On risk and return

Objective Learn the math of portfolio diversification Measure relative risk Estimate required return as a function of relative risk

Important observation  (portfolio) <  (r A )w A +  (r B ) w B In general, the standard deviation of the portfolio is less than the average of the individual standard deviations

The standard deviation of a portfolio return Number of stocks in the portfolio  (portfolio)

The standard deviation of expected return: A summary Standard deviation and variance measure the variability of the return Standard deviation is a measure of absolute risk The standard deviation of a portfolio is less than the weighted average of individual standard deviations This is true because the returns of various securities are not perfectly correlated, i.e. changes in returns are not perfectly synchronized. By adding individual securities to a portfolio, the overall standard deviation of the portfolio is likely to decrease.

More on the standard deviation of a portfolio return Bundling stocks & bonds into portfolios is called diversification Diversification is useful because it reduces risk The amount of risk (standard deviation) that can be eliminated is called diversifiable or non-systematic risk.

More on the standard deviation of a portfolio return Number of securities in the portfolio  (portfolio) Non-systematic risk Systematic (market) risk

Remember E(r A ) = 6.6% E(r B ) = 5.2%  (r A ) = 2.94%  (r B ) = 0.979%

Standard deviation Expected return  (r A ) = 2.94  (r B ) = 0.98 E(r B ) = 5.2% E(r A ) = 6.6% A B  (p) = 1.64 E(p) = 5.9 P Portfolio P when  ( A,B ) = 0.2

Standard deviation % Expected return %  (r A ) = 2.94  (r B ) = 0.98 E(r B ) = 5.2 E(r A ) = 6.6 A B  (p) = 1.64 E(p) = 5.9 All possible portfolio combinations of A and B when  ( A,B ) = 0.2

Variations What if the returns of A and B were perfectly correlated?  ( A,B ) = 1

Standard deviation % Expected return %  (r A ) = 2.94  (r B ) = 0.98 E(r B ) = 5.2 E(r A ) = 6.6 A B Portfolio P when  (r A,B ) = 1  (p) = 1.96 E(p) = 5.9 P

Standard deviation % Expected return %  (r A ) = 2.94  (r B ) = 0.98 E(r B ) = 5.2 E(r A ) = 6.6 A B  (p) = 1.96 E(p) = 5.9 All possible portfolio combinations of A and B when  (r A,B ) = 1 P

More variations What if the returns of A and B were perfectly negatively correlated?  (r A,B ) = - 1

Standard deviation % Expected return %  (r A ) = 2.94  (r B ) = 0.98 E(r B ) = 5.2 E(r A ) = 6.6 A B All possible portfolio combinations of A and B when  (r A,B ) = -1  (p) = 1.45 E(p) = 5.9 P

Standard deviation Expected return A B All possible portfolio combinations of A and B, for all possible correlations between the return of A and B

Reality check There are thousands of securities in the market Their returns are highly correlated, but not perfectly correlated 0 <  < 0.8 There are benefits from diversification!

Standard deviation Expected return All possible portfolio combinations in a world with n securities

Question Of all possible combinations, which portfolios would you rather hold?

Answer It is expected that you would want to hold the portfolios that have: the highest expected return for a given standard deviation, or the lowest standard deviation for a given level of expected return

Standard deviation Expected return All possible portfolio combinations in a world with n securities The efficient set

Question From the efficient set, which portfolios would you rather hold?

Answer It depends on your risk preference.

Yet another reality check Individuals can borrow and lend money fairly easily...

Question(s) How many individuals/families have a savings account/GIC? How many individuals/families invest in the stock market directly, or through mutual funds, pension plans etc?

Facts Almost everyone holds (directly or indirectly) a combination of risky assets and risk-free investments. Risky assets: Stocks, bonds, etc.

A portfolio of risky assets and risk-free investments Risky assets: A and B Risk-free investment: T-bill E(r A ) = 6.6% E(r B ) = 5.2%  (r A ) = 2.94%  (r B ) = 0.979%  A,B = 0.2 E(r T ) = 3%  (r T ) = 0

Portfolio P Weights: A (50%) and B (50%)  (portfolio) = 1.64% ER(p) = 5.9%

Standard deviation % Expected return %  (r A ) = 2.94  (r B ) = 0.98 E(r B ) = 5.2 E(r A ) = 6.6 A B  (p) = 1.64 E(p) = 5.9 Portfolio P when  (A,B) = 0.2 P E(r T ) = 3

Various combinations between P and T Combination C 1 : Invest $5,000 in T and $5,000 in P E(C 1 ) = (1/2)3% + (1/2)5.9% = 4.45%  (C 1 ) = (1/2)1.64% = 0.82%

Standard deviation % Expected return %  (p) = 1.64 E(p) = 5.9 P E(r T ) = 3  (C 1 ) =0.82% E(C 1 ) =4.45 C1C1

Various combinations between P and T Combination C 2 : Invest $2,500 in T and $7,500 in P E(C 2 ) = (1/4)3% + (3/4)5.9% = 5.175%  (C 2 ) = (3/4)1.64% = 1.23%

Standard deviation % Expected return %  (p) = 1.64 E(p) = 5.9 P E(r T ) = 3 C1C1  (C 2 ) = 1.23 E(C 2 ) =5.175 C2C2

Various combinations between P and T Combination C 3 : Invest $7,500 in T and $2,500 in P E(C 3 ) = (3/4)3% + (1/4)5.9% = 3.725%  (C 3 ) = (1/4)1.64% = 0.41%

Standard deviation % Expected return %  (p) = 1.64 E(p) = 5.9 P E(r T ) = 3 C1C1 C2C2  (C 3 ) = 0.41 E(C 3 ) = C3C3

Important Combinations among risky assets lie on a curved line Combinations between risky assets and the risk-free investment lie on a straight line

Question How many possible combinations of risky assets and risk free investments are there?

Standard deviation Expected return All possible portfolio combinations in a world with n securities and a risk-free investment Risk-free return

Standard deviation Expected return Risk-free return Question: Of all possible portfolios in the world, which ones would you rather hold?

Standard deviation Expected return Answer: Efficient portfolios only! Risk-free return The efficient set

Important Again, note that all portfolios from the efficient set have: - The highest expected return for a given level of risk - The lowest level of risk for a given level of expected return

Standard deviation Expected return Risk-free return The efficient set Question: Of all the efficient portfolios, which ones would you hold?

Answer The choice is dictated by individual risk preferences

Standard deviation Expected return Question: Of all possible portfolios of risky assets, which one(s) would you rather hold? Risk-free return The efficient set

Standard deviation Expected return Answer: Of all possible combinations of risky assets, investor would want to hold only M Risk-free return The efficient set M

Question Why only M?

Answer M is the only portfolio of risky assets that produces efficient portfolios when combined with the risk-free investment

Important All investors should buy the same portfolio of risky assets, regardless of their risk preference Adjusting for risk: In order to reflect individual risk preferences, each investor would combine M with the risk-free asset: - More audacious investors would borrow money to buy more of M - More prudent investors would park a fraction of their wealth in the risk-free investment

Consequences M is very important ! Out of respect, let’s call it The Optimal Portfolio. Optimal portfolio aka Market portfolio Due to its importance, M becomes the yardstick for risk in the marketplace

More consequences If M is the yardstick for risk, we should compare each risky security/portfolio to M The result of the comparison would yield the relative risk of any given security

Comparing risky securities to M Comparison by regression: R i =  +  R M +e  i = the relative risk of security “i” In other words,  measures the contribution of each stock to the volatility of the market portfolio

Comparing risky securities to M Convention:  M = 1  i < 1, the security is less risky than the market  i > 1, the security is riskier than the market

Risk and return: The climax We want to find how to estimate the expected return that would compensate for bearing the aforementioned risk Again, use M

The Facts of life On average, M earns a return above and beyond the risk free rate. In other words, M earns a risk premium, which is the reward for bearing risk. return M = risk free rate + risk premium M

Risk and return: The climax Using algebra, we can prove that: (R i - R f )/  = (R M - R f )/1 Interpretation: The required risk premium per unit of relative risk is constant among all securities in this world

Summary Diversification reduces absolute risk Some combinations of risky securities result in efficient portfolios When there is a risk-free investment, only one efficient portfolio of risky assets is desirable: M Investors combine M with the risk-free asset in different proportions M is the yardstick for risk (CAPM) The risk premium per unit of relative risk is constant across all securities (CAPM)