## Probabilistic Returns¶

Since we don’t know future returns, we will treat them as random variables.

• We can model them as discrete random variables, taking one of a finite set of possible values in the future: $$r(s)$$, $$s = 1, \ldots, S$$.
• In this case the probability of each value is $$p(s)$$, $$s=1,\ldots,S$$.
• We can model them as continuous random variables, taking one of an infinite set of possible values in the future: $$r(s)$$, $$s \in \mathcal{S}$$ (e.g. $$\mathcal{S} = (-\infty, \infty)$$).
• In this case the probability of each value (kind of) is $$f(s)$$, $$s \in \mathcal{S}$$.

## Expected Returns¶

Our best guess for the future return is the expected value:

$\begin{split}E[r] & \equiv \mu = \sum_{s=1}^S r(s) p(s),\end{split}$

or

$\begin{split}E[r] & \equiv \mu = \int_{s \in \mathcal{S}} r(s) f(s) dr(s).\end{split}$

## Return Volatility¶

The amount of uncertainty in potential returns can be measured by the variance or standard deviation.

• Volatility of returns specifically refers to standard deviation, NOT VARIANCE.
$\begin{split}Std(r) & \equiv \sigma = \sqrt{\sum_{s=1}^S (r(s) - \mu)^2 p(s)},\end{split}$

or

$\begin{split}Std(r) & \equiv \sigma = \sqrt{\int_{s \in \mathcal{S}} (r(s) - \mu_r)^2 f(s) dr(s)}.\end{split}$

## Expectation and Variance Example¶

State Probability Return
Severe Recession 0.05 -0.37
Mild Recession 0.25 -0.11
Normal Growth 0.40 0.14
Boom 0.30 0.30

What are $$\mu$$ and $$\sigma$$?

$\begin{split}\mu & = 0.05*(-0.37) + 0.25*(-0.11) \\ & \qquad \qquad + 0.40*0.14 + 0.30*0.30 = 0.10\end{split}$
$\begin{split}E[r^2] & = 0.05*(-0.37)^2 + 0.25*(-0.11)^2 \\ & \qquad \qquad + 0.40*(0.14)^2 + 0.30*(0.30)^2 = 0.04471\end{split}$
$\begin{split}\sigma & = \sqrt{E[r^2] - \mu^2} = 0.04471 - 0.10^2 = 0.03471\end{split}$

## Assumption of Normality¶

It will often be convenient to assume asset returns are normally distributed.

• In this case, we will treat returns as continuous random variables.
• We can use the normal density function to compute probabilities of possible events.
• We will not assume that returns of different assets come from the same normal, but instead FROM DIFFERENT normal distributions.

## Differing Normal Distributions¶

As an example, suppose that

• Amazon stock (AMZN) has an expected monthly return of 3% and a volatility (standard deviation) of 8%.
• Coca-Cola stock (KO) has an expected monthly return of 1% and a volatility (standard deviation) of 4%.

What do their probability distributions look like?

## Implications of Normality¶

The assumption of normality is convenient because

• If we form a portfolio of assets that are normally distributed, then the distribution of the portfolio return is also normally distributed.
• Recall that if $$X_i \sim \mathcal{N}(\mu_i, \sigma_i)$$, $$i = 1,\ldots,N$$, then $$W = \sum_{i=1}^N w_i X_i$$ is also normally distributed (where $$w_i$$ are constant weights).
• The mean and the variance (or standard deviation) fully characterize the distribution of returns.
• The variance or standard deviation alone is an appropriate measure of risk (no other measure is needed).

## Estimating Means and Volatilities¶

Typically we don’t know the true mean and standard deviation of Amazon and Coca-Cola. What do we do?

• Use historical data to estimate them.
• Collect $$N+1$$ past prices of each asset for a particular interval of time (daily, monthly, quarterly, annually).
• Compute $$N$$ returns using the formula
$\begin{split}r_t & = \frac{P_t - P_{t-1}}{P_{t-1}}.\end{split}$

We don’t include dividends in the return calculation above, because we use ADJUSTED closing prices, which account for dividend payments directly in the prices.

## Estimating Means and Volatilities¶

Compute the sample mean of returns

$\begin{split}\hat{\mu} & = \frac{1}{N} \sum_{t=1}^N r_t.\end{split}$

Compute the sample standard deviation of returns

$\begin{split}\hat{\sigma}^2 & = \frac{1}{N-1} \sum_{t=1}^N (r_t - \hat{\mu})^2.\end{split}$

The “hats” indicate that we have estimated $$\mu$$ and $$\sigma$$: these are not the true, unknown values.

## Estimating Means and Volatilities - Example¶

Let’s collect the $$N = 13$$ closing prices for Amazon and Coca-Cola between 3 Jan 2012 and 2 Jan 2013.

• We will only keep the first closing price on the first trading day of each month.
• We can then compute 12 monthly returns by computing the difference in month prices at the beginning of each month, dividing by the price of the previous month.
• This will give us 12 returns that we can use to estimate the means and standard deviations.

## Risk-Free Returns¶

We will typically assume that a risk-free asset is available for purchase.

• We will denote the risk-free return as $$r_f$$.
• If an asset is risk free, its return is certain and has no variability:
$\begin{split}E[r_f] & = r_f \\ Var(r_f) & = 0.\end{split}$

## T-Bills as Risk-Free Assets¶

The return on a short-term government t-bill is usually considered risk free:

• Although the price changes over time, the risk of default is extremely low.
• Also, the holding period return can be determined at the beginning of the holding period (unlike other risky assets).

## Compensation for Risk¶

If you can invest in a risk-free asset, why would you purchase a risky asset instead?

• Risky assets compensate for risk through higher expected return.
• If risky assets didn’t offer higher expected return, everyone would sell them, leading to a price decline today and a higher expected return:
$\begin{split}\uparrow E[r_t] & = \frac{E[P_t] - \downarrow P_{t-1}}{\downarrow P_{t-1}}\end{split}$
• There is no guarantee that the actual return will be higher – only its expected value.

## Risk Premium & Excess Returns¶

The amount by which the expected return of some risky asset $$A$$ exceeds the risk-free return is known as the risk premium:

$\begin{split}\text{rp}_{A,t} & = E[r_{A,t}] - r_{f,t}.\end{split}$

The excess return measures the difference between a previously observed holding period return of $$A$$ and the risk-free:

$\begin{split}\text{er}_{A,t-1} & = r_{A,t-1} - r_{f,t-1}.\end{split}$

## Risk Premium & Excess Returns¶

• Note that excess returns can only be computed with past returns.
• We estimate risk premia with the sample mean of historical excess returns.

## Sharpe Ratio¶

The Sharpe Ratio is a measure of how much risk premium investors require, per unit of risk:

$\begin{split}\text{SR}_{A,t} & = \frac{\mu_{A,t} - r_{f,t}}{\sigma_{A,t}}\end{split}$
• The Sharpe Ratio is a measure of risk aversion.
• It is often referred to as the price of risk.
• The Sharpe Ratio for a broad market index of assets (like the S&P 500) is referred to as the market price of risk.
• The true Sharpe Ratio is unknown, since we don’t know $$\mu_{A,t}$$ and $$\sigma_{A,t}$$, but we can estimate these with historical returns.

$rp_{AMZN} = 0.03 - 0.002 = 0.028$
$SR_{AMZN} = \frac{rp_{AMZN}}{0.08} = 0.35$