.. slideconf:: :slide_classes: appear ============================================================================== Risk and Risk Premiums ============================================================================== Probabilistic Returns ============================================================================== Since we don't know future returns, we will treat them as random variables. .. rst-class:: to-build - We can model them as discrete random variables, taking one of a finite set of possible values in the future: :math:`r(s)`, :math:`s = 1, \ldots, S`. .. rst-class:: to-build - In this case the probability of each value is :math:`p(s)`, :math:`s=1,\ldots,S`. .. rst-class:: to-build - We can model them as continuous random variables, taking one of an infinite set of possible values in the future: :math:`r(s)`, :math:`s \in \mathcal{S}` (e.g. :math:`\mathcal{S} = (-\infty, \infty)`). .. rst-class:: to-build - In this case the probability of each value (kind of) is :math:`f(s)`, :math:`s \in \mathcal{S}`. Expected Returns ============================================================================== Our best guess for the future return is the expected value: .. rst-class:: to-build .. math:: E[r] & \equiv \mu = \sum_{s=1}^S r(s) p(s), .. rst-class:: to-build or .. rst-class:: to-build .. math:: E[r] & \equiv \mu = \int_{s \in \mathcal{S}} r(s) f(s) dr(s). Return Volatility ============================================================================== The amount of uncertainty in potential returns can be measured by the variance or standard deviation. .. rst-class:: to-build - Volatility of returns specifically refers to standard deviation, NOT VARIANCE. .. rst-class:: to-build .. math:: Std(r) & \equiv \sigma = \sqrt{\sum_{s=1}^S (r(s) - \mu)^2 p(s)}, .. rst-class:: to-build or .. rst-class:: to-build .. math:: Std(r) & \equiv \sigma = \sqrt{\int_{s \in \mathcal{S}} (r(s) - \mu_r)^2 f(s) dr(s)}. 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 ================= ============ ======= .. rst-class:: to-build What are :math:`\mu` and :math:`\sigma`? .. rst-class:: to-build .. math:: \mu & = 0.05*(-0.37) + 0.25*(-0.11) \\ & \qquad \qquad + 0.40*0.14 + 0.30*0.30 = 0.10 .. rst-class:: to-build .. math:: 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 .. rst-class:: to-build .. math:: \sigma & = \sqrt{E[r^2] - \mu^2} = 0.04471 - 0.10^2 = 0.03471 Assumption of Normality ============================================================================== It will often be convenient to assume asset returns are normally distributed. .. rst-class:: to-build - In this case, we will treat returns as continuous random variables. .. rst-class:: to-build - We can use the normal density function to compute probabilities of possible events. .. rst-class:: to-build - 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 .. rst-class:: to-build - Amazon stock (AMZN) has an expected monthly return of 3\% and a volatility (standard deviation) of 8\%. .. rst-class:: to-build - Coca-Cola stock (KO) has an expected monthly return of 1\% and a volatility (standard deviation) of 4\%. .. rst-class:: to-build What do their probability distributions look like? Amazon Distribution ============================================================================== .. ifslides:: .. image:: /_static/amazon.png :width: 7.5in :align: center .. ifnotslides:: .. image:: /_static/amazon.png :width: 6in Coca-Cola Distribution ============================================================================== .. ifslides:: .. image:: /_static/amazon_coke.png :width: 7.5in :align: center .. ifnotslides:: .. image:: /_static/amazon_coke.png :width: 6in Implications of Normality ============================================================================== The assumption of normality is convenient because .. rst-class:: to-build - If we form a portfolio of assets that are normally distributed, then the distribution of the portfolio return is also normally distributed. .. rst-class:: to-build - Recall that if :math:`X_i \sim \mathcal{N}(\mu_i, \sigma_i)`, :math:`i = 1,\ldots,N`, then :math:`W = \sum_{i=1}^N w_i X_i` is also normally distributed (where :math:`w_i` are constant weights). .. rst-class:: to-build - The mean and the variance (or standard deviation) fully characterize the distribution of returns. .. rst-class:: to-build - 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? .. rst-class:: to-build - Use historical data to estimate them. .. rst-class:: to-build - Collect :math:`N+1` past prices of each asset for a particular interval of time (daily, monthly, quarterly, annually). .. rst-class:: to-build - Compute :math:`N` returns using the formula .. rst-class:: to-build .. math:: r_t & = \frac{P_t - P_{t-1}}{P_{t-1}}. .. rst-class:: to-build 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 .. rst-class:: to-build .. math:: \hat{\mu} & = \frac{1}{N} \sum_{t=1}^N r_t. .. rst-class:: to-build Compute the sample standard deviation of returns .. rst-class:: to-build .. math:: \hat{\sigma}^2 & = \frac{1}{N-1} \sum_{t=1}^N (r_t - \hat{\mu})^2. .. rst-class:: to-build The "hats" indicate that we have estimated :math:`\mu` and :math:`\sigma`: these are not the true, unknown values. Estimating Means and Volatilities - Example ============================================================================== Let's collect the :math:`N = 13` closing prices for Amazon and Coca-Cola between 3 Jan 2012 and 2 Jan 2013. .. rst-class:: to-build - We will only keep the first closing price on the first trading day of each month. .. rst-class:: to-build - 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. .. rst-class:: to-build - This will give us 12 returns that we can use to estimate the means and standard deviations. Amazon Monthly Prices ============================================================================== .. ifslides:: .. image:: /_static/amzn-monthly.png :width: 7.5in :align: center .. ifnotslides:: .. image:: /_static/amzn-monthly.png :width: 6in Coca-Cola Monthly Prices ============================================================================== .. ifslides:: .. image:: /_static/ko-monthly.png :width: 7.5in :align: center .. ifnotslides:: .. image:: /_static/ko-monthly.png :width: 6in Computing Returns and Moments ============================================================================== .. ifslides:: .. image:: /_static/amzn-coke-xls.png :width: 8.5in :align: center .. ifnotslides:: .. image:: /_static/amzn-coke-xls.png :width: 6in Risk-Free Returns ============================================================================== We will typically assume that a risk-free asset is available for purchase. .. rst-class:: to-build - We will denote the risk-free return as :math:`r_f`. .. rst-class:: to-build - If an asset is risk free, its return is certain and has no variability: .. rst-class:: to-build .. math:: E[r_f] & = r_f \\ Var(r_f) & = 0. T-Bills as Risk-Free Assets ============================================================================== The return on a short-term government t-bill is usually considered risk free: .. rst-class:: to-build - Although the price changes over time, the risk of default is extremely low. .. rst-class:: to-build - 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? .. rst-class:: to-build - Risky assets compensate for risk through higher expected return. .. rst-class:: to-build - If risky assets didn't offer higher expected return, everyone would sell them, leading to a price decline today and a higher expected return: .. rst-class:: to-build .. math:: \uparrow E[r_t] & = \frac{E[P_t] - \downarrow P_{t-1}}{\downarrow P_{t-1}} .. rst-class:: to-build - 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 :math:`A` exceeds the risk-free return is known as the *risk premium*: .. rst-class:: to-build .. math:: \text{rp}_{A,t} & = E[r_{A,t}] - r_{f,t}. .. rst-class:: to-build The *excess return* measures the difference between a previously observed holding period return of :math:`A` and the risk-free: .. rst-class:: to-build .. math:: \text{er}_{A,t-1} & = r_{A,t-1} - r_{f,t-1}. Risk Premium \& Excess Returns ============================================================================== - Note that excess returns can only be computed with past returns. .. rst-class:: to-build - 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: .. rst-class:: to-build .. math:: \text{SR}_{A,t} & = \frac{\mu_{A,t} - r_{f,t}}{\sigma_{A,t}} .. rst-class:: to-build - The Sharpe Ratio is a measure of risk aversion. .. rst-class:: to-build - It is often referred to as the price of risk. .. rst-class:: to-build - The Sharpe Ratio for a broad market index of assets (like the S\&P 500) is referred to as the market price of risk. .. rst-class:: to-build - The true Sharpe Ratio is unknown, since we don't know :math:`\mu_{A,t}` and :math:`\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? .. rst-class:: to-build .. math:: rp_{AMZN} = 0.03 - 0.002 = 0.028 .. rst-class:: to-build .. math:: SR_{AMZN} = \frac{rp_{AMZN}}{0.08} = 0.35