CHAPTER 5 Risk and Rates of Return Stand-alone risk Portfolio risk Risk & return: CAPM / SML

Return & Risk Return is the annual income received plus any change in market price of an asset or investment. Risk is the variability of actual return from the expected return associated with a given asset.

Rate of Return The rate of return on an investment for a period (which is usually a period of one year) is defined as follows, Annual Income + (Ending price – Beginning price) Beginning price

Rate of Return Price at the beginning of the year = Tk. 60.00 Dividend paid toward the end to the year = Tk. 2.40 Price at the end of the year = Tk. 66.00 2.40 + (66.00 – 60.00) 60.00 = 0.14 or 14%.

Rate of Return Current Yield & Capital gains/Losses Capital gains/Losses Yield

Current Yield & Capital gains/Losses Annual Income Ending – Beginning Price + Beginning Price Beginning Price (Current Yield) (Capital Gain/Losses)

Current Yield & Capital gains/Losses 2.40 (66.00 – 60.00) + 60.00 60.00   = 4% + 10% Current Yield Capital gains/Losses

Current Yield & Capital gains/Losses If the price of a share on April 1 is TK. 25, the annual dividend received at the end of the year is TK. 1 and the year end price on March 31 is TK 30. Find the Rate of Return Find the Current Yield Find the Capital gains/Losses Yield.

Current & Capital gain/Losses 1 (30.00 – 25.00) + 25 25.00 = 4% + 20% 1 + (30.00 – 25.00) Rate of return = 25.00 = 0.24 or 24%. Current & Capital gain/Losses 1 (30.00 – 25.00) + 25 25.00   = 4% + 20% Current Yield Capital gains/Losses

What is investment risk? Two types of investment risk Stand-alone risk Portfolio risk 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.

Measurement of Risk: Single Asset The risk associated with single asset is assessed from both, Behavioral point of view Sensitivity Analysis Probability Distribution Statistical point of view Standard Deviation Coefficient of Variation

Measurement of Risk: Single Asset Behavioral Point of View This approach is to estimate the worst (pessimistic), the expected (most likely) and the best (optimistic) return associated with the asset. The level of outcome may be related to the economic conditions namely, recession, growth and Boom. The difference between pessimistic and optimistic outcome is the RANGE which is the measurement of RISK. The greater the RANGE, the more RISKY the Asset.

Sensitivity Analysis Particular Asset X Asset Y Initial Outlay 50 50 Annual Return (%) Pessimistic 14 8 Most Likely 16 16 Optimistic 18 24 RANGE = 4 16 (optimistic – Pessimistic)

The probability of an event represent the % chance of its occurrence. Measurement of Risk: Single Asset Behavioral Point of View Probability Distribution The probability of an event represent the % chance of its occurrence. Probability Distribution is model that relates probabilities to the associated outcome.

Probability Distribution : Asset X Possible Outcome (1) Probability (2) Returns (3) Expected Returns (2)X(3)=4 Pessimistic Most Likely Optimistic 0.20 0.60 1.00 14 16 18 2.8 9.6 3.6 16.00

Probability Distribution : Asset Y Possible Outcome (1) Probability (2) Returns (3) Expected Returns (2)X(3)=4 Pessimistic Most Likely Optimistic 0.20 0.60 1.00 8 16 24 1.6 9.6 4.8 16.00

Measurement of Risk: Single Asset Statistical Point of View Standard Deviation Risk refers to the dispersion of returns around an expected value. The most common statistical measure of risk of an asset is the standard deviation from the mean/expected value of return.  = (R-R)2 X pr

Standard Deviation Asset X i R R - R (R – R)2 Pr (R – R)2x Pr 1 14% 16% -2% 4% .20 .80 2 0% .60 3 18% 2% 1.6

Standard Deviation  = (R-R)2 X pr = 1.6 = 1.26%

Standard Deviation Asset Y i R R - R (R – R)2 Pr (R – R)2x Pr 1 8% 16% -8% 64% .20 12.80 2 0% .60 3 24% 25.6

Standard Deviation  = (R-R)2 X pr = 25.6 = 5.06% = 25.6 = 5.06% The greater the Standard Deviation of Returns, the greater the risk.

Measurement of Risk: Single Asset Statistical Point of View Coefficient of Variation it is the measure of relative dispersion used in comparing the risk of assets with differing expected returns.  CV = R

The coefficient of variation of assets X & Y are respectively, Measurement of Risk: Single Asset Statistical Point of View Coefficient of Variation The coefficient of variation of assets X & Y are respectively, Asset X = ( 1.26% / 16%) = 0.079 Asset Y = (5.06 / 16% ) = 0.316 The larger the CV, the larger the relative risk of the asset.

Risk & Return of PORTFOLIO Portfolio means a combination of two or more Assets. Each portfolio has risk return characteristics of its own. Portfolio theory developed by Harry Markowitz, shows that portfolio risk, unlike portfolio return, is more than simple aggregation of the risks of individual assets. This depends on the interplay between the returns on assets comprising the portfolio.

Portfolio Expected return E (rp) = wi E(ri) E (rp) = Expected return from portfolio Wi = Proportion invested in asset i E(ri) = Expected return for asset i n = number of assets in portfolio

Portfolio Expected return The expected return on two assets L and H are 12% & 16% respectively. If the corresponding weights are 0.65 & 0.35. Calculate Portfolio Expected return E (rp) = wi E(ri) = [0.65 x 0.12 + 0.35 x 0.16] = 0.134 = 13.4%.

Portfolio Risk:Two Asset portfolio 2p = w2121 + w2222 + 2 w1 w2 (12) Alternatively, 2p = (w11)2 + (w22 )2 + 2 w1 w2 (P 12 1  2) W1 = fraction of total portfolio invested in Asset 1 W2 = fraction of total portfolio invested in Asset 2 21 = Variance of asset 1 1 = Standard deviation of Asset 1 22 = Variance of asset 2 2 = Standard deviation of Asset 2 12 = Covariance between returns of two assets (P 12 1  2) P 12 = Coefficient of correlation between the returns of two asset.

Portfolio Risk:Two Asset portfolio The expected return on two assets L and H are 12% & 16% respectively. The standard deviations of assets L & H are 16% and 20% respectively. If the coefficient of correlation between their returns is 0.6 and the two assets are combined in the ratio of 3:1. (1) Calculate expected rate of return (2) variance of Portfolio (3) Standard Deviation

Portfolio Expected return E (rp) = wL E(rL) + wH E(rH) = (0.75 x 0.12) + (0.25 x 0.16) = 9%+4% = 13%.

The Variance of the Portfolio [2] 2p = (w11)2 + (w22 )2 + 2 w1 w2 (P 12 1  2) = (0.75 x 16)2 + (0.25 x 20) 2 + 2 (0.75) (0.25) [(0.06) (16 x 20) = 144 + 25+ (0.375)(192) = 144 + 25+72 = 241 [3] p = 241 = 15.52

Portfolio Risk The above discussion shows that the portfolio risk depends on 3 factors [1] Variance or Standard deviation of each asset in portfolio. [2] Relative importance or weight of each asset in the portfolio [3] Interplay between returns on two assets Among these only weights can be controlled by the portfolio managers. Therefore his/her primary task is to decide the proportion of each security in the portfolio.

Investor attitude towards risk Risk aversion – assumes investors dislike risk and require higher rates of return to encourage them to hold riskier securities. Risk premium – the difference between the return on a risky asset and less risky asset, which serves as compensation for investors to hold riskier securities.

Breaking down sources of risk Stand-alone risk = Market risk + Firm-specific risk Market risk – portion of a security’s stand-alone risk that cannot be eliminated through diversification. Measured by beta. Firm-specific risk – portion of a security’s stand-alone risk that can be eliminated through proper diversification.

Capital Asset Pricing Model (CAPM) Model based upon concept that a stock’s required rate of return is equal to the risk-free rate of return plus a risk premium that reflects the riskiness of the stock after diversification. It is the logical & major extension of the portfolio theory of Markowitz by william Sharpe (1964), John Linter ( 1965) & Jan Mossin (1967).

Capital Asset Pricing Model (CAPM) CAPM is a theory that explains how asset prices are formed in the market place. CAPM provides the framework for determining the equilibrium expected return for risky return. It uses the results of capital market theory to derive the relationship between expected return and systematic risk of individual assets/securities and portfolio.

Capital Asset Pricing Model (CAPM) The CAPM has implication for Risk-Return relationship for an efficient Portfolio Risk-Return relationship for an individual asset Identification of over valued or under valued assets traded in in the market Pricing of assets not yet traded in the market Effect of leverage on cost of equity.

Capital Asset Pricing Model (CAPM) Capital budgeting decision & cost of capital Risk of the firm through diversification of the project portfolio.

Capital Asset Pricing Model (CAPM) : Assumption All investors are price takers. There number is so large that no single investor can affect prices Assets/securities are perfectly divisible All investors plan for one identical holding period Investors can lend or borrow at an identical risk-free rate. There is no transaction costs & income Tax

Capital Asset Pricing Model (CAPM) The elements of the model: K = K RF + (KM - K RF) β Where, K RF = Risk Free Return KM = required rate of return of market β = Beta (systematic risk of the asset)

Beta It measure the risk of an individual asset relative to the market portfolio. Beta shows how the price of securities responds to market force. In practice, the more responsive the price of security is to changes in the market, the higher will be its beta. The beta for the overall market is equal to 1.00 Beta can be positive or negative. Investors will find beta helpful in assessing systematic risk and understanding the impact the market movement can have on the return expected from a share or stocks.

Calculating betas The ABC Company is considering a new capital investment proposal. The project’s risk structure is very similar to that of the company’s existing business. Return for this company’s stocks for the past ten years are given in the following table together with returns for a country’s stock market index. The Govt. Treasury Bill return (Risk Free Return) was around 5.6% per annum.  

Company’s Stock Return Stock Market Index Return Calculating betas Year Company’s Stock Return Stock Market Index Return 1992 0.09 0.07 1993 0.10 1994 1995 0.11 0.12 1996 1997 1998 1999 2000 0.08 2001

Calculating betas Required Calculate the [1] Beta [2] Required Return according to CAPM model

Comments on beta If beta = 1.0, the security is just as risky as the average stock. If beta > 1.0, the security is riskier than average. If beta < 1.0, the security is less risky than average. Most stocks have betas in the range of 0.5 to 1.5.

Problem: Assume a security with beta of 1.2 being considered at a time when the risk free rate is 4% and the market return is expected to be 12%. Substitute those data by using CAPM equation. Calculate Expected Return according to CAPM model

Problem: There are three assets- X, Y & Z with beta value of 0.5, 1.0 & 1.5 respectively. The risk free rate is assumed to be 5% and the market return is expected to be 15%. calculate the expected return

Illustrating the calculation of beta . ki _ kM -5 0 5 10 15 20 20 15 10 5 -5 -10 Regression line: ki = -2.59 + 1.44 kM ^ Year kM ki 1 15% 18% 2 -5 -10 3 12 16

The Security Market Line (SML): Calculating required rates of return SML: ki = kRF + (kM – kRF) βi Assume kRF = 8% and kM = 15%. The market (or equity) risk premium is RPM = kM – kRF = 15% – 8% = 7%.

Illustrating the Security Market Line . Coll. HT T-bills USR SML kM = 15 kRF = 8 -1 0 1 2 SML: ki = 8% + (15% – 8%) βi ki (%) Risk, βi

Factors that change the SML What if investors raise inflation expectations by 3%, what would happen to the SML? ki (%) SML2 D I = 3% SML1 18 15 11 8 Risk, βi 0 0.5 1.0 1.5

Factors that change the SML What if investors’ risk aversion increased, causing the market risk premium to increase by 3%, what would happen to the SML? ki (%) SML2 D RPM = 3% SML1 18 15 11 8 Risk, βi 0 0.5 1.0 1.5

Verifying the CAPM empirically The CAPM has not been verified completely. Statistical tests have problems that make verification almost impossible. Some argue that there are additional risk factors, other than the market risk premium, that must be considered.

More thoughts on the CAPM Investors seem to be concerned with both market risk and total risk. Therefore, the SML may not produce a correct estimate of ki. ki = kRF + (kM – kRF) βi + ??? CAPM/SML concepts are based upon expectations, but betas are calculated using historical data. A company’s historical data may not reflect investors’ expectations about future riskiness.