- Define, calculate, and distinguish between the following portfolio ๐๐๐ measures: individual ๐๐๐ , incremental ๐๐๐ , marginal ๐๐๐ , component ๐๐๐ , undiversified portfolio ๐๐๐ , and diversified portfolio ๐๐๐ .
- Explain the role of correlation on portfolio risk.
- Describe the challenges associated with ๐๐๐ measurement as portfolio size increases.
- Apply the concept of marginal ๐๐๐ to guide decisions about portfolio ๐๐๐ .
- Explain the risk-minimizing position and the risk and return-optimizing position of a portfolio.
- Explain the difference between risk management and portfolio management, and describe how to use marginal ๐๐๐ in portfolio management.

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- A portfolio can be characterized by positions on a certain number of constituent assets, expressed in the base currency, say, dollars. If the positions are fixed over the selected horizon, the portfolio rate of return is a linear combination of the returns on underlying assets, where the weights are given by the relative amounts invested at the beginning of the period

\(R_{P,t+1} = \sum_{i=1}^{N} w_i R_{i,t+1}\)

The rate of return is defined as the change in the dollar value, or dollar return, scaled by the initial investment. This is a unitless measure.

In matrix notation

\(R_P = w_1 R_1 + w_2 R_2 + \cdots + w_N R_N = \begin{bmatrix} w_1 & w_2 & \cdots & w_N \end{bmatrix} \begin{bmatrix} R_1 \\ R_2 \\ \vdots \\ R_N \end{bmatrix} = wโR \)

- The portfolio expected return is

\(E(R_P) = \mu_P = \sum_{i=1}^{N} w_i \mu_i \)

and the Variance is

As the number of assets increases, it becomes difficult to keep track of all covariance terms, which is why it is more convenient to use matrix notation. The variance can be written as

Defining โ as the covariance matrix, the variance of the portfolio rate of return can be written more compactly as

This also can be written in terms of dollar exposures ๐ฅ as

**Diversified**๐ฝ๐๐น is the portfolio ๐๐๐ , taking into account diversification benefits between components

\( \text{Portfolio VaR} = \text{VaR}_P = \alpha \sigma_P W = \alpha \sqrt{xโ \Sigma x} \)

**Individual**๐ฝ๐๐น**is t**he ๐๐๐ of one component taken in isolation. At this point, the individual risk of each component can be defined as

\( \text{VaR}_i = \alpha \sigma_i \left| x_i \right| = \alpha \sigma_i \left| w_i \right| W \)

The correlation coefficient is a more convenient, scale-free measure of linear dependence:

\( \rho_{12} = \frac{\sigma_{12}}{\sigma_1 \sigma_2} \)

- Risk tends asymptotically to zero. More generally, portfolio risk is

\( \sigma_P = \sigma \sqrt{\frac{1}{N} + \left(1 โ \frac{1}{N}\right) \rho } \)

which tends to \( \sigma \sqrt{\rho} \) as ๐ increases.

- For a two asset portfolio, the โdiversifiedโ portfolio variance is

\( \sigma_P^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \rho_{12} \sigma_1 \sigma_2 \)

The portfolio ๐๐๐ is then

\( VaR_P = \alpha \sigma_P W = \alpha \sqrt{ w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \rho_{12} \sigma_1 \sigma_2 } W \)

- ๐๐๐ is a coherent risk measure for normal and, more generally, elliptical distributions.
- When the correlation is exactly unity and ๐ค
_{1}and ๐ค_{2}are both positive, **Undiversified**๐ฝ๐๐น The sum of individual ๐๐๐ s, or the portfolio ๐๐๐ when there is no short position and all correlations are unity.- This interpretation differs when short sales are allowed. Suppose that the portfolio is long asset 1 but short asset 2 (๐ค
_{1}is positive, and ๐ค_{2}is negative). This could represent a hedge fund that has $1 in capital and a $1 billion long position in corporate bonds and a $1 billion short position in Treasury bonds, the rationale for the position being that corporate yields are slightly higher than Treasury yields. If the correlation is exactly unity, the fund has no risk because any loss in one asset will be offset by a matching gain in the other. The portfolio ๐๐๐ then is zero. - Instead, the risk will be greatest if the correlation is -1, in which case losses in one asset will be amplified by the other.

Consider a portfolio with two foreign currencies, the Canadian dollar (๐ถ๐ด๐ท) and the euro (๐ธ๐๐ ). Assume that these two currencies are uncorrelated and have a volatility against the dollar of 5 and 12 percent, respectively. The first step is to mark to market the positions in the base currency. The portfolio has ๐๐ $2 million invested in the ๐ถ๐ด๐ท and ๐๐ $1 million in the ๐ธ๐๐ . Find the portfolio ๐๐๐ at the 95 percent confidence level.

**Marginal**๐๐๐ is**t**he change in portfolio ๐๐๐ resulting from taking an additional dollar of exposure to a given component. It is also the partial (or linear) derivative with respect to the component position.

\( \Delta VaR_i = \frac{\partial VaR_P}{\partial x_i} = \frac{\partial VaR_P}{\partial w_i W} = \alpha \frac{\partial \sigma_P}{\partial w_i} = \alpha \frac{cov(R_i, R_P)}{\sigma_P} \)

- This marginal ๐๐๐
is closely related to the
**beta***,*defined as

\( \beta_i = \frac{cov(R_i, R_P)}{\sigma_P^2} = \rho_{iP} \frac{\sigma_i}{\sigma_P} \)

- Using matrix notation, the vector ๐ฝ, including all assets, can be written as

\( \beta = \frac{\Sigma w}{wโ \Sigma w} \)

The relationship between the ฮ๐๐๐ and ๐ฝ is

\( \Delta VaR_i = \alpha (\beta_i \times \sigma_P) = \frac{VaR_P}{W} \times \beta_i \)

**Incremental**๐ฝ๐๐น is the change in the ๐๐๐ of a portfolio from the addition of a new position in a portfolio. The incremental ๐๐๐ then as described in this Figure, as

๐ผ๐๐๐๐๐๐๐๐ก๐๐ ๐๐๐
=ย ๐๐๐
_{P+a} โ ๐๐๐
_{P}

- Incremental ๐๐๐ differs from the marginal ๐๐๐ in that the amount added or subtracted can be large, in which case ๐๐๐ changes in a nonlinear fashion.

- The main drawback of this approach is that it requires a full revaluation of the portfolio ๐๐๐ with the new trade. This can be quite time-consuming for large portfolios.

- A shortcut can be taken to get an approximation

\( Incremental \, VaR \approx (\Delta VaR)โ \times a \)

This measure is much faster to implement because the ฮ๐๐๐
vector is a by-product of the initial ๐๐๐
_{P} computation. The new process is described in this Figure.

- Going back to the previous two-currency example, now consider increasing the ๐ถ๐ด๐ท position by ๐๐ $10,000.

**Component**๐ฝ๐๐น is a partition of the portfolio ๐๐๐ that indicates how much the portfolio ๐๐๐ would change approximately if the given component was deleted. By construction, component ๐๐๐ s sum to the portfolio ๐๐๐ .

\( Component \, VaR_i = (\Delta VaR_i) \times w_i W = \frac{VaR \beta_i}{w_i W} = VaR \beta_i w_i \)

The quality of this linear approximation improves when the ๐๐๐ components are small. Hence this decomposition is more useful with large portfolios, which tend to have many small positions.

\( CVaR_1 + CVaR_2 + \cdots + CVaR_N = VaR_P \left( \sum_{i=1}^{N} w_i \beta_i \right) = VaR_P \)

because the term between parentheses is simply the beta of the portfolio with itself, which is unity.

- The component ๐๐๐ can be simplified further.

\( CVaR_i = VaR_P w_i \beta_i = (\alpha \sigma_P W) w_i \beta_i = (\alpha \sigma_i w_i W) \rho_i = VaR_i \rho_i \)

This conveniently transforms the individual ๐๐๐ into its contribution to the total portfolio simply by multiplying it by the correlation coefficient.

- Percent contribution to ๐๐๐ of component

\( i = \frac{CVaR_i}{VaR} = w_i \beta_i \)

- Continuing with the previous two-currency example, find the component ๐๐๐ for the portfolio.

- Marginal ๐๐๐ and component ๐๐๐ are useful tools, best suited to small changes in the portfolio. This can help the portfolio manager to decrease the risk of the portfolio. Positions should be cut first where the marginal ๐๐๐ is the greatest, keeping portfolio constraints satisfied. For example, if the portfolio needs to be fully invested, some other position, with the lowest marginal ๐๐๐ , should be added to make up for the first change.
- This process can be repeated up to the point where the portfolio risk has reached a global minimum. At this point, all the marginal ๐๐๐ s, or the portfolio betas, must be equal:

\( \Delta VaR_1 = \frac{VaR}{W} \times \beta_i = constant \)

- This table illustrates this process with the previous two currency portfolio.

- The next step is to consider the portfolio expected return as well as its risk. Indeed, the role of the portfolio manager is to choose a portfolio that represents the best combination of expected return and risk.
- For simplicity, all returns are defined in excess of the risk- free rate. In the figure, this translates all the points down by the same amount so that the risk-free asset is at the origin.

- Suppose now that the objective function is to maximize the ratio of expected return to risk. This Sharpe ratio is

\( SR_P = \frac{E_P}{\sigma_P} \)

- This can be modified and written with ๐๐๐ in the denominator instead of standard deviation.

- At the optimum,

\( \frac{E_i}{\Delta VaR_i} = \frac{E_i}{\beta_i} = \text{constant} \)