45.1.1 Domestic Equity: SIM

Consider a portfolio consisting of n-stocks held in a specific country (e.g. USA). Forecasting the variances and particularly the correlations (covariances) across all stocks is computationally extremely burdensome, particularly for the covariance terms as there are so many of them. To ease the computational burden, we use the SIM since this allows all of the variances and covariances between returns to be subsumed into the n-asset betas (and a forecast for the variance of the market return). The SIM is:

(45.1)

where is the return on stock-i, is the market return, is the beta of stock-i. The are (usually) assumed to be niid and in addition we make the crucial assumption that there is no contemporaneous correlation across the random error terms for different stocks (at time t), . Hence, we assume that firm ‘specific risk’ (e.g. due to patent applications, IT failures, strikes, reputational effects, etc.) are uncorrelated across different firms. (Also, is independent of .) An estimate of can be obtained either from a ‘risk measurement service’ (e.g. investment bank) or by running a time series regression of on (using say one year of daily returns data). Using (45.1) the return on a portfolio of n-stocks can be represented as:

(45.2)

where , , and is the proportion of total portfolio value held in each stock. From Equation (45.2), the portfolio return depends linearly on the market return (given the portfolio beta ). It is easily shown that if , the variance of the portfolio return is:

(45.3a)

where .

The specific risk of each stock can be ‘diversified away’ when stocks are held as part of a well-diversified portfolio (with small amounts held in each stock). Hence the term is small and can be ignored (see Appendix 45.B):

(45.3b)

Thus when using the SIM we only require estimates of for the n-stocks and a forecast of , in order to calculate . We therefore dispense with the need to estimate n-values for the standard deviation of individual stock returns and more importantly the n(n –1)/2 values for the correlation coefficients between all of the stock returns. This considerably reduces the computational burden of estimating the portfolio-VaR, which is given by:

(45.4)

The SIM is a ‘factor model’ with only one factor, the market return . However, this approach could be extended within the VaR framework, by assuming several factors affect each stock return. For example, a widely used model is the so-called Fama–French three-factor model, where the return on any stock is determined by the market return , the return on small (market cap) stocks minus large (market cap) stocks and the return on high book-to-market minus low book-to-market stocks . Applying the SIM and the Fama–French three-factor model to each of the n-stock returns, implies that the forecast of portfolio variance only depends on estimates of the three factor betas, and forecasts of the three variances and six distinct covariances between the ‘three factors’ of the Fama–French model.

45.1.2 Foreign Assets

How do we deal with exchange rate risk when calculating VaR for a portfolio that contains foreign assets? Suppose a US investor (Mr Trump) holds a €100m diversified portfolio in German stocks which exactly mirror movements in the German stock market index, the DAX. So, Trump's German stock portfolio has a beta of 1 with respect to the DAX market index. The dollar value of Mr Trump's German portfolio is subject to changes in the DAX and changes in , the Dollar-Euro spot FX-rate. In effect Mr Trump holds a two-asset portfolio, the foreign stock itself and the foreign currency. The percentage return on the German portfolio (in USD) is¹:

(45.5)

where = return on the spot-exchange rate, that is . If stocks in the DAX fall at the same time as the euro depreciates against the US dollar, then Mr Trump would lose in USDs on both counts – a ‘double whammy’ of losses on the DAX itself plus losses when euros are converted into dollars. This is because after depreciation of the euro, for every euro that Mr Trump has in the DAX, he obtains fewer dollars.

Hence a positive correlation between the DAX index and the euro exchange rate increases the riskiness of the US investor's portfolio, measured in US dollars. Conversely, a negative correlation between the DAX and the euro exchange rate implies a lower dollar risk for Mr Trump's German stock portfolio.

To measure the dollar-VaR for a US investor with stocks held in the DAX we have to modify our previous VCV approach. First, because we are interested in the dollar-VaR we have to convert Trump's initial position in Euros, into USD. If Mr Trump has €100m invested in the DAX and the current Dollar-Euro spot FX rate is (USD per Euro), then the initial USD position is . The key equation (see Appendix 45.A) is:

It follows directly that the dollar-VaR is then given by the usual formula:

(45.6)

where

The Z-vector ‘contains’ the same $-value, for both the DAX and the FX position. This is because for a US investor, holding the DAX is equivalent to holding both in the DAX plus in foreign exchange (euros). The correlation matrix C contains the correlation coefficient between the return on the DAX and the Euro-dollar spot exchange rate.

45.1.3 German and US Stock Portfolios

Suppose (as above) a Mr Trump (a US investor) holds a portfolio of German stocks but the beta of the German stocks is and Mr Trump also holds a portfolio of US stocks with . Assume the US investor holds dollars in US stocks and dollars in German stocks (that is, , where is the Euro-value of the German stock portfolio and is the current USD-Euro exchange rate). Assume the SIM holds within any single country, hence:

(45.7)

where = standard deviation of the S&P 500 US stock market index. The change in the dollar value of Mr Trump's portfolio is (approximately) linear in returns:

(45.8)

where = return on German stock portfolio and , the proportionate change in the spot FX rate. The USD-VaR for this portfolio is:

(45.9)

This approach is easily extended to holding several foreign portfolios where for each foreign stock portfolio there are two entries in the Z vector – the VaR of the foreign stock portfolio and the VaR of the US-foreign currency, FX rate.

Suppose Mr Trump has a US stock portfolio as well as German and Brazilian stock portfolios (which are unhedged). Then there may still be some risk reduction (and hence a low portfolio VaR) if some of the spot FX rates have low (or negative) correlations with either the US-dollar exchange rate, or with stock market returns in different countries. If Mr Trump hedges his spot FX positions (using the forward market) then clearly any spot-FX volatilities and the correlation between spot FX-rates and foreign stock returns do not enter the calculation of VaR. But we cannot say unequivocally whether an unhedged or a FX-hedged ‘world portfolio’ has a lower risk – it depends on the interplay of the correlation coefficients in the two scenarios.

45.2.1 VaR: Non-parallel Shifts in the Yield Curve

Can we improve on this approach and calculate the VaR for a portfolio of coupon paying bonds when there are non-parallel shifts in the yield curve? The way to do this is to consider a single coupon paying bond, with n-years to maturity, as a series of zero-coupon bonds. (This is extended to a portfolio of coupon paying bonds, below.) For a zero-coupon bond which has single cash flow at , the current value (price) is:

(45.11)

= (continuously compounded) spot yield for period . The proportionate change in value (price) of the zero-coupon bond is the ‘return’ on the bond, that is . Differentiating (45.11) we have a linear relationship between the return on the zero-coupon bond and the (absolute) change in yield:

(45.12)

where for a zero-coupon bond (i.e. a zero-coupon bond's duration equals its maturity – measured in years). From (45.12):

A coupon paying bond is just a series of zero-coupon bonds and the value/price of a coupon bond with n-cash flows at time is²:

(45.13)

Hence:

(45.14)³

The dollar change in value of the coupon paying bond is (approximately) linear in the returns of its constituent zero-coupon bonds (for small changes in yields). We have ‘mapped’ the non-linear ‘yield-bond price relationship’ into an (approximate) linear framework. In addition, if we assume yield changes are normally distributed, we can use Equation (45.14)³ to calculate the VaR (at the 5th percentile) for a coupon paying bond:

(45.15)

where , and C is the (n × n) correlation matrix of zero-coupon bond returns.

In general we do not forecast the volatility of bond returns directly, but instead forecast using the EWMA model. Then provides a forecast of the standard deviation of (zero-coupon) bond returns for use in Equation (45.15). Similarly, we use the EWMA model to forecast the covariance between changes in spot yields from which we obtain the correlation coefficients (for use in the correlation matrix ):

All of the above equations can be applied to a portfolio consisting of coupon paying bonds. The cash flows from the m-bonds at each time period are aggregated to give the total coupon payments at this date, . The current value of these coupons from all the bonds is then and the analysis follows as above.

45.2.2 Mapping Cash Flows

A large investment bank may have coupon receipts on its bond portfolio virtually every week for the next 20–30 years. We cannot use current spot yields for every horizon over the next 30 years – there would be too many cash flows to deal with. To reduce the scale of the problem ‘standard vertices’ of say 1, 3, 6, and 12 months and 2, 3, 4, 5, 7, 9, 10, 15, 20, and 30 years, are used. All cash flows are ‘mapped’ onto these standard vertices and bond return volatilities and correlations are provided only for these chosen vertices.

For example, suppose an actual cash flow of $100m at (Figure 45.1) has to be apportioned between the standard vertices at and . One approach (see Appendix 45.C) is to allocate cash flows to adjacent standard vertices to ensure that:

• Total market value of the cash flows allocated to adjacent vertices (e.g. 5 and 7 years) equals the market value of the original cash flow (i.e. $100m at years).
• The two cash flows at and both have the same sign as the original cash flow.
• The volatility of (present) value of the two cash flows at and equal the volatility of the (present) value of the original cash flow at .

45.2.3 Swaps

A pay-fixed, receive-floating interest rate swap is equivalent to a short position in a fixed rate bond and a long position in a FRN. The VaR for the fixed leg can therefore be analysed as a standard coupon paying bond. The value of the floating leg can be treated as a zero-coupon bond with a single known payment at the next payment date, discounted by the time to the next payment. Hence the VaR of the floating leg can also be calculated using the cash-flow mapping approach.

45.2.4 Principal Components Analysis

A useful alternative way of parsimoniously representing changes in say different spot yields is to use principal components analysis (PCA). PCA ‘represents’ the changes in all the spot yields by a few (usually about three) new variables called principal components (PCs).⁴ When computing the VaR of a bond portfolio we then use these three principal components, in place of the many correlations between the original spot yields.

PCA is a statistical technique which takes a matrix of changes in k-spot yields X, and seeks to ‘explain’ the movement in all these yields using just the three principal components, q_i (i = 1, 2, 3). The q's are linear combinations (i.e. weighted averages) of the ‘raw’ spot rate data in the X-matrix:

(45.16)

where c_i is an estimated constant. (It is in fact the eigenvalue corresponding to the largest eigenvector of the (X^′X) matrix.) A major benefit of PCA is that the q's are orthogonal (uncorrelated) with each other and hence their correlation coefficients are zero (by construction).

It is generally found that about 95% of the variation in, say, 15 spot yields (e.g. for years 1, 2, 3, … 15) can be ‘empirically explained’ by about three PCs. The first PC is interpreted as representing parallel shifts in the yield curve, the second represents a twist in the yield curve, whereas the third (which explains the smallest proportion of the variation in the yield data) represents a ‘bowing’ in the yield curve (which occurs relatively infrequently). What is important for our VaR calculation is that the volatility of these 15 spot yields can be ‘linearly mapped’ into only three variables – the three PCs, . The three PCs are uncorrelated with each other and this considerably reduces the dimensionality of the VaR calculations.

A US investor holds Euros in German stocks, which in USD is , where is the current Euro-USD spot exchange rate. The USD change in value of this portfolio is:

(45.A.1)

Let be the (proportionate) change in the spot exchange rate (i.e. the return on foreign exchange). Substituting and rearranging:

(45.A.2a)

(45.A.2b)

where we have ignored the cross product terms, which are small. In (45.A.2b) we see that the USD investor is effectively holding equal dollar amounts in German stocks and in the Euro-USD exchange rate. Hence it follows directly from (45.A.2):

(45.A.3)

Therefore the USD-VaR for the US investor who holds the German portfolio is:

(45.A.4)

(45.A.5)

In addition, if we use the SIM then where is the portfolio-beta of the German stock portfolio and is the volatility of the German stock market index, the DAX.

Using the single index model (SIM) we show that the standard deviation of a portfolio of n-stocks is:

(45.B.1)

where = portfolio standard deviation, = standard deviation of the market portfolio and is the ‘beta’ of the stock portfolio. The SIM for each stock return is:

(45.B.2)

Assumptions:

(45.B.3a)

(45.B.3b)

(45.B.3c)

(45.B.3d)

It follows that :

(45.B.4a)

(45.B.4b)

(45.B.4c)

The portfolio return and variance are:

(45.B.5a)

(45.B.5b)

The formula for portfolio variance requires n-variances and n(n – 1)/2 covariances. For example for this amounts to 11,325, inputs to estimate. To reduce the number of inputs required we utilize the SIM. Substituting from Equation (45.B.2) in (45.B.5a):

(45.B.6)

(45.B.7)

Equation (45.B.7) may be interpreted as, ‘Total Portfolio Risk = Market Risk + Specific Risk’. Examining the specific risk term more closely and assuming , that is an equally weighted (i.e. well diversified) portfolio:

(45.B.8)

where we have used the key assumption of the SIM namely, . The average specific risk is defined as and as this term goes to zero (n = 35 randomly selected stocks is usually sufficient to ensure this last term is relatively small). Hence under the SIM and assuming a well-diversified portfolio:

(45.B.9)

A crucial assumption in the SIM is – the covariance of firm specific ‘random shocks’ to stock-i and stock-j are contemporaneously uncorrelated, across all stocks. This is reasonable for stocks in different sectors (e.g. oil and IT) but not for stocks in the same sector (e.g. Shell and BP). However, in a well-diversified portfolio even though there may be some small positive correlations between some and for daily returns, nevertheless portfolio specific risk still falls quite rapidly as increases and it is small relative to market (beta) risk. To see this note that from (45.B.8) that if then:

(45.B.10)

where is the average variance of the firm's specific risks and is the average covariance of the (contemporaneous) specific risks across firms:

(45.B.11a)

and

(45.B.11b)

From (45.B.11a) the variance term goes to zero as n increases so for large , the variance of portfolio specific risk equals the average covariance between specific risks across different stocks, . In a well-diversified portfolio where stocks are held across many different sectors (and countries), the average covariance of ‘specific risk’ across firms is likely to be small relative to market risk, . Hence Equation (45.B.9) is a reasonable approximation. Of course, if our portfolio is not well diversified (e.g. we hold only energy stocks) then the assumption of the SIM that will be incorrect and may be quite large. In this case our simplified expression for portfolio risk ceases to hold and all the covariances are required for an accurate calculation of .