### Portfolio Risk: Analytical Methods

In this chapter, the role of correlation in portfolio risk will be explained. The chapter further expounds on the challenges of measuring VaR when increasing the size of the portfolio. The concept of marginal VaR will then be applied.

Portfolio VaR measures, including incremental VaR, undiversified portfolio VaR, individual VaR, diversified portfolio VaR, marginal VaR, and component VaR, will all be defined, computed, and distinguished.

A portfolio’s risk minimizing position, and its risk and return-optimizing position will be explained in detail. Finally, the chapter will give an explanation on the differences between risk management and portfolio management, and a further description of the application of marginal VaR when managing a portfolio.

# Portfolio VaR

Positions on a certain number of constituent assets can be used to characterize a portfolio. The expression should be in the base currency, e.g., dollars. The portfolio return rate becomes a linear combination of returns on underlying assets in case the positions are fixed over the selected horizon. The relative amounts invested at the start of the period gives the weights.

The portfolio return rate is defined from $$t$$ to $$t + 1$$, as follows:

$${ R }_{ p,t+1 }=\sum _{ i=1 }^{ N }{ { W }_{ i }{ R }_{ i,t+1 } }$$

The number of assets is $$N$$, the rate of return of asset $$i$$ is given as $${ R }_{ i,t+1 }$$, and the weight is given as $${ W }_{ i }$$. The change in dollar value, or the dollar return, scaled by the initial investment is termed as the rate of return and is a unitless measure.

Each constituent asset is a security in a traditional mean/variance analysis. Contrary to that, the component is defined by VaR as a risk factor and is defined as the linear exposure to the said risk factor.

A matrix notation can be used to write the portfolio return so as to shorten the notation, and a string of numbers can be replaced by a single vector:

$${ R }_{ P }={ w }_{ 1 }{ R }_{ 1 }+{ w }_{ 2 }{ R }_{ 2 }+\cdots +{ w }_{ N }{ R }_{ N }=\left[ { w }_{ 1 }w_{ 2 }\dots { w }_{ N }\left[ \begin{matrix} { R }_{ 1 } \\ \vdots \\ { R }_{ N } \end{matrix} \right] ={ w }^{ \prime }R \right]$$

Where the transported vector of weights is denoted by $${ w }^{ \prime }$$, and the vertical vector with individual asset returns is denoted as $$R$$.

The following expression gives the portfolio expected return:

$$E\left( { R }_{ P } \right) ={ \mu }_{ P }=\sum _{ i=1 }^{ N }{ { w }_{ i }{ \mu }_{ i } }$$

The variance is given as:

$$V\left( { R }_{ P } \right) ={ \sigma }_{ P }^{ 2 }={ \Sigma }_{ i=1 }^{ N }{ w }_{ i }^{ 2 }{ \sigma }_{ i }^{ 2 }+{ \Sigma }_{ i=1 }^{ N }{ \Sigma }_{ j=1 }^{ N }{ w }_{ i }{ w }_{ j }{ \sigma }_{ ij }={ \Sigma }_{ i=1 }^{ N }{ w }_{ i }^{ 2 }{ \sigma }_{ i }^{ 2 }+2{ \Sigma }_{ i=1 }^{ N }{ \Sigma }_{ j<1 }^{ N }{ w }_{ i }{ w }_{ j }{ \sigma }_{ ij }\quad \quad \quad \left( a \right)$$

This summation adds up to a total of $${ N\left( N-1 \right) }/{ 2 }$$ different terms and does not only account for the risk of the individual securities $${ \sigma }_{ i }^{ 2 }$$.

The variance can be expressed in the following way:

$${ \sigma }_{ P }^{ 2 }=\left[ { W }_{ 1 }\dots { W }_{ N } \right] \left[ \begin{matrix} { \sigma }_{ 1 }^{ 2 } & \cdots & { \sigma }_{ 1N } \\ \vdots & \ddots & \vdots \\ { \sigma }_{ N1 } & \cdots & { \sigma }_{ NN } \end{matrix} \right] \left[ \begin{matrix} { W }_{ 1 } \\ \vdots \\ { W }_{ N } \end{matrix} \right]$$

Let the covariance matrix be defined as $$\Sigma$$, the following is a more compact way of writing the variance of the portfolio:

$${ S }_{ P }^{ 2 }={ w^{ \prime } }\sum { w }$$

Let $$W$$ define the unitless weights. Writing this in terms of dollar exposure can be as follows:

$${ S }_{ P }^{ 2 }={ x^{ \prime } }\sum { x }$$

A confidence level $$c$$ can be translated into a standard normal deviate $$\alpha$$ making the likelihood of observing a loss that is worse than $$-\alpha$$ to $$c$$. If $$W$$ is defined as the portfolio value, the portfolio VaR is:

$$Portfolio\quad VaR={ VaR }_{ P }=\alpha { \sigma }_{ P }W=\alpha \sqrt { { x }^{ \prime }\Sigma x }$$

Each component’s individual risk can at this point be defined as:

$${ VaR }_{ j }=\alpha { \sigma }_{ j }|{ W }_{ j }|=\alpha { \sigma }_{ j }|{ W }_{ j }|W$$

It should be noted that since the value of the risk weight, $${ W }_{ j }$$, can be negative, then its absolute value is taken and the risk measure should be positive.

A more convenient, scale-free measure of linear dependence is the correlation coefficient:

$${ \rho }_{ 12 }={ { \sigma }_{ 12 } }/{ \left( { \sigma }_{ 1 }{ \sigma }_{ 2 } \right) }$$

Where the correlation coefficient, $$\rho$$, must always lie between -1 and +1. The two variables are considered perfectly correlated when close to 1, and uncorrelated when they are zero.

The general representation of portfolio risk is as follows:

$${ \sigma }_{ P }=\sigma \sqrt { \frac { 1 }{ N } +\left( 1-\cfrac { 1 }{ N } \right) \rho }$$

This risk tends to $$\sigma \sqrt { \rho }$$ as $$N$$ increases. The portfolio risk is usually diversified by low correlations. The diversified portfolio variance for portfolio with only two assets is:

$${ \sigma }_{ P }^{ 2 }={ w }_{ 1 }^{ 2 }{ \sigma }_{ 1 }^{ 2 }+{ w }_{ 2 }^{ 2 }{ \sigma }_{ 2 }^{ 2 }+2{ w }_{ 1 }{ \sigma }_{ 1 }{ \rho }_{ 12 }{ \sigma }_{ 1 }{ \sigma }_{ 2 }\quad \quad \quad \quad I$$

And therefore, the portfolio VaR is given as:

$${ VaR }_{ P }=\alpha { \sigma }_{ P }W=\alpha \sqrt { { w }_{ 1 }^{ 2 }{ \sigma }_{ 1 }^{ 2 }+{ w }_{ 2 }^{ 2 }{ \sigma }_{ 2 }^{ 2 }+2{ w }_{ 1 }{ \sigma }_{ 1 }{ \rho }_{ 12 }{ \sigma }_{ 1 }{ \sigma }_{ 2 } } W\quad \quad \quad \quad II$$

The portfolio VaR reduces to the following expression in case the correlation $$\rho$$ is zero:

$${ VaR }_{ P }=\sqrt { { \alpha }^{ 2 }{ w }_{ 1 }^{ 2 }{ \sigma }_{ 1 }^{ 2 }+{ { \alpha }^{ 2 }w }_{ 2 }^{ 2 }{ \sigma }_{ 2 }^{ 2 }{ \sigma }_{ 2 }^{ 2 } } =\sqrt { { VaR }_{ 1 }^{ 2 }+{ VaR }_{ 2 }^{ 2 } } \quad \quad \quad \quad III$$

The sum of the individual VaRs should be higher than the portfolio risk: $${ VaR }_{ P }<{ VaR }_{ 1 }+{ VaR }_{ 2 }$$.

$${ w }_{ 1 }$$ and $${ w }_{ 2 }$$ are both positive if the correlation is exactly 1, and thus equation $$II$$ becomes:

$${ VaR }_{ P }=\sqrt { { VaR }_{ 1 }^{ 2 }+{ VaR }_{ 2 }^{ 2 }+2{ VaR }_{ 1 }\times { VaR }_{ 2 } }$$

$$={ VaR }_{ 1 }+{ VaR }_{ 2 }$$

The difference between diversified VaR and undiversified VaR can be applied in the evaluation of the diversification benefit. This is typically shown in VaR reporting systems.

# VAR Tools

VaR was initially developed as a methodology through which portfolio risk can be evaluated. In order to most effectively modify the VaR, the information on the positions has to be altered, which is quite crucial since portfolios are typically traded incrementally owing to the costs of transactions. This is the role of the VaR tools: marginal, component, and incremental VaRs.

## Marginal VaR

Individual VaRs are insufficient in the evaluation of the impact of the variation of positions on portfolio risk. Starting with an existing portfolio comprised of $$N$$ securities, enumerated as $$j=1,\dots ,N,$$ by adding a unit of security, a new portfolio will be obtained.

Its marginal contribution to risk is evaluated by increasing $$w$$ by a minute amount or a differentiation of equation $$\left( a \right)$$ with respect to $${ w }_{ i }$$, for the assessment of the effect of this trade:

$$\frac { \partial { \sigma }_{ P }^{ 2 } }{ \partial { w }_{ i } } =2{ w }_{ i }+2\sum _{ j=1,j\neq i }^{ N }{ { w }_{ j }{ \sigma }_{ ij } }$$

$$=2cov\left( { R }_{ i },{ w }_{ i }{ R }_{ i }+\sum _{ j\neq i }^{ N }{ { w }_{ j }{ R }_{ j } } \right) =2cov\left( { R }_{ i },{ R }_{ p } \right)$$

The derivative of the volatility is needed instead of that of the variance. The sensitivity of the portfolio volatility to a change is given as follows, having noticed that $$\frac { \partial { \sigma }_{ P }^{ 2 } }{ \partial { w }_{ i } } =\frac { 2{ \sigma }_{ P }\partial { \sigma }_{ P } }{ \partial { w }_{ i } }$$:

$$\frac { \partial { \sigma }_{ P } }{ \partial { w }_{ i } } =\frac { cov\left( { R }_{ i },{ R }_{ P } \right) }{ { \sigma }_{ P } }$$

The expression of the marginal VaR can be determined by converting it into a VaR number. The following is the vector component for the expression of the marginal VaR:

$$\Delta { VaR }_{ 1 }=\frac { \partial VaR }{ \partial { X }_{ i } } =\frac { \partial VaR }{ \partial { w }_{ i }W } =\alpha \frac { \partial { \sigma }_{ P } }{ \partial { w }_{ i } } =\alpha \frac { cov\left( { R }_{ i },{ R }_{ P } \right) }{ { \sigma }_{ P } }$$

This marginal VaR measure is unitless due to the fact that this was defined as a ratio of the dollar amounts.

The following defined beta is closely related to this marginal VaR:

$${ \beta }_{ i }=\frac { cov\left( { R }_{ i },{ R }_{ P } \right) }{ { \sigma }_{ P }^{ 2 } } =\frac { { \sigma }_{ ip } }{ { \sigma }_{ P }^{ 2 } } =\frac { { \rho }_{ iP }{ \sigma }_{ i }{ \sigma }_{ P } }{ { \sigma }_{ P }^{ 2 } } ={ \rho }_{ iP }\frac { { \sigma }_{ i } }{ { \sigma }_{ P } }$$

This is an evaluation of the contribution of one security to total portfolio risk. Another terminology given to beta is systemic risk of security $$i$$ vis-à-vis portfolio $$P$$, and its evaluation can be from the slope coefficient in a regression of $${ R }_{ i }$$ on $${ R }_{ P }$$:

$${ R }_{ i,t }={ \alpha }_{ i }+{ \beta }_{ i }{ R }_{ P,t }+{ \epsilon }_{ i,t }\quad \quad \quad \forall t=1,\dots ,T$$

The vector beta can be written as follows, applying matrix notation and including all assets:

$$\beta =\frac { \Sigma W }{ { W }^{ \prime }\Sigma W }$$

The relationship between marginal VaR and beta can be summarized as follows:

$$\Delta { VaR }_{ 1 }=\frac { \partial VaR }{ \partial { X }_{ i } } =\alpha \left( { \beta }_{ i }\times { \sigma }_{ P } \right) =\frac { VaR }{ W } \times { \beta }_{ i }$$

## Incremental VaR

The total effect of a proposed trade on a portfolio $$P$$ can also be evaluated with this methodology. A position $$a$$ is used to represent the new trade. $$a$$ is a vector of additional exposures to our risk factors, measured in dollars.

The following expression is applied to obtain the incremental VaR:

$$Incremental\quad VaR={ VaR }_{ P+a }-{ VaR }_{ P }$$

The change in VaR owing to a new position is called incremental VaR. The difference between incremental VaR and marginal VaR is that the added or subtracted amount can be large, thus changing the VaR in a nonlinear fashion.

$${ VaR }_{ P-a }$$ can be expanded in a series around the original point:

$${ VaR }_{ P-a }=VaR_{ P }+{ \left( \Delta VaR \right) }^{ \prime }\times a+\cdots$$

In case the deviations are small, the second order terms are ignored. Therefore the incremental VaR can be approximated as:

$$Incremental\quad VaR\approx { \left( \Delta VaR \right) }^{ \prime }\times a$$

The application of the incremental VaR can be to the general case where a new set of exposures are involved in a trade. The variance of the dollar returns on the new portfolio can be written as:

$${ \sigma }_{ P+a }^{ 2 }{ W }_{ p+a }^{ 2 }={ \sigma }_{ P }^{ 2 }{ W }^{ 2 }+2aW{ \sigma }_{ ip }+{ a }^{ 2 }{ \sigma }_{ i }^{ 2 }$$

Differentiating with regards to $$a$$, we get:

$$\frac { { \partial \sigma }_{ P+a }^{ 2 }{ W }_{ P+a }^{ 2 } }{ \partial a } =2W{ \sigma }_{ ip }+2a{ \sigma }_{ i }^{ 2 }$$

And it attains a value of zero for:

$${ a }^{ \ast }=-W\frac { { \sigma }_{ iP } }{ { \sigma }_{ i }^{ 2 } } =-W{ \beta }_{ i }\frac { { \sigma }_{ P }^{ 2 } }{ { \sigma }_{ i }^{ 2 } }$$

This is the best hedge and is the variance minimizing-position.

## Component VaR

Having a decomposition of the current portfolio is of extreme importance, for risk to be managed. An additive VaR decomposition that gives recognition to the power of decomposition is needed.

For this reason, we count on marginal VaR to evaluate the contribution of each asset to the existing portfolio risk. The marginal VaR is multiplied by the current dollar position in asset or risks factor $$i$$ such that:

$$Component\quad VaR_{ i }=\left( \Delta { VaR }_{ i } \right) \times { w }_{ i }W=\frac { VaR{ \beta }_{ i } }{ W } \times { w }_{ i }W=VaR{ \beta }_{ i }{ w }_{ i }$$

Therefore, the portfolio VaR is an indication of the approximate change of portfolio VaR in case the component was deleted from the portfolio.

The total portfolio VaR is precisely a summation of these component VaRs, that is:

$${ CVaR }_{ 1 }+{ CVaR }_{ 2 }+\cdots { CVaR }_{ N }=VaR\left( \sum _{ i=1 }^{ N }{ { w }_{ i }{ \beta }_{ i } } \right) =VaR$$

This is due to the fact that the term between the parentheses happens to be the beta of the portfolio with itself, and is 1.

The fact that $${ \beta }_{ i }$$ equals the product of the correlation $${ \rho }_{ i }$$ and $${ \sigma }_{ i }$$ divided by the portfolio $${ \sigma }_{ P }$$ is taken into account to further simplify the portfolio VaR:

$${ CVaR }_{ i }=VaR{ w }_{ i }{ \beta }_{ i }=\left( \alpha { \sigma }_{ P }W \right) { w }_{ i }{ \beta }_{ i }=\left( \alpha { \sigma }_{ P }{ w }_{ i }W \right) { \rho }_{ i }={ VaR }_{ i }{ \rho }_{ i }$$

Through this equation, the individual VaR is conveniently transformed into its contribution to the total portfolio by simply getting its product with the correlation coefficient.

Finally, normalization can be done by the total portfolio VaR to report the percent contribution to VaR of the component:

$$i=\frac { { CVaR }_{ i } }{ VaR } ={ w }_{ i }{ \beta }_{ i }$$

Applying the desired criterion, a breakdown of the contribution to risk can be provided by the VaR systems.

## VaR Tools for General Distributions

The portfolio return can be considered as a function of the positions on the individual components $${ R }_{ P }=f\left( { w }_{ 1 },\dots, { w }_{ N } \right)$$. If all positions are multiplied by a vector $$k$$, then the portfolio return will be enlarged by a similar amount.

$$k{ R }_{ P }=f\left( k{ w }_{ 1 },\dots, k{ w }_{ N } \right)$$

This function is considered to be homogeneous of degree one, and the Euler’s Theorem can be applied. The Euler’s theorem states that:

$${ R }_{ P }=f\left( { w }_{ 1 },\dots ,{ w }_{ N } \right) =\sum _{ i=1 }^{ N }{ \frac { \partial f }{ \partial { w }_{ i } } } { w }_{ i }$$

Setting $${ R }_{ P }$$ to the portfolio VaR, the following expression can be obtained:

$$VaR=\sum _{ i=1 }^{ N }{ \frac { \partial VaR }{ \partial { w }_{ i } } } \times { w }_{ i }=\sum _{ i=1 }^{ N }{ \frac { \partial VaR }{ \partial { X }_{ i } } } \times { X }_{ i }$$

$$=\sum _{ i=1 }^{ N }{ \left( \Delta { VaR }_{ i } \right) } \times { X }_{ i }$$

With a normal distribution, the marginal VaR, which is proportional to $${ \beta }_{ i }$$, can be expressed as follows:

$$\Delta { VaR }_{ i }={ \beta }_{ i }\left( \alpha { \sigma }_{ P } \right)$$

Supposing that a distribution of returns $${ R }_{ P,1 },\dots ,{ R }_{ P,T }$$ has been generated by a manager, and it cannot be estimated by an elliptical distribution perhaps due to an irregular shape owing to option positions, then the observation $${ R }_{ P }^{ \ast }$$ is used to estimate the VaR, which can lead to the following equation when the Euler’s Theorem has been applied:

$${ R }_{ P }^{ \ast }=\sum _{ i=1 }^{ N }{ E\left( { R }_{ i }|{ R }_{ P }={ R }_{ P }^{ \ast } \right) { w }_{ i } }$$

The expectation of the risk factor conditional on the portfolio having a return equivalent to VaR is given as $$E\left( . \right)$$.

Therefore, it is true to assert that $${ CVaR }_{ i }$$ can be estimated when $${ R }^{ \ast }$$ is decomposed into the realized value of each component. Since the basis of the estimates is on one data only, the estimates are less reliable.

# From VaR to Portfolio Management

## From Risk Measurement to Risk Management

Since the marginal VaR and component VaR are best suited to small variations in the portfolio, the portfolio manager can monitor the risk of the portfolio. Keeping the portfolio constraints satisfied, the cutting of the position should be where the marginal VaR is the greatest.

We can repeat this process until the portfolio risk has reached a global minimum. At this point, we must have equality in all the marginal VaRs or portfolio betas:

$$\Delta { VaR }_{ 1 }=\frac { VaR }{ W } \times { \beta }_{ i }=constant$$

## From Risk Management to Portfolio Management

The expected return of the portfolio and its risk are considered in the next step. Choosing a portfolio representing the best combination of expected return and risk is the portfolio manager’s duty. Therefore, we are shifting from risk management to portfolio management.

The expected return of the portfolio is defined as:

$${ E }_{ P }=\sum _{ i=1 }^{ N }{ { w }_{ i }{ E }_{ i } }$$

The definition of all returns, for the sake of simplicity, is in excess of the risk-free rate. The best portfolio combinations can then be defined as the set of optimal portfolios that offers the highest expected return for a defined level of risk or the lowest risk for a given level of expected return; this defines the efficient frontier.

Assuming that maximizing the ratio of expected return to risk is the objective function. This Sharpe ratio is:

$${ SR }_{ P }=\frac { { E }_{ P } }{ { \sigma }_{ P } }$$

Our wish now is to increase the portfolio’s expected return while moving to the portfolio whose Sharpe ratio is the highest. This portfolio is called the optimal portfolio. At this point, we must have an equal ratio of all expected returns to marginal VaRs.

The following is the relationship at the optimum:

$$\frac { { E }_{ i } }{ \Delta { VaR }_{ i } } =\frac { { E }_{ i } }{ { \beta }_{ i } } =constant$$

According to Roll (1977), the expected return on any component asset must be proportional to its beta relative to the portfolio. This is what is implied when any portfolio is said to be efficient, that is:

$${ E }_{ i }={ E }_{ m }{ \beta }_{ i }$$

Therefore, we must have a constant ratio between the excess return $${ E }_{ i }$$ and the beta.

# Practice Questions

1) The portfolio of X&M Bank is comprised of two foreign currencies, the Euro (EUR) and Sterling Pound (GBP). Suppose that there is no correlation between these two currencies and that they have a 5% and 9% probability of default against the dollar, respectively. The portfolio has USD 2.1 million invested in the EUR and USD 1.9 million invested in the GBP. X&M is considering increasing the GBP position by USD 12,500. Using the marginal VaR method, calculate the incremental VaR with the assumption that $$\alpha=1.65$$.

1. USD 1,578.75
2. USD 2,317.85
3. USD 1,432.65
4. USD 2,561.25

We determine the dollar volatility by calculating the product:

$$\sum { x } =\begin{bmatrix} { 0.05 }^{ 2 } & 0 \\ 0 & { 0.09 }^{ 2 } \end{bmatrix}\begin{bmatrix} 2.1 \\ 1.9 \end{bmatrix}=\begin{bmatrix} { 0.05 }^{ 2 }\times 2.1+0\times 1.9 \\ 0\times 2.1+{ 0.09 }^{ 2 }\times 1.9 \end{bmatrix}=\begin{bmatrix} 0.00525 \\ 0.01539 \end{bmatrix}$$

Therefore, the portfolio value in dollar returns is:

$${ \sigma }_{ P }^{ 2 }{ W }^{ 2 }={ x }^{ \prime }\sum { x } =\begin{bmatrix} \begin{matrix} 2.1 & \quad 1.9 \end{matrix} \end{bmatrix}\begin{bmatrix} 0.00525 \\ 0.01539 \end{bmatrix}=2.1\times 0.00525+1.9\times 0.01539$$

$$=0.040266$$

The dollar volatility is:

$$\sqrt { 0.040266 } =0.201\quad million$$

Therefore, the marginal VaR is given as:

$$\Delta VaR=\alpha \frac { cov\left( R,{ R }_{ P } \right) }{ { \sigma }_{ P } } =1.65\times \frac { \left[ \begin{matrix} 0.00525 \\ 0.01539 \end{matrix} \right] }{ 0.201 } =\left[ \begin{matrix} 0.0432 \\ 0.1263 \end{matrix} \right]$$

As the second position is increased by USD 12,500, the incremental VaR becomes:

$${ \left( \Delta VaR \right) }^{ \prime }\times a =\left[ \begin{matrix} 0.0432\quad & 0.1263 \end{matrix} \right] \left[ \begin{matrix} 0 \\ 12,500 \end{matrix} \right]$$

$$=0.0432\times 0+0.1263\times 12,500$$

$$=1,578.75$$