Risk and Return – Part 3

Risk and Return – Part 3 For 9.220, Term 1, 2002/03 02_Lecture14.ppt Student Version

Outline • Introduction • The Markowitz Efficient Frontier • The Capital Market Line (CML) • The Capital Asset Pricing Model (CAPM) • Summary and Conclusions

Introduction • We have seen that holding portfolios of more than one asset gives the potential for diversification. • We will now look at what might be an optimal strategy for portfolio construction – being well diversified. • We extend the results from this into a model of Risk and Return called the Capital Asset Pricing Model (CAPM) that theoretically holds for individual securities and for portfolios.

The Opportunity Set and The Efficient Set 100% Stock 1 The portfolios in this area are all dominated. 100% Stock 2

The Opportunity Set when considering all risky securities Consider all the risky assets in the world; we can still identify the Opportunity Set of risk-return combinations of various portfolios. E[R] Individual Assets 

The Efficient Set when considering all risky securities The section of the frontier above the minimum variance portfolio is the efficient set. It is named the Markowitz Efficient Frontier after researcher Harry Markowitz (Nobel Prize in Economics, 1990) who first discussed it in 1959. E[R] efficient frontier minimum variance portfolio Individual Assets 

Optimal Risky Portfolio with a Risk-Free Asset • In addition to risky assets, consider a world that also has risk-free securities like T-bills. • We can now consider portfolios that are combinations of the risk-free security, denoted with the subscript f and risky portfolios along the Efficient Frontier. E[R] 

The riskfree asset: riskless lending and borrowing • Consider combinations of the risk-free asset with a portfolio, A, on the Efficient Frontier. • With a risk-free asset available, taking a long f position (positive portfolio weight in f) gives us risk-free lending combined with A. • Taking a short f position (negative portfolio weight in f) gives us risk-free borrowing combined with A. E[R] Portfolio A Rf P

The riskfree asset: riskless lending and borrowing • Which combination of f and portfolios on the Efficient Frontier are best? E[R] Rf P What is the optimal strategy for every investor?

M: The Market Portfolio CML The combination of f and portfolios on the Efficient Frontier that are best are… All investors choose a point along the line… In a world with homogeneous expectations, M is the same for all investors. E[R] M CML stands for the Capital Market Line Rf P

A new separation theorem CML This separation theorem states that the market portfolio, M, is the same for all investors. They can separate their level of risk aversion from their choice of the risky component of their total portfolio. All investors should have the same risky component, M! E[R] M Rf P

Given Separation, what does an investor choose? CML While all investors will choose M for the risky part of their portfoio, the point on the CML chosen depends on their level of risk aversion. E[R] M Rf P

The Capital Market Line (CML) Equation The CML equation can be written as follows: Where • EPi = efficient portfolio i (a portfolio on the CML composed of the risk-free asset, f, and M) • E[ ] is the expectation operator • R = return • σ = standard deviation of return • f denotes the risk-free asset • M denotes the market portfolio Note: the CML is our first formal relationship between risk and expected return. Unfortunately it is limited in its use as it only works for perfectly efficient portfolios: composed of f and M.

The Capital Asset Pricing Model (CAPM) • If investors hold the market portfolio, M, then the risk of any asset, j, that is important is not its total risk, but the risk that it contributes to M. • We can divide asset j’s risk into two components: the risk that can be diversified away, and the risk that remains even after maximum diversification. • The division is found by examining ρjM, the correlation between the returns of asset j and the returns of M. • Asset j’s total risk is defined by σj • The part of σj that can be diversified away is (1-ρjM)● σj • The part of σj that remains is ρjM● σj

Non-diversifiable risk and the relation to expected return. We can extend the CML to a single asset by substituting in the asset’s non-diversifiable risk for σEPi: SML stands for Security Market Line. It relates expected return to β and is the fundamental relationship specified by the CAPM.

The Security’s Beta • The important measure of the risk of a security in a large portfolio is thus the beta (b)of the security. • Beta measures the non-diversifiable risk of a security – i.e., the risk related to movements in the market portfolio.

Characteristic Line Estimating b with regression Security Returns Return on market

Know your betas! • The possible range for β is -∞ to +∞ • The value of βM is… • The value of βf is… • For a portfolio, if you know the individual securities’ β’s, then the portfolio β is… where the xi values are the security weights.

Estimates of b for Selected Stocks

Examples • What would be your portfolio beta, βp, if you had weights in the first four stocks of 0.2, 0.15, 0.25, and 0.4 respectively. • What would be E[Rp]? Calculate it two ways. • Suppose σM=0.3 and this portfolio had ρpM=0.4, what is the value of σp? • Is this the best portfolio for obtaining this expected return? • What would be the total risk of a portfolio composed of f and M that gives you the same β as the above portfolio? • How high an expected return could you achieve while exposing yourself to the same amount of total risk as the above portfolio composed of the four stocks. What is the best way to achieve it?

Summary and Conclusions • The CAPM is a theory that provides a relation between expected return and an asset’s risk. • It is based on investors being well-diversified and choosing non-dominated portfolios that consist of combinations of f and M. • While the CAPM is useful for considering the risk/return tradeoff, and it is still used by many practitioners, it is but one of many theories relating return to risk (and other factors) so it should not be regarded as a universal truth.

Risk and Return – Part 3