Finance - XII
Financial Return and Risk
17 Jan 2025
Measuring return and risk
We can compute the monthly returns for a stock by taking the difference between the stock price at the end of the month and the stock price at the beginning of the month (plus any dividends), and then dividing by the stock price at the beginning of the month. Then, we can measure the risk of the stock by computing the variability of these monthly returns. In general, the higher the variability, the higher the risk.
The arithmetic average return is the sum of the returns over a period divided by the number of periods. This metric measures the average return of the stock over the period, but it does not account for the variability of the returns.
Standard deviation is a measure of the dispersion of a set of values. In the context of stock returns, it measures the variability of the returns, which we can use as a proxy for risk. The higher the standard deviation, the higher the fluctuations in the stock price, and the higher the risk of investing in the stock.
Sources of risk
Each component of a firm's income statement is subject to its own set of risks. These risks can be broadly categorized into three types:
Expected measures of return and risk
We need to estimate the expected return and risk of an investment in order to decide whether to invest in it. One way is to consider the possible outcomes of the investment and assign probabilities to each outcome. Then, we can compute the expected return and risk of the investment based on these probabilities. For example, the investor may expect a:
For each of these scenarios, the investor can estimate the expected stock returns. For instance, the investor may expect a 20% return in a boom economy, a 10% return in a normal economy, and a -5% return in a recession. Then, the investor can compute the expected return of the stock by weighting these returns by the probabilities of each scenario. Having computed the expected return, the investor can then compute the investment's risk by computing the standard deviation of the returns.
All this seems hand-wavy, but there is evidence indicating the practicality of this approach. For example, around 2004, when the Fed is expected to change short-term interest rates, journalists conducted surveys of practitioners on what they expected the Fed to do. They predicted that the Fed would likely cut rates by half a percentage point (unlikely that they would cut by 3 quarters of a point, very unlikely that they would not cut), and the Fed did exactly that.
Some analysts run simulations to estimate the expected return and risk of an investment.
Generally speaking, higher expected returns are associated with higher risk. When investors decide whether to invest in an asset, they set an expected rate of return (or discount rate) that they require in order to invest in the asset. The expected rate of return is the sum of the risk-free rate, inflation expectations, and a risk premium that compensates the investor for the risk of the investment. The first two components are the same for all investment opportunities, and they can be approximated by the yield on a short-term government bond. The risk premium is specific to the investment opportunity, and it depends on the risk of the investment. For example, bonds have lower risk premiums than stocks, because bondholders have a higher claim on the firm's assets in the event of bankruptcy, for instance.
Efficient capital markets
Prices on securities change over time as new information becomes available. When new information becomes available and an investor starts expecting a lower discount rate (e.g. interest rates are expected to decrease), then the price of a security increases, because the same cash flows are discounted at a lower rate, i.e. the present value of the cash flows increases.
An efficient market is one in which prices reflect all available information, and where prices adjust quickly to new information. Note that new information should be unexpected. If investors already expect the information, then the price of the security would have already adjusted to reflect this expectation.
In a perfectly efficient market, prices would adjust instantaneously to new information. There would be no trend in stock prices. Real markets are not perfectly efficient, thus, there is a slight time lag between the release of new information and the adjustment of prices. This time lag is due to the time it takes for investors to process the information and adjust their expectations. Price reversals, on the other hand, indicate an overreaction to new information.
In an efficient market, it is difficult to consistently earn above-average returns. Thus, instead of buying individual stocks to try to beat the market, some investors opt to invest in a diversified portfolio of stocks. One popular strategy is to invest in an index fund, which is a fund that tracks a stock market index, such as the S&P 500.
Managers of firms can also look at the stock market to determine how investors perceive the firm's prospects.
Since it is difficult to consistently earn above-average returns in an efficient market, the best strategy for investors is to establish the amount of risk they are willing to take, and then build an investment portfolio that matches their risk tolerance, while at the same time removing all unsystematic risk through diversification.
Investment portfolios
A portfolio is a collection of investments. When we build a portfolio, we are removing unsystematic risk by diversifying across different investments.When one investment performs poorly, other investments in the portfolio can help offset the losses. A well-diversified portfolio is one that contains investments that are not perfectly correlated with each other. Thus, a good strategy is to invest in assets that cut across different asset classes and industries.
Simply adding more investments to a portfolio does not necessarily reduce risk, as the added investments may be highly correlated with the existing investments.
Note that diversification only removes unsystematic risk. Systematic risk are risks that affect the entire market, and they usually have to do with macroeconomic factors such as interest rates, inflation, and political instability. A well-diversified portfolio minimizes unsystematic risk, but it may still be exposed to systematic risk.
Capital asset pricing model
Since unsystematic risk can be diversified away, investors (or at least the prudent ones) are only concerned with systematic risk. Thus, when building a portfolio, investors are concerned with how different asset classes with different levels of systematic risk contribute to the overall risk of the portfolio. For example, T-bills are considered to be risk-free. Thus, adding T-bills to a portfolio (that contains at least one risky asset) can help reduce the overall risk of the portfolio. Although gold is a risky investment on its own, it can help reduce the overall risk of a portfolio, because gold prices are typically negatively correlated with stock prices (investors prefer gold when stock prices are falling, e.g. when inflation and interest rates are high).
We can estimate the systematic risk of an asset by comparing its returns to the returns of the market. A market portfolio is a portfolio that contains all risky assets in the market. An asset can have different levels of systematic risk:
In general, introducing assets with higher systematic risk into a portfolio increases the expected return of the portfolio, but it also increases the risk of the portfolio. The capital asset pricing model (CAPM) is a model that describes the relationship between the expected return of an asset and its systematic risk, there systematic risk is measured relative to the market portfolio (also called the beta). A beta of 1 indicates that the asset moves with the market; a beta greater than 1 indicates that the asset moves more than the market; a beta less than 1 indicates that the asset moves less than the market.