Portfolio Return and Variance (Calculations for CFA® and FRM® Exams)

Portfolio Expected Return

Portfolio expected return is the sum of each of the individual asset’s expected return multiplied by its associated weight. Thus:

$${E(R_p)}=\sum{w_ir_i}$$

Where:

$$i$$ = 1, 2, 3, …, n;

$$w_i$$ = the weight attached to asset i; and

$$r_i$$ = the asset’s return.

The weight attached to an asset = market value of asset/market value of the portfolio

Example 1: Portfolio Expected Return

Assume we have a simple portfolio of two mutual funds, one invested in bonds and the other invested in stocks. Let’s further assume that we expect a stock return of 8% and a bond return of 6% and our allocation is equal in both funds.

The expected portfolio is closest to:

A. 7%.

B. 8%.

C. 14%.

Solution

We know that:

\begin{align*}{E(R_p)}=&\sum{w_ir_i}\end{align*}

Therefore,

\begin{align*}{E(R_p)}=&(0.5\times0.08)+(0.5\times0.06) \\ =& 0.07 \ or \ 7\% \end{align*}

Standard Deviation of a Portfolio

The standard deviation of a portfolio of assets, or portfolio risk, is NOT simply the sum of the underlying securities’ risk. Due to the correlation between securities, the computation of the portfolio risk must incorporate this correlation relationship.

The portfolio standard deviation or variance comprises two key parts: the variance of the underlying assets plus the covariance of each underlying asset pair.

It is a crucial tool that helps match the risk level of a portfolio with a client’s risk appetite, and it measures the total risk in the portfolio comprising both the unsystematic risk and systematic risk.

A larger standard deviation suggests more volatility and more dispersion in the returns, and riskier. It assists in measuring the consistency in which returns are generated and is a good measure to evaluate the consistency of returns on Hedge Funds and the performance of Mutual funds

Computing Standard Deviation of Portfolio

For a portfolio with two underlying assets, X and Y, we can compute the portfolio variance as follows:

\begin{align*}\text{Portfolio variance}=&{W_X^2\sigma_X^2}+{W_Y^2\sigma_Y^2}+2W_XW_Y{\sigma_X}{\sigma_Y}{\rho_{XY}}\end{align*}

Therefore,

\begin{align*}\text{Portfolio standard deviation}=&\sqrt{{W_X^2\sigma_X^2}+{W_Y^2\sigma_Y^2}+2W_XW_Y\sigma_X\sigma_Y\rho_{XY}}\end{align*}

Where:

$$W$$ = the weight of the asset within the portfolio;

$$\sigma$$ = the standard deviation; and

$$\rho$$ = the correlation coefficient.

Note that $$\sigma_X\sigma_Y\rho_{XY}$$ = the covariance between asset X and asset Y.

A Quick Trick to Remember this Formula

\begin{align*}\text{Portfolio variance}=&{W_X^2\sigma_X^2}+{W_Y^2\sigma_Y^2}+2W_XW_Y{\sigma_X}{\sigma_Y}{\rho_{XY}}\end{align*}

Recall from algebra:

\begin{align*}{(a+b)^2}=&a^2+b^2+2ab\end{align*}

Where,

\begin{align*}a=&W_X\sigma_X\end{align*}

\begin{align*}b=&W_Y\sigma_Y\end{align*}

\begin{align*}=&{W_X^2\sigma_X^2}+{W_Y^2\sigma_Y^2}+2W_XW_Y{\sigma_X}{\sigma_Y}{\rho_{XY}}\end{align*}

Example 1: Standard Deviation of Portfolio

A portfolio is made up of two assets. Asset A has an allocation of 80% and a standard deviation of 16%, and asset B has an allocation of 20% and a standard deviation of 25%.

If the correlation coefficient between assets A and B is 0.6, the portfolio standard deviation is closest to:

A. 16.3%.

B. 2.7%.

C. 22%.

Solution

We know that:

\begin{align*}\text{Portfolio variance}=&{W_X^2\sigma_X^2}+{W_Y^2\sigma_Y^2}+2W_XW_Y{\sigma_X}{\sigma_Y}{\rho_{XY}}\\=&{(0.8)^2}\times{(0.16)^2}+{(0.2)^2}\times{(0.25)^2}+2(0.8)(0.2)(0.16)(0.25)(0.6)\\=&2.66\%\end{align*}

Thus,

\begin{align*}\text{Portfolio standard deviation}=&\sqrt{{W_X^2\sigma_X^2}+{W_Y^2\sigma_Y^2}+2W_XW_Y\sigma_X\sigma_Y\rho_{XY}}\\=&\sqrt{2.66\%}\\=&16.3\%\end{align*}

Example 2: Standard Deviation of Portfolio

Assume we have equally invested in two different companies; ABC and XYZ. We anticipate that there is a 15% chance that next year’s stock returns for ABC Corp will be 6%, a 60% probability that they will be 8% and a 25% probability that they will be 10%. In addition, we already know the expected value of returns is 8.2%, and the standard deviation is 1.249%.

We also anticipate that the same probabilities and states are associated with a 4% return for XYZ Corp, a 5% return, and a 5.5% return. The expected value of returns is 4.975% and the standard deviation is 0.46%.

Calculate the portfolio standard deviation:

A. 0.0000561

B. 0.00001

C. 0.00851

Solution

First, we must calculate the covariance between the two stocks:

$$Cov(R_{ABC},R_{XYZ})$$

\begin{align*}=&0.15(0.06-0.082)(0.04-0.04975)+0.6(0.08-0.082)(0.05-0.04975)+0.25(0.10-0.082)(0.055-0.04975)\\=&0.0000561\end{align*}

Then, we calculate the portfolio variance:

\begin{align*}\text{Portfolio standard deviation}=&\sqrt{{W_X^2\sigma_X^2}+{W_Y^2\sigma_Y^2}+2W_XW_Y\sigma_X\sigma_Y\rho_{XY}}\\=&0.5^2\times0.01249^2+0.5^2\times0.0046^2+2\times0.5\times0.5\times0.0000561\\=&0.00007234\end{align*}

Therefore,

\begin{align*}\text{Portfolio standard deviation}=&\sqrt{{W_X^2\sigma_X^2}+{W_Y^2\sigma_Y^2}+2W_XW_Y\sigma_X\sigma_Y\rho_{XY}}\\=&\sqrt{0.00007234}\\=&0.00851\end{align*}

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