FIN 300

Risk and Return
Lecture 10

Topics Covered



  • Expected Return
  • Variance and Standard Deviation
  • Portfolios
  • Portfolio Variance
  • Diversification and Risk

Expected Return

  • Point-estimate of average belief
  • Many possible versions of the future

$E(R) = \sum_{s=1}^S Pr(R_s) \times R_s$

  • $S$ possible states of the world (s)
  • $R_s$ is the return to the stock in state s
  • $Pr(R_s)$ probability of $R_s$
Example

Suppose there was a $50\%$ probability that a stock has a return of $5\%$ and a $50\%$ probability the stock had a return of $15\%$. The expected return $E(R)$ would be:


$E(R) = 50\% \times 5\% + 50\% \times 15\% = 10\%$

$E(R) = 0.5 \times 0.05 + 0.5 \times 0.15 = 0.1$

We could also have calculated the expected return in a similar way if there was a $99\%$ probability of a $5\%$ return and a $1\%$ probability of a $15\%$ return:



$E(R) = 0.99 \times 0.05 + 0.01 \times 0.15 = 0.051$

Consider three states of the world (for next year): good, bad, and ugly. The probability that we have a good state is $50\%$. There is a $25\%$ probability for bad and $25\%$ for ugly. Suppose our stock earns $30\%$ in the good state, $0\%$ in the bad state, and $-30\%$ in the ugly state. What is the expected return on our stock?



$E(R) = 0.5 \times 0.30 + 0.25 \times 0 + 0.25 \times -0.30 = 0.075$

Variance

  • Variance is a measure of dispersion for the possible future returns
  • This tells us, overall, how far away the various possible outcomes are from the expected value

$Var(R)=\sigma^2_R= \sum_{s=1}^S Pr(R_s) \times \left(R_s-E(R)\right)^2$
Example

From our previous example, we can calculate the value of the return's variance:


$\sigma^2_R= \, 0.5 \times (0.30-0.075)^2 \\ \space \space \space \space \space \space \space \space \space \space + 0.25 \times (0-0.075)^2 \\ \space \space \space \space \space \space \space \space \space \space + 0.25 \times (-0.30-0.075)^2 \space \space \space \space \space \space \space \space \space \space \\ \space \space \space \space \space = \, 0.0619$
Standard Deviation

To calculate standard deviation, we simply take the square root of the variace:


$\sigma_R = \sqrt{\sigma^2_R} = \left[\sum_{s=1}^S Pr(R_s) \times \left(R_s-E(R)\right)^2 \right]^{\frac{1}{2}}$

From our example:

$\sqrt{0.0619} = 0.249$

Portfolios


  • Collection of investments
  • Multiple assets
  • Portfolio weights are asset value divided by total value

$w_i = \dfrac{Value_i}{\sum^N_{i=1}Value_i}$
Example

Suppose I had money invested in two assets: $\$300$ in company X and $\$100$ in company Y.

The total value of my portfolio would be $\$400$.

My weight in company X would be $\dfrac{\$300}{\$400}$, or $0.75$.


$w_X = 0.75$ ; $w_Y=0.25$
Expected Return
  • Relatively straightforward
  • Sum of each stock's weight multiplied by expected return:

$E(R_p) = \sum_i^N w_i \times E(R_i)$
Example

From our example, suppose companies X and Y had expected returns of 0.125 and 0.06 respectively. Our portfolio's expected return would be:


$E(R_p) = 0.75 \times 0.125 + 0.25 \times 0.06 = 0.1088$

Portfolio Variance

  • Slightly more complicated
  • Volatility can be partially cancelled out by diversification
  • Intuition:
    • 5 friends play poker
    • Each person's balance fluctuates
    • Total cash at table is constant
    • Very negative correlation in returns here
  • Calculating portfolio variance requires that we know an additional piece of information
  • Covariance
  • Calculation of covariance:

$\sigma_{x,y} = \sum_s^S Pr(s)\left[(R_{s,x}-E(R_x)) \times (R_{s,y}-E(R_y)) \right]$
Example (Covariance)

Let's say that there are two states of the world "Bull Market" and "Bear Market" with equal probability (0.5). Let's say we also know the returns to stocks X and Y in these two states:

$R_X$ $R_Y$
Bull Market .35 .08
Bear Market -.10 .04

First, we should calculate $E(R_X)$ and $E(R_Y)$:


$E(R_X) = 0.5 \times 0.35 + 0.5 \times (-0.10) = 0.125$

$E(R_Y) = 0.5 \times 0.08 + 0.5 \times 0.04 = 0.06$

At this point, we can apply the covariance formula directly:


$\sigma_{X,Y} = 0.5 [(0.35 - 0.125) \times (0.08 - 0.06)] \\ \space \space \space \space \space \space \space \space \space \space \space + 0.5 [(-0.10 - 0.125) \times (0.04 - 0.06)]$

$= 0.0045$
Portfolio Variance Formula

The variance portfolio of a portfolio with two stocks
(X and Y) can be calculated in the following way:



$\sigma_P^2 = w_X^2 \sigma_X^2 + w_Y^2 \sigma_Y^2 + 2 w_X w_Y \sigma_{x,y}$

Returning to our example with stocks X and Y, we'll also need to calculate their respective return variances:

$\sigma_X^2 = 0.5 \times (0.35-0.125)^2 + 0.5 \times (-0.10 - 0.125)^2 = 0.0506$

$\sigma_Y^2 = 0.5 \times (0.08-0.06)^2 + 0.5 \times (0.04-0.06)^2= 0.0004$

Once we know, $\sigma_X^2$, $\sigma_Y^2$, $\sigma_{x,y}$ and the portfolio weights (0.75, and 0.25 from our example), we can finally calculate the portfolio's return variance:



$\sigma_P^2 = (0.75^2)\times (0.0506) + (0.25^2)\times (0.0004) \\ \space \space \space \space \space \space \space \space + 2 \times (0.75) \times (0.25) \times (0.0045)$

$\sigma_P^2 = 0.030189$; $\sigma_P= \sqrt{0.030189}=0.17375$

Diversification and Risk

  • Don't put all your eggs in one basket
  • Not perfectly correlated
  • Generalize the portfolio variance formula:


$\sigma^2_P = \sum_{i=1} w_i^2 \sigma^2_X + \sum_{i=1} \sum_{i \neq j} w_i w_j \sigma_{i,j}$

Two Types of Risk

  • Diversifiable Risk
    • Idiosyncratic Risk
    • Non-Systematic
  • Non Diversifiable Risk
    • Market Risk
    • Systematic Risk