| Monthly Estimate (%) | Annualized Estimate (%) | |
|---|---|---|
| Mean | 2.24 | 26.91 |
| St. Dev. | 12.49 | 43.27 |
Statistics of Asset Returns
These notes introduce the basic building blocks used to measure and describe the performance of an investment. We start by defining what it means to hold an asset and the cash flows it can generate, and use this to define the holding-period return (HPR), the basic unit of profitability in finance. We then treat returns as random variables and introduce their two most important moments, the expected return and the volatility, first using scenario analysis and then using historical stock and ETF data. The notes close by extending these ideas to portfolios of several assets, and by introducing the law of one price, the principle underlying portfolio construction, arbitrage, and the pricing of Exchange-Traded Funds.
Financial and Real Assets
Investing involves purchasing an asset today, which you may sell the next period, sell at some later date, or hold indefinitely. Even if you hold the asset indefinitely, we can always take the perspective of an investor who sells it and immediately repurchases it every period, which lets us evaluate its performance one period at a time regardless of your actual holding horizon.
The asset you purchase can be a real asset, like a house, or a financial asset like a stock or a bond. Financial assets are claims to parts of real assets, or to the income they generate. Because every financial asset is also someone else’s liability, financial assets net to zero across the economy as a whole. Real assets are the only source of society’s net wealth.
When you purchase an asset today and hold it for one period, you might also be entitled to a cash flow along the way. Many stocks in the U.S. pay dividends quarterly, while bonds typically pay coupons semi-annually. Real assets can generate cash flows too. For example, if you purchase a house and rent it, you will collect a monthly rent from your tenant.
The Rate of Return
In order to analyze the performance of your investment, it is useful to think in terms of profitability per dollar invested. Say you purchase an asset at time 0 for P_{0} and sell it a period later for P_{1}. If the asset pays a cash flow, we assume it is paid just before you purchase or sell the asset. The holding-period return (HPR) from time 0 to time 1 is defined as \text{HPR} = \frac{P_{1} - P_{0} + D_{1}}{P_{0}}. The HPR can be decomposed as \text{HPR} = \frac{P_{1} - P_{0}}{P_{0}} + \frac{D_{1}}{P_{0}}. The first term in the previous expression denotes the capital gain and the second term represents the income yield. Thus, the total return of investing in the asset depends on how much the price appreciates or depreciates, and the amount of cash flow paid during the holding period.
Example 1 Suppose you purchase today an index fund for $100 per share. The fund will pay dividends over a year of $4. If the price per share next year is $110, your HPR amounts to \text{HPR} = \frac{110 - 100 + 4}{100} = 14\%. The total return of 14% consists of 10% in capital gains and 4% of dividend yield.
To ease notation, in the following we denote by r_{i} the HPR of investing in asset i. Since there is considerable uncertainty about the future price and cash flows of the asset, its HPR is unknown today. A useful way to think about the uncertainty of r_{i} is to model it as a random variable in certain probability space (\Omega, \operatorname{P}).
The expectation of r_{i} is called the expected return of asset i, whereas the standard deviation of r_{i} is typically called the volatility or standard deviation of returns. I usually write \begin{aligned} \mu_{i} & = \operatorname{E}(r_{i}), \\ \sigma_{i} & = \sqrt{\operatorname{V}(r_{i})}. \end{aligned}
Example 2 Suppose your expectations regarding a stock price are as follows:
| State of the Market | Probability | HPR |
|---|---|---|
| Boom | 0.35 | 44.5% |
| Normal | 0.30 | 14% |
| Recession | 0.35 | -16.5% |
We can compute the mean and standard deviation of the stock’s HPR as \begin{aligned} \mu & = 0.35 \times 0.445 + 0.3 \times 0.14 + 0.35 \times (-0.165) = 14\%, \\ \sigma & = \sqrt{0.35 \times (0.445 - 0.14)^{2} + 0.3 \times (0.14 - 0.14)^{2} + 0.35 \times (-0.165 - 0.14)^{2}} \\ & = 25.52\%. \end{aligned}
Stock Returns
The previous example assumed we knew the probabilities of each state of the market. In practice, we rarely know these probabilities, so we instead estimate the mean and volatility of returns from historical data.
Figure 1 plots the monthly returns of Apple (AAPL) since the stock was listed in December 1980. The figure shows that returns are volatile with periods in which volatility spikes and other periods of lower volatility. We can compute the average monthly return as \bar{r} = \frac{1}{N} \sum_{t = 1}^{N} r_{t}, and sample standard deviation as \hat{\sigma} = \sqrt{\frac{1}{N - 1} \sum_{t = 1}^{N} (r_{t} - \bar{r})^{2}}. If we assume that the monthly returns are independent and identically distributed, we can annualize the average monthly return and volatility as: \begin{aligned} \bar{r}_{\text{Annual}} & = 12 \times \bar{r}_{\text{Monthly}}, \\ \hat{\sigma}_{\text{Annual}} & = \sqrt{12} \times \hat{\sigma}_{\text{Monthly}}. \end{aligned}
Using these expressions, we get for Apple that
Because the volatility is so high compared to the average return, estimates of average returns are typically very noisy and not very useful in practice to proxy for expected returns. Also, an average of monthly returns going back to 1980 is probably not a very good proxy of your expectation for returns going forward.
Portfolios
In finance, a portfolio consists of allocating a certain amount of wealth into different assets such as stocks, bonds, and real estate. Say you have two assets A and B. Denote by r_{A} and r_{B} the HPR of each asset, respectively. If you have a certain amount of wealth W, you can split it and invest W_{A} in asset A and the rest, W_{B} = W - W_{A}, in B. Each dollar invested in A yields 1 + r_{A} dollars next period whereas each dollar invested in B generates 1 + r_{B}.
Denote by r the HPR of your portfolio. Using this notation, the total return of your investment next period is (1 + r) W. But this amount can be computed by considering the returns of the individual investments in A and B. Therefore, we must have that (1 + r) W = (1 + r_{A}) W_{A} + (1 + r_{B}) W_{B}. Since W = W_{A} + W_{B}, the previous expression can be simplified as r = \frac{W_{A}}{W} r_{A} + \frac{W_{B}}{W} r_{B}. \tag{1} In equation (1), the fractions W_{A} / W and W_{B} / W denote the proportion of wealth allocated to each asset. In finance, we call these fractions the portfolio weights. If we denote by w_{A} and w_{B} these portfolio weights, we can write expression (1) as r = w_{A} r_{A} + w_{B} r_{B}. An important thing to note is that the sum of the weights is by construction always equal to one.
Example 3 You have the following scenario analysis for the HPR of stocks X and Y:
| Market | Probability | Stock X | Stock Y |
|---|---|---|---|
| Bull | 0.3 | 40% | 10% |
| Normal | 0.5 | 15% | 20% |
| Bear | 0.2 | -18% | -5% |
Assume that of your $10,000 portfolio, you invest $8,000 in Stock X and $2,000 in Stock Y.
To compute the expected return and standard deviation of the portfolio, we can first compute the HPR in each state of the world:
| Market | Probability | Invested in X | Invested in Y | Total | HPR |
|---|---|---|---|---|---|
| Bull | 0.3 | 11,200 | 2,200 | 13,400 | 34.0% |
| Normal | 0.5 | 9,200 | 2,400 | 11,600 | 16.0% |
| Bear | 0.2 | 6,560 | 1,900 | 8,460 | -15.4% |
Thus, \begin{aligned} \mu_{P} & = 0.3 \times 0.34 + 0.5 \times 0.16 + 0.2 \times (−0.154) = 15.12\%, \\ \sigma_{P} & = \sqrt{0.3 \times (0.34 − 0.1512)^{2} + 0.5 \times (0.16 − 0.1512)^{2} + 0.2 \times (−0.154 − 0.1512)^{2}} \\ & = 17.14\%. \end{aligned}
The Law of One Price
The portfolio mathematics we just did relies on a very important assumption called the law of one price (LOOP). In competitive markets, LOOP guarantees that the price of a basket of stocks is equal to the sum of the prices of its constituents. This logic is at the heart of how Exchange-Traded Funds (ETF) operate, as the next example shows.
Example 4 An ETF is a type of investment fund that is traded on stock exchanges, similar to individual stocks. ETFs hold a diversified portfolio of assets, such as stocks, bonds, or commodities, which provides investors with broad exposure to specific markets or investment strategies.
ETF arbitrage is the mechanism that helps keep the market price of an ETF in line with its Net Asset Value (NAV). Authorized Participants (APs), typically large financial institutions, have the ability to create or redeem ETF shares in large blocks called creation units.
When the ETF market price is higher than the NAV, APs can buy the underlying securities of the ETF in the open market and then deliver them to the ETF issuer in exchange for new ETF shares. The AP can then sell these ETF shares at the higher market price, making a profit. This buying of underlying securities pushes their prices up, while the selling of new ETF shares pushes the ETF price down, bringing the two prices closer together.
When the ETF market price is lower than the NAV, APs can buy ETF shares in the open market and deliver them to the ETF issuer in exchange for the underlying securities. The AP can then sell these underlying securities at the higher NAV price, making a profit. This buying of ETF shares pushes their price up, while the selling of the underlying securities pushes their prices down, again bringing the two prices closer together.
This creation and redemption process happens continuously and helps to keep the ETF price in line with the NAV.
You can find more information in the BlackRock white paper on ETF primary trading and the role of authorized participants (opens in new tab).
ETF arbitrage generally keeps an ETF’s market price close to its NAV, which is reassuring evidence that LOOP is a reasonable axiom to start working from. That said, the mechanism cannot be perfect: if there were never a gap between price and NAV, APs would have no profit incentive to engage in creation and redemption in the first place, a version of the Grossman-Stiglitz paradox for informationally efficient markets. In practice these deviations are usually small, but they can widen substantially when the underlying assets are illiquid or hard to trade, as happened with some bond ETFs during the March 2020 market stress.
One of the most important ETFs out there is the SPDR S&P 500 ETF Trust which seeks to provide investment results that, before expenses, correspond generally to the price and yield performance of the S&P 500 Index. The figure below plots the monthly returns of SPDR (Ticker: SPY) since the ETF was listed in 1993.
Practice Problems
These problems give you a chance to practice the concepts introduced in this chapter. Try to solve each one on your own before expanding the solution.
Problem 1 (Holding-Period Return) You buy a share of a REIT for $50. Over the year it pays $2 in dividends, and you sell it a year later for $53. Compute the capital gain, the income yield, and the HPR.
Solution
The capital gain is \frac{53 - 50}{50} = 6\%, and the income yield is \frac{2}{50} = 4\%. Thus, \text{HPR} = 6\% + 4\% = 10\%.Problem 2 (Expected Return and Volatility) A stock’s HPR over the next year depends on the state of the economy according to the scenario below.
| State | Probability | HPR |
|---|---|---|
| Boom | 0.25 | 20% |
| Normal | 0.50 | 5% |
| Bust | 0.25 | -10% |
Compute the expected return \mu and the volatility \sigma of the stock.
Solution
\mu = 0.25 \times 20 + 0.50 \times 5 + 0.25 \times (-10) = 5\%. The variance is \sigma^{2} = 0.25 (20 - 5)^{2} + 0.50 (5 - 5)^{2} + 0.25 (-10 - 5)^{2} = 112.5, so \sigma = \sqrt{112.5} \approx 10.61\%.Problem 3 (Annualizing Monthly Statistics) Using several years of data, you estimate that a stock’s average monthly return is 0.9\% with a monthly standard deviation of 5.5\%. Assuming monthly returns are independent and identically distributed, compute the annualized mean and standard deviation.
Solution
Annualizing the mean simply scales it by 12, \bar{r}_{\text{Annual}} = 12 \times 0.9\% = 10.8\%. The standard deviation, however, scales by the square root of 12, \hat{\sigma}_{\text{Annual}} = \sqrt{12} \times 5.5\% \approx 19.05\%. Notice that volatility does not scale linearly with the horizon: quadrupling the horizon only doubles the standard deviation.Problem 4 (Portfolio Weights and Return) You have $20,000 to invest. You put $12,000 in asset A, which returns 8%, and the remaining $8,000 in asset B, which returns -3\%. Compute the portfolio weights w_{A} and w_{B}, and the portfolio’s HPR.
Solution
The portfolio weights are w_{A} = \frac{12{,}000}{20{,}000} = 0.6, \qquad w_{B} = \frac{8{,}000}{20{,}000} = 0.4, which sum to one, as they must. Using (1), r = 0.6 \times 8\% + 0.4 \times (-3\%) = 3.6\%.Problem 5 (Portfolio Scenario Analysis) You invest 60% of your wealth in stock C and 40% in stock D. Their HPRs depend on the state of the economy as follows.
| Market | Probability | Stock C | Stock D |
|---|---|---|---|
| Bull | 0.4 | 25% | 5% |
| Normal | 0.4 | 10% | 8% |
| Bear | 0.2 | -20% | -2% |
Compute the portfolio’s HPR in each state of the market, and then its expected return and standard deviation.
Solution
Using r = 0.6 \, r_{C} + 0.4 \, r_{D}, the portfolio’s HPR in each state is
| Market | Probability | Portfolio HPR |
|---|---|---|
| Bull | 0.4 | 17% |
| Normal | 0.4 | 9.2% |
| Bear | 0.2 | -12.8% |
Problem 6 (The Law of One Price and ETF Arbitrage) Suppose an ETF’s market price is currently below its Net Asset Value (NAV). Describe the trade an Authorized Participant (AP) would execute to profit from this discrepancy, and explain how this trade affects the ETF’s market price and its NAV.
Solution
Since ETF shares are cheap relative to the value of the underlying securities, the AP buys ETF shares in the open market and delivers them to the ETF issuer in exchange for the underlying securities at NAV. The AP then sells the underlying securities, which are worth more than what it paid for the ETF shares, locking in a profit.
This trade pushes the ETF’s market price up, since the AP is buying ETF shares, and pushes down the prices of the underlying securities, since the AP is selling them. Both forces narrow the gap between the ETF’s market price and its NAV, consistent with the law of one price.Problem 7 (Holding-Period Return on a Bond) An investor purchased a bond one year ago for $980. He received $17 in interest and sold the bond for $987. What is the holding-period return on his investment?
Solution
The capital gain is \frac{987 - 980}{980} \approx 0.71\%, and the income yield is \frac{17}{980} \approx 1.73\%. Thus, \text{HPR} = \frac{987 - 980 + 17}{980} = \frac{24}{980} \approx 2.45\%.Solution
The expected end-of-year cash flow is \operatorname{E}(CF) = 0.5 \times 70{,}000 + 0.5 \times 200{,}000 = \$135{,}000.
Discounting this expected cash flow at 14%, the most you should be willing to pay is its present value, P = \frac{135{,}000}{1.14} \approx \$118{,}421.05.
If you pay exactly P = \$118{,}421.05 for the portfolio, its expected rate of return is, by construction, \operatorname{E}(r) = \frac{\operatorname{E}(CF)}{P} - 1 = \frac{135{,}000}{135{,}000 / 1.14} - 1 = 1.14 - 1 = 14\%, exactly the discount rate you used to price it. This is not a coincidence: whenever you price an asset at the present value of its expected cash flow using a given required return, the expected return you earn at that price equals the required return you used.
With a 12% risk premium, the discount rate rises to 6\% + 12\% = 18\%, so P = \frac{135{,}000}{1.18} \approx \$114{,}406.78. Demanding a higher risk premium makes you willing to pay less for the same risky cash flow.
Problem 9 (Mean and Standard Deviation of Stock Returns) The stock of company XYZ currently trades at $100 and just paid a dividend of $1.8. Suppose that your expectations regarding the stock price and dividends next year are as follows:
| State of the Market | Probability | Dividend | Price |
|---|---|---|---|
| Boom | 0.3 | $3 | $120 |
| Normal growth | 0.5 | $2 | $100 |
| Recession | 0.2 | $1 | $80 |
Compute the mean and standard deviation of the returns for company XYZ.
Solution
The dividend of $1.8 already paid is sunk and irrelevant; only next year’s price and dividend matter for the HPR, computed off today’s price of $100. In each state, \text{HPR} = \frac{P_{1} - 100 + D_{1}}{100}.
| State of the Market | Probability | HPR |
|---|---|---|
| Boom | 0.3 | 23% |
| Normal growth | 0.5 | 2% |
| Recession | 0.2 | -19% |
Problem 10 (Covariance and Correlation of Two Securities) Suppose the economy can only be in one of the following two states: (i) Boom or “good” state and (ii) Recession or “bad” state. Each of the states can occur with an equal probability. At the beginning of a month, you can purchase the following two securities in the market:
- Security 1 is currently trading at $4. At the end of the month, the security price is expected to increase by $10 in the good state, and expected to remain unchanged in the bad state.
- Security 2 is currently trading at $5. At the end of the month, the price of security 2 is expected to remain unchanged in the good state and expected to increase by $10 in the bad state.
- Compute the expected returns of securities 1 and 2.
- Compute the standard deviations of returns for securities 1 and 2.
- Compute the covariance and the correlation between the returns of the two securities.
Solution
The HPRs in each state are r_{1}^{\text{good}} = \frac{10}{4} = 250\%, \quad r_{1}^{\text{bad}} = 0\%, \qquad r_{2}^{\text{good}} = 0\%, \quad r_{2}^{\text{bad}} = \frac{10}{5} = 200\%.
Since each state has probability 0.5, \mu_{1} = 0.5 \times 250\% + 0.5 \times 0\% = 125\%, \qquad \mu_{2} = 0.5 \times 0\% + 0.5 \times 200\% = 100\%.
The variances are \sigma_{1}^{2} = 0.5 (250 - 125)^{2} + 0.5 (0 - 125)^{2} = 15{,}625, \qquad \sigma_{2}^{2} = 0.5 (0 - 100)^{2} + 0.5 (200 - 100)^{2} = 10{,}000, so \sigma_{1} = 125\% and \sigma_{2} = 100\%.
The covariance is \sigma_{12} = 0.5 (250 - 125)(0 - 100) + 0.5 (0 - 125)(200 - 100) = -6{,}250 - 6{,}250 = -12{,}500, so the correlation is \rho_{12} = \frac{\sigma_{12}}{\sigma_{1} \sigma_{2}} = \frac{-12{,}500}{125 \times 100} = -1. The two securities move in exactly opposite directions across states — whichever state occurs, the security priced to gain in that state gains while the other stays flat — so they are perfectly negatively correlated.