# Risk and Risk Premiums¶

## 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:

or

## 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.

or

## 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 \(\sigma\)?

## 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?

## Amazon Distribution¶

## Coca-Cola Distribution¶

## 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

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

Compute the sample standard deviation of returns

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.

## Amazon Monthly Prices¶

## Coca-Cola Monthly Prices¶

## Computing Returns and Moments¶

## 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:

## 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:

- 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*:

The *excess return* measures the difference between a previously
observed holding period return of \(A\) and the risk-free:

## 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:

- 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.

## Risk Premium Example¶

Suppose the monthly risk-free rate is 0.2%. What is the estimated risk premium and Sharpe Ratio for Amazon stock?