Finance chapter 8

Calculating Beta

A typical approach to estimate (future) β is to run a regression of the security's past returns against the past returns of the market. The slope of the regression line is defined as the beta coefficient of the security. In the next slide, RM are the past returns of the stock market, and Ri are the past returns of stock i. The beta of stock i = 1.44, though in practice, this calculation would be done with more than 3 data points.

Security Market Line

Closely related to the Capital Asset Pricing Model, is the "Security Market Line", which is a graph of the relationship between the equilibrium required rate of return of a stock (measured on the Y-axis) and the amount of risk taken (as measured by beta on the X-axis). See Figure 8.8, page 285. Y-intercept = RRF Slope = (RM - RRF)

An asset's risk can be divided into 2 parts.

Diversifiable risk: can be diversified away; does not concern an investor who holds a diversified portfolio Market risk: the risk of a general market decline; can not be diversified away; does concern all investors

SPY and TLT, Stand-Alone Basis

SPY is an exchange traded fund that tracks the S&P 500 index of large capitalization U.S. stocks. TLT is an exchange-traded fund that tracks long-term Treasury bonds with 20-30 years to maturity. Average returns ≈ 8-9% Risk (σ) ≈ 18%

Assets Risk

Stand-alone basis (asset considered by itself) Portfolio basis (asset is held as one of a number of assets in a portfolio)

Standard Deviation

Standard deviation is a statistical measure of the variability of a set of observations and is used to measure stand-alone risk. The tighter the probability distribution of expected future returns (the smaller the standard deviation), the smaller the risk of a given investment. In this case, U.S. Water is less risky than Martin Products

Return and Risk on the Calculator

Step 1. Enter the data (e.g., Martin Products). [2nd] [Data, above 7] to access the data-entry part of the Statistics worksheet, then [2nd] [CLR WRK, above CE/C] to clear the worksheet Step 2. Enter the data. X01 is the 1st rate of return. Key in 80, [Enter], [↓] Y01 is the probability of the 1st rate of return. Key in 30, [Enter], [↓] [Note: you can enter X as a whole number or as a decimal, but be sure to enter Y as a whole number.] Step 3. Enter the data. X02 is the 2nd rate of return. Key in 10, [Enter], [↓] Y02 is the probability of the 2nd rate of return. Key in 40, [Enter], [↓] Step 4. n = 100, the sum of the probabilities xbar = 10, the expected rate of return Sx = 54.49, the sample standard deviation σx = 54.22, the population standard deviation, which we will use as our measure of stand-alone risk After viewing these, [2nd] [Quit]

Measuring Return

The "total return" of a stock has 2 pieces . Dividend (usually paid in quarterly installments) Capital gain (change in price between when you buy and when you sell) Similarly, we saw in chapters 6 and 7 that the total return of a bond has 2 pieces. Interest (usually paid in semi-annual installments) Capital gain (change in price between when you buy a when you sell)

Capital Asset Pricing Model

The CAPM applies to all securities, but it is most commonly used to calculate the required return of an equity security such as common stock. Rstock = [ (RM - RRF) × β] + RRF RM = required rate of return of a portfolio consisting of all stocks, which is known as the [M]arket portfolio RRF = nominal [R]isk-[F]ree rate (see chapter 6), usually measured by the yield of Treasury securities (RM - RRF) = known as the "market risk premium", which is the additional return above the risk-free rate that investors require to invest in the market portfolio, usually estimated in the range of 4% to 8% β = beta, which is the stock's relative exposure to the market risk premium, as discussed in §8-3

Beta of a Portfolio

The beta of a portfolio is the weighted average of the betas of the individual assets in the portfolio. βportfolio = W1β1 + W2β2 + W3β3 + ... + WNβN W1, W2, and so on are [W]eights, or the percentage of the total $ value of the portfolio invested in each security (expressed as a decimal) β1, β2, and so on are betas, either historical or expected in the future N is the number of assets in the portfolio

Expected Rate of Return

The expected rate of return to be realized from an investment = weighted average of the probability distribution of the possible outcomes. Martin: (0.3×0.80) + (0.4×0.10) + (0.3×-0.60) = 0.10 Water: (0.3×0.15) + (0.4×0.10) + (0.3×0.05) = 0.10

Measuring Risk in a Portfolio

The proper measure of the risk of a single asset is its contribution to the risk of a diversified portfolio. Only market risk is relevant because it can not be diversified. Therefore, only market risk will be compensated with a higher return. The most appropriate measure of market risk is denoted by the Greek letter β, which is beta. Measures a stock's volatility relative to the market. Indicates how risky a stock is if the stock is held in a well-diversified portfolio.

Return of a Portfolio

The return of a portfolio is the weighted average of the returns of the individual assets in the portfolio. Rportfolio = W1R1 + W2R2 + W3R3 + ... + WNRN W1, W2, and so on are [W]eights, or the percentage of the total $ value of the portfolio invested in each security (expressed as a decimal) R1, R2, and so on are [R]eturns, either historical or expected in the future N is the number of assets in the portfolio

Risk of a Portfolio

The risk of a portfolio is NOT the weighted average of the risks of the individual assets in the portfolio. The risk of a portfolio is less than the weighted average of the risks of the individual assets in the portfolio, if the assets in the portfolio do not all move up or down in value in perfect unison. The tendency of two variables to move together is called "correlation", denoted by the Greek letter ρ. Correlation coefficient can range from maximum of +1.00 (two variables move up and down in perfect unison) to minimum of -1.00 (two variables move in exactly opposite directions).

SPY and TLT, Portfolio Basis

The typical endowment fund puts 60% of its money into stocks and 40% into bonds. Note that portfolio standard deviation is much lower than that of either SPY or TLT on its own. Return about the same. Risk much lower!!!! How is this possible?

Expected versus Required Returns

We can compare expected versus required returns to decide whether to buy or sell a stock. If the expected return > required return, then the stock is a good deal (under valued) and the analyst will issue a "buy" recommendation. If the expected return = required return, then the stock is a fair deal (fairly valued) and the analyst will issue a "hold" recommendation. If the expected return < required return, then the stock is a bad deal (over valued) and the analyst will issue a "sell" recommendation.

Risk in a Portfolio Context

An asset held as part of a portfolio (a collection of assets) is less risky than the same asset held on a stand-alone basis, if the assets in the portfolio do not all move up or down in value in perfect unison. Therefore, the risk and return of an individual asset should be analyzed in terms of how that security affects the risk and return of the portfolio in which it is held.

Assets

Assets can be categorized as "financial assets" (e.g., bonds, stocks) or as "real assets" (e.g., land, buildings, machines, equipment).

Calculation Exercises

Be able to calculate the value of any missing variable (Rstock, RM, RRF, β), given values for the other variables. What is the required return of a stock, if the required return of the market is 10%, the risk-free rate is 6%, and the beta of the stock is 1.25? Rstock = ( (10 - 6) × 1.25 ) + 6 Rstock = 11 What is the required return of a stock, if the market risk premium is 5%, the risk-free rate is 6%, and the beta of the stock is 1.20? Rstock = ( (5) × 1.20 ) + 6 Rstock = 12 What is the beta of a stock, if the required return of the stock is 3%, the required return of the market is 6%, and the risk-free rate is 2%? 3 = ( (6 - 2) × β ) + 2 β = 0.25 What is the risk-free rate, if the required return of the stock is 9%, the required return of the market is 10%, and the beta of the stock is 0.80? 9 = ( (10 - RRF) × 0.80 ) + RRF RRF = 5

Return Formula

Example: ABC's stock price started the period at $50 per share and ended the period at $58. ABC paid a dividend of $2 per share during the period. $ total return = $ capital gain + $ dividend Let P0 = price at time 0 Let P1 = price at time 1 Let D1 = dividend during time period 1 Then, $ total return = (P1 - P0) + D1 $ total return = ($58 - $50) + $2 = $10

Measuring % Total Return

Example: ABC's stock price started the period at $50 per share and ended the period at $58. ABC paid a dividend of $2 per share during the period. % total return = capital gain yield + dividend yield Capital gains yield = (P1 - P0) / P0 Dividend yield = D1 / P0 Then, % total return = [ ($58-$50) / $50 ] + ($2 / $50) = 16% + 4% = 20%

Creating a Portfolio: Adding Randomly Selected Stocks to Portfolio

Expected portfolio return would remain relatively constant. Most stocks are positively, though not perfectly, correlated with the overall market (i.e., ρ between 0 and +1), so portfolio risk decreases as stocks are added, because they would not be perfectly correlated with the existing portfolio. Additional stocks do little to reduce risk once the portfolio holds 40-50 stocks. For portfolios of large U.S. stocks, σportfolio tends to converge to 20%.

Risk-Return Trade-Off

Good (return) versus Evil (risk) Investors like return Investors do not like risk Fundamental trade-off between risk and return!!!! If you want to take less risk, then you must be willing to accept a lower return. If you want a higher return, then you must be willing to take more risk.

Comments on Beta

If beta = 1.0, the security is just as risky as the average stock or the overall stock market. If beta > 1.0, the security is riskier than average. Example: if stock beta = 1.50 and market ↑ (or ↓) 2%, then stock ↑ (or ↓) by 1.50×2% = 3% on average If beta < 1.0, the security is less risky than average. Example: if stock beta = 0.50 and market ↑ (or ↓) 2%, then stock ↑ (or ↓) by 0.50×2% = 1% on average Most stocks have betas in the range of 0.5 to 1.5, though β can even be negative

Coefficient of Variation

Most investors are "risk averse". When choosing between 2 investments that have equal expected returns, but different risk, most will choose the investment with the lower risk. When choosing between 2 investments that have equal risk, but different expected returns, most will choose the investment with the higher return. What if one investment has higher expected return and higher risk than the other investment? Coefficient of variation: standardized measure of dispersion about the expected value, that shows the risk per unit of return. 𝐶𝑉= 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 / 𝑅𝑎𝑡𝑒 𝑜𝑓 𝑅𝑒𝑡𝑢𝑟𝑛

Two Types of Risk

Not all risk can be eliminated! Stand-alone risk = diversifiable risk + market risk "Diversifiable risk", also known as "unsystematic risk", also known as "company-specific risk": part of a security's stand-alone risk associated with unique events; can be eliminated with proper diversification. "Non-diversifiable risk", also known as "systematic risk", also known as "market risk": part of a security's stand-alone risk that cannot be eliminated through diversification.

More about Risk Aversion

Other things held constant, the higher (lower) a security's risk, the higher (lower) its required return. If this relationship does not hold, then market prices will adjust to bring about the required condition. Risk premium: the difference between the expected rate of return on a given risky asset and that on a less risky asset.

Stand-Alone Risk

the risk that an investor would face if they held only the 1 asset being analyzed. Investment risk is related to the probability of earning a low or negative actual return. The greater the chance of lower than expected, or negative returns, the riskier the investment. How do we measure return? See next few slides.