve – or the zero curve. The mapping will be to the standard tenors in a yield curve.
This approach requires the sensitivity of the portfolio to interest rates which is represented by the DV01 of the positions. Let us look at an example.
– Let us say we have a cash flow that will occur in exactly 2.75 years, of 1000,000.
– We would then create 2 cash flows to replicate the risk profile of the cash flow. In the above case as we only have possible mappings to 2 years and 3 years, so we map to both.
– Next we can interpolate the 2.75 year rate from the above table :
r2.75 =r2 *(3 – 2.75) + r3 *(2.75 – 2.00) = 0.045 * 0.25 + 0.051 * 0.75 = 0.0495.
– Increase the 2 year rate by 1 bp to get 4.9525
– Increase the 3 year rate by 1 bp to get 4.9575
– Next we can calculate the DV01.
– PV012=1000,000 * (exp(-2.75 * 4.9525% ) – exp(-2.75 * 4.95% ) = -60
– PV013=1000,000 * (exp(-2.75 * 4.9575% ) – exp(-2.75 * 4.95% ) = -180
– The next step is to find the flows at 2 and 3 years that have the same DV01. That is to says satisfy the equations :
– We solve by rearranging :
– So now we know that the risk sensitivity of the position is effectively the same as the cash flows of 328,285 at 2 years and 699,299 at 3 years.
– The next step to calculate VaR is to know the volatilities of the 2 year and the 3 year buckets. Let us assume that the 2 year daily volatility is 1% and the 3 year daily volatility is 1.2%.
– With interest rate volatilities you have to be careful a 1% daily volatility means that you have multiply that by the rate to get the move : 4.5% * 1% = 4.5 basis points
– The 3 year shift is : 1.25% * 5.1% = 6.38 basis points
– Based on the sensitivities from before we can now calculate the VaR. Further assume that the 2 and 3 year rates have a correlation of 80%.
– Based on the Equation for portfolio volatility we can calculate the VaR:
– We can now work out the VaR based on the the figures we have and we will get a portfolio variance of 1,373 for the position above. At 95% Confidence Interval we will multiply by 1.645 to get 2,260.
We have now calculated the VaR by working out the sensitivity from scratch. On a practical level in most investment banks and financial firms the sensitivities will be calculated by the trading system. These sensitivities can then be fed into a risk system in order to calculate the VaR. So while this is not needed it is still useful to know how these calculations are done especially to check and verify figures.
Mapping Futures & Forwards
While we know that there are differences in how futures and forwards work on a practical level, for risk purposes we can treat them as analogous. The position from a futures of a forward can be mapped to an appropriate risk factor.
One problem with a futures position is that quite often you cannot get a full time series , especially if the futures contract is quite new. To counter these problems it is possible to get a time series for the generic contract from most data providers , which maintain a long and detailed time series of the first few futures contracts. This can be quite useful as it is often necessary to also capture the spread between different month futures contracts sometimes. That is to say an investor may be short the March Brent contract and long the June (the first and the second with respect to today lets assume). If we assumed that both had the same spot risk factor then we would see this position as risk flat (assuming the positions are the same size). However the risk is not the same and in fact the two may have basis risk which should be taken into account. Hence using the generic futures contracts should be able to capture the risk much more closely.
The VaR would be given by :
You can watch this interesting video on youtube on Risk Factor mapping