Financial Management (Lecture 5)

Financial Management(N12403)

Lecture 5 The Capital Asset Pricing Model, Arbitrage Pricing Theory & Cost of Capital

Lecturer: Farzad Javidanrad

(Autumn 2014-2015)

Risk-Return Analysis • In the previous lecture we talked about the risk associated to a security and how variance (or standard deviation) of returns of the security can measure this risk. • We also mentioned that a rational investor hold a well diversified portfolio of different securities in order to reduce the level of unique (unsystematic, diversifiable) risk associated to individual securities, but market (systematic, non-diversifiable) risk cannot be eliminated. • The idea that how diversification leads to the reduction of risk was formulated by Harry Markowitz in 1952 as he believed risk, as well as the highest expected return, is another element that investors care about and the variance of a portfolio does not only depend on the variance of individual assets in the portfolio but also on covariance between them.

• The risk of a portfolio is measured by its variance (or its standard deviation) but the risk of an individual asset (security) in the portfolio is measured by its covariance with other assets (securities) in the portfolio.

Risk-Return Analysis • Imagine two portfolios A and B plotted based on their expected returns and their levels of risk (following diagrams). In each extreme case the best portfolio is: Expected Return %

A

𝑟𝐴

A because it has more expected return at a specific level of risk

B

𝑟𝐵

𝜎𝐴 = 𝜎𝐵

Risk (Standard Deviation)

Expected Return %

𝑟𝐴 = 𝑟𝐵

A

𝜎𝐴

B

𝜎𝐵

A because its risk is lower at a given level of return

Risk (Standard Deviation)

Efficient Portfolios • Now, suppose A is a portfolio with low risk and low return, compare to B with high return and high risk. • If we imagine portfolios A and B have a perfect positive correlation (𝒓𝑨𝑩 = 𝟏) then there is a linear relationship between them (they are connected through the green line) and any linear combination between them cannot bring the variance of the portfolio lower than the average of two variances. Therefore, it is not possible to reduce the risk by diversification.  In reality, perfect positive/negative Expected correlation (𝒓𝑨𝑩 = ±𝟏 ) between two Return % securities or two portfolios does not exist. B  when 𝒓𝑨𝑩 < 𝟏, diversification can reduce the 𝒓𝑨𝑩 < 𝟏 level of risk and any combination under the 𝒓𝑨𝑩 = 𝟏 red curve is feasible but a rational investor A will find the best combination on the red curve and not below that. Why? Risk (Standard Deviation)

Efficient Portfolios

• The best portfolio is an efficient portfolio which gives the highest expected return at a given level of risk or lowest level of risk at a given expected return.

• Therefore, between portfolio A and C, a rational investor choose C as it gives the highest return at a certain level of risk. Expected Return %

C

𝒓𝑨𝑩 < 𝟏

B

𝒓𝑨𝑩 = 𝟏 Portfolio with minimum risk

A Risk (Standard Deviation)

• How can we find the best portfolios?

Efficient (Markovitz) Frontier • In reality, we are facing with so many different portfolios not just two of them in our example.

• Let’s imagine that in the return-risk space we are dealing with some portfolios, such as the following scatter plot, which shows a set of investment opportunities. Each point shows a portfolio which is a combination of different assets with different levels of risk and return. Obviously, the correlation should be positive. Why? Expected Return %

• At any level of expected return or at any level of risk the green portfolios are the best. The curve which connects all the best portfolios at different levels of return & risk is called efficient frontier or Markovitz efficient frontier. So, the efficient Risk (Standard Deviation) frontier is the set of all efficient portfolios.

Risk-Return Indifference Curves • Which portfolio should an investor choose? What is the optimal or the best portfolio for an investor? • Using the risk-return analysis, risk-averse investors choose a portfolio on the lowerleft of the Markovitz frontier while risk-taker investors choose a portfolio on the upper-right of the curve. • But the best portfolio for each group of investors can be found when we have information about their risk-return indifference curves. Any indifference curve represents a unique level of satisfaction that an investor gets by combining different assets (securities) in the form of a portfolio. So, on an indifference curve there are different portfolios which all give the same satisfaction to the portfolio holder; so he/she is indifferent between them.

P4

Adopted from http://www.emeraldinsight.com/journals.htm?articleid=1558121&show=html

𝑃1 is a better portfolio compare to 𝑃3 because it is on higher indifference curve but 𝑃3 and 𝑃4 give the same satisfaction to the investor.

Optimal Portfolio • The investor's optimal portfolio is at the tangency point of the efficient frontier with the indifference curve. • The tangency point M represents the highest level of satisfaction which is on the efficient frontier. An indifference curve suggests all those portfolios (combinations of expected returns and risks) that provide the same level of satisfaction for a specific investor.

Expected Return % M

Risk (Standard Deviation)

• All portfolios on efficient frontier curve include risky assets. As soon as, risk-free assets (such as government securities) are introduced all investors will find a new efficient frontier which is dominant on the Markovitz efficient frontier as it enables investors to achieve a higher rate of return on a higher level of satisfaction.

Introducing Risk-Free Assets • Risk-free (risk-less) assets (such as government bonds, treasury bills) have some interesting characteristics, including: 1. Their expected return is equal to their actual return; 𝐸 𝑟𝑓 = 𝑟𝑓 . This means that the return from risk-free assets is with certainty. 2. For the above reason, the variance of their return is zero;

𝜎𝑟2𝑓

= 𝐸 𝑟𝑓 − 𝐸(𝑟𝑓 )

2

= 0.

3. The covariance between them and other risky assets are zero too; 𝑐𝑜𝑣 𝑟𝑓 , 𝑟 = 0. • Combination of risk-free assets and risky assets provides higher level of return (at any specific level of risk) in compare with a portfolio of just risky assets.

• To show that, imagine an investor who is going to invest 𝜔 proportion of his money on risky assets (for example, portfolio A on the efficient frontier) and the rest of that, (1 − 𝜔), on risk free assets.

Expected Return of a Portfolio Including Risk-Free Assets • If 𝑟𝑓 is the return of risk-free assets and 𝐸(𝑟𝐴 ) is the expected return on the risky portfolio A, the expected return and the standard deviation of the new combined portfolio (𝐸 𝑟𝑝 and 𝜎𝑝 ) are: 𝐸 𝑟𝑝 = 𝝎𝐸(𝑟𝐴 ) + 𝟏 − 𝝎 𝑟𝑓

R

And

𝜎𝑝 =

𝝎𝟐 𝜎𝐴2 + 𝟏 − 𝝎 𝟐 𝜎𝑓2 + 2𝝎 𝟏 − 𝝎 𝑐𝑜𝑣(𝑟𝑓 , 𝐸(𝑟𝐴 ))

But we know that 𝜎𝑓2 = 0 and 𝑐𝑜𝑣 𝑟𝑓 , 𝐸(𝑟𝐴 ) = 0, therefore: 𝜎𝑝 𝜎𝑝 = 𝝎𝜎𝐴 → 𝝎 = 𝜎𝐴

Substituting this in 𝐸 𝑟𝑝

R

we have:

𝜎𝑝 𝜎𝑝 𝐸(𝑟𝐴 ) − 𝑟𝑓 = 𝐸(𝑟𝐴 ) + 1 − 𝑟𝑓 → 𝐸 𝑟𝑝 = 𝑟𝑓 + 𝜎𝑝 𝜎𝐴 𝜎𝐴 𝜎𝐴

Capital Allocation Line (CAL) • The linear equation 𝐸 𝑟𝑝

𝐸(𝑟𝐴 ) − 𝑟𝑓 = 𝑟𝑓 + 𝜎𝑝 𝜎𝐴

is the risk-return trade-off between the expected return 𝐸 𝑟𝑝 and the risk 𝜎𝑝 for efficient portfolios. The line is called the Capital Allocation Line (CAL). • The intercept of the line is 𝑟𝑓 , which is the point of risk-free rate on vertical axis and the slope of the line is that higher risk comes with higher expected return.

𝐸(𝑟𝐴 )−𝑟𝑓 𝜎𝐴

, which is positive and reflects the fact

𝐸 𝑟𝑝 𝑪𝑨𝑳: 𝑬 𝒓𝒑 = 𝒓𝒇 + 𝐸(𝑟𝐴 ) 𝑟𝑓

𝐴

𝜎𝑝

𝑬(𝒓𝑨 ) − 𝒓𝒇 𝝈𝒑 𝝈𝑨

Capital Market Line and Sharpe Ratio • Any portfolio resulted from the combination of risk-free assets and risky portfolio A , i.e. any point on the line between 𝑟𝑓 and A , are superior to the points on the efficient frontier curve and under the portfolio A, because the combination with free-risk assets provides a higher expected return. • So, the new efficient frontier starts from 𝑟𝑓 to A and then continues on the previous Markovitz frontier. CML:

𝐸 𝑟𝑝 M

𝐸 𝑟𝑀

𝐸 𝑟𝑝 = 𝑟𝑓 +

𝐸(𝑟𝑀 ) − 𝑟𝑓 𝜎𝑝 𝜎𝑀

A

𝑟𝑓

𝜎𝑀

𝜎𝑝

Market price of risk = additional expected return that investors would require to compensate them for incurring additional risk.

 Portfolio A composed of only risky assets but a wise investor can buy some risk-free assets at rate 𝒓𝒇 to move up on new efficient frontier, which introduce higher expected return. This new efficient frontier is called Capital Market Line (CML).

Market Portfolio & Sharpe Ratio • By taking more risks (moving along the efficient frontier) an investor can increase the expected return even higher and find better combinations like M.

• Point M is the tangency point and is the most efficient portfolio known as market portfolio as it is the most diversified portfolio and it includes just risky assets. • This point also offers the highest slope that is; the highest ratio of risk premium to standard deviation which is called Sharpe ratio of the market portfolio. Sharpe Ratio of (M)=

𝐸(𝑟𝑀 )−𝑟𝑓 𝜎𝑀

Sharpe Ratio represents the excess return per unit of risk

Alternatively, the Sharpe ratio of any portfolio 𝐴 (not on CML) can be written as: 𝑆ℎ𝑎𝑟𝑝𝑒 𝑅𝑎𝑡𝑖𝑜 𝑜𝑓 𝐴 =

𝐸(𝑟𝐴 )−𝑟𝑓

𝜎𝐴

Borrowing/Lending at Risk-Free Rate • All investors now have two jobs: 1. Finding the best portfolio (point M in our example) 2. Lend or borrow in order to reach to a suitable risk level they can bear in which higher expected return is embedded. 𝑪𝑴𝑳: 𝑬 𝒓𝒑 = 𝒓𝒇 +

𝑬(𝒓𝑴 ) − 𝒓𝒇 𝝈𝒑 𝝈𝑴

𝐸 𝑟𝑝 M

𝐸 𝑟𝑀

𝑟𝑓 𝜎𝑀

• With the lending/borrowing opportunity at the risk-free rate, an investor is no longer restricted to holding a portfolio on the efficient frontier. • They can now invest in combinations of risky and risk-free assets in accordance with their risk preferences. • A risk-averse investor will put part of their money in the efficient portfolio M and part of that in the risk-free assets. A risk-taker investor may put all their money in the best portfolio M or may even borrow and invest all of them on M.

Borrowing/Lending in the Absence of Risk-Free Rate • When risk-averse investors buy risk-free assets such as government securities at rate 𝑟𝑓 , in fact, they lend their money to the government. So, the efficient frontier line before the point M (the line between 𝑟𝑓 and M) is called lending portfolio as it includes a portion (1 − 𝜔) of the government securities. • In case, that risk-taker investors are able to borrow at the same rate (𝑟𝑓 ), in order to invest more at point M, the efficient frontier after M is called borrowing portfolio. But in reality, risk-taker investors are not able to find such a good rate to borrow and need to accept higher rates of borrowing (such as 𝑟𝑏 ). Therefore, the efficient frontier will be different: Z

𝐸 𝑟𝑝 B

𝑟𝑏

𝑟𝑓

M

𝜎𝑝

New efficient frontier when there is two different rates of borrowing/lending: 𝑟𝑓 𝑀𝐵𝑍

The Capital Asset Pricing Model (CAPM) • Capital market line (CML) introduces efficient portfolios but does not say anything about the relationship between expected return and risk for individual assets or even inefficient portfolios.  The Capital Asset Pricing Model (CAPM):

• The capital asset pricing model (CAPM) is a linear relationship between the expected rate of return of an individual asset (which is going to be added to an already welldiversified portfolio) and its contribution to the portfolio’s risk. • CAPM is a model for pricing an individual asset 𝑖 by connecting its expected rate of return (𝐸(𝑟𝑖 )) to the risk-free rate of interest (𝑟𝑓 ), expected return on market portfolio 2 ) and the risk (𝐸 𝑟𝑚 ), the variance of the return on the market portfolio (𝜎𝑚 contributed to the portfolio by the risky assets (𝑐𝑜𝑣 𝑟𝑖 , 𝑟𝑚 = 𝜎𝑖𝑚 = 𝜌𝜎𝑖 𝜎𝑚 ).

The Capital Asset Pricing Model (CAPM) • The idea behind pricing is that no individual security should be over-valued or undervalued in compare to the market values. So, the reward-to-risk ratio for an individual risky asset should be equal to the reward-to-risk ratio of the overall market; i.e. 𝐸 𝑟𝑖 −𝑟𝑓 𝜎𝑖𝑚

Risk Premium

=

𝐸 𝑟𝑚 −𝑟𝑓 2 𝜎𝑚

Market Premium

Q

Where risk here is expressed in terms of the variance and not standard deviation. • By re-writing equation

Q

and make 𝐸 𝑟𝑖 as the subject, we have: 𝐸 𝑟𝑖 = 𝑟𝑓 + (𝐸 𝑟𝑚

𝜎𝑖𝑚 − 𝑟𝑓 ) × 2 𝜎𝑚

= 𝑟𝑓 + (𝐸 𝑟𝑚 − 𝑟𝑓 ) × 𝛽 Where 𝛽 measures the sensitivity of the asset to the market movements and it is representative of the systematic (market) risk.

Security Market Line • The re-written form of the equation Q is called Security Market Line (SML), which shows the relation between the expected return of an asset and its systematic risk 𝛽. • SML states that an individual asset risk premium is equal to the market premium times the systematic risk 𝛽 and it is a useful tool in determining whether an individual asset offers a rational expected return for the risk contributed to the whole portfolio. SML: 𝑬 𝒓𝒊 = 𝒓𝒇 + 𝑬 𝒓𝒎 − 𝒓𝒇 . 𝜷

0.5 Adopted from http://en.wikipedia.org/wiki/File:SML-chart.png

2

Securities with 𝜷>𝟏 amplify the overall movements of the market; with 0