Capital Market Theory

Chapter 7

Efficient Frontier Investors should select portfolios on the basis of

expected return and risk. A portfolio is efficient if:

1. It has the smallest level of risk for a given return.

or

2. The largest return for a given level of risk. To select efficient portfolios, investors should find out

all portfolios opportunities set i.e. find out risk and return set for all portfolios.

Efficient Frontier

Change Weights

Which combination is superior:

Portfolios PIA POL SD(P) ER(P)

A 25% 75% 4.59% 11.4%

B 50% 50% 3.55% 11.2%

C 75% 25% 5.71% 10.9%

D 100% 0 9.01% 10.67%

Efficient frontier It is a line that shows risk and return combination of all

efficient portfolios. Any portfolio that is below this frontier is considered less

efficient because it has:− Higher risk for the same level of return.− Lower return for the same level of risk.

All rational investors will invest only in portfolios that are on the efficient frontier.

Aggressive investors will invest in portfolios on the upper right of the efficient frontier.

Portfolio Weights with Minimum Variance

Suppose POL is Security A; to get minimum variance, we should invest 55% of our funds in POL, and the rest in PIA.

WA= (σ 2B-COV)/(σ 2

A+ σ 2B-2COV)

= (9.02)2 – (-44.44) / (7.64)2 + (9.02)2 – 2 x (-44)

= (81+44) / (58+81+88)

= 125 / 227

= 0.55

Modern Portfolio Theory Markowitz laid the foundation of modern portfolio

theory. Whatever we have studied so far with regard to risk,

return and diversification of portfolio are part of Markowitz portfolio theory.

Before Markowitz there was no formal answer to the question whether combing more than two assets reduce the risk?

The general perception was, “ Do not put all of your eggs in one basket.”

Modern Portfolio Theory

Markowitz was the first to develop measure of portfolio risk based on the covariance of assets and the expected return.

He proved that return of a portfolio is the weighted average return of the assets.

But risk of the portfolio is not the weighted average risk of individual assets, it will be less if there is no perfect positive correlation between the two assets.

Capital Market Theory

William Sharp presented capital market theory, after Markowitz.

He analyzed the effect of risk free asset on a portfolio.

He proved that when a risk free asset is combined with a risky portfolio, the risk of the new portfolio considerably declines without significant decline in the return of the new portfolio.

Risk Free Asset

The return on a risk free asset is fixed hence the standard deviation of risk free asset is zero.

The return on risk free asset does not change with change in return of other securities, so the covariance of risk free asset with any other asset will be zero.

Adding Risk Free Assets to Portfolio

• Suppose risk free asset is taken as A so SD of A =0• The first term of the equation will be removed• The COV term will also be 0, so the last term will also be removed• Risk of the portfolio will be

• The weight of B is equal to WB = (1-WA)• So we can write the above formula as:

2/12222 ]2[ ABBABBAAp Covwwww

2/122 ]00[ BBp w

2/122 ])1[( BAp w

BAp w )1[(

Example Suppose we made a portfolio of POL, MCB, and

Engro. After trying different weights, we found out that one efficient portfolio give SD of 8%; can we reduce this SD further?

YES by investing some portion of our funds in the above risky portfolio and remainder in risk-free asset.

Suppose we invest 30% in risk free and 70% in the risky asset, the SD of our new portfolio will be?

Example (Continued)

IF we invest 30% in risk free and 70% in risky SD of portfolio = (1-WA) σ B = (1 -.3) x 8 = 0.7 x 8 = 5.6

IF we invest 50% in risk free and 50% in risky SD of portfolio = (1-WA) σ B = (1 - .5) x 8 = 0.5 x 8 = 4

If we invest 80% in risk free, SD of portfolio = (1-WA) σ B = (1 - .8) x 8 = 0. 2x 8 = 1.6

Interpretation The formula shows that the risk of the risky portfolio

falls proportionately with weight of the risk free security

In other words, if we add 50% risk free asset to our portfolio, the overall risk of the risky portfolio will decrease by 50%

The return of the portfolio will also decrease but not by 50%: WHY?

Portfolio SD Return

Risk Free 0 4

Risky 6 10

Risk Free Assets Risky Assets New Portfolio

WA WB SD Return

0 1 6 10

0.1 0.9 5.4 9.4

0.2 0.8 4.8 8.8

0.3 0.7 4.2 8.2

0.4 0.6 3.6 7.6

0.5 0.5 3 7

0.6 0.4 2.4 6.4

0.7 0.3 1.8 5.8

0.8 0.2 1.2 5.2

0.9 0.1 0.6 4.6

1 0 0 4

Capital Market Line

If 100% of the fund is invested in portfolio A, the investor earn a return equal to R1 with 3% risk.

Contrary to this, if all the funds are invested in Risk free assets, the investor earns a return equal to RFR with no risk.

If the funds are invested in both Portfolio A and risk free asset, the investors earns a return equal to some value on the line between RFR and portfolio A.

Efficient Frontier

A risk averse investor who wants to earn a return equal to R1 can invest in portfolio A.

The investor can earn the same return R1, with a lower level of risk by investing half his funds portfolio B and the rest in risk-free assets.

Efficient Frontier

Instead of investing in portfolio B, the investor can make a combination of portfolio C and risk-free assets, thus further reducing the risk from 1.5% to 1% without effecting the return.

Efficient Frontier

Why is portfolio D not preferred investment in comparison to the other portfolios?

Efficient Frontier

Capital Market Line and Portfolio M

The straight line from RFR to portfolio C is called Capital Market Line (CML)

Any portfolio on this line dominates all other portfolio in terms of risk and return combinations.

Question:1. Why all investor will invest only in portfolio C?

2. Why all risky assets will be included in this portfolio?

3. Why it is a completely diversified portfolio?

1. Why all investors invest only portfolio C?

Because no other portfolio gives the best risk-return combination as compared to portfolio C when combined with risk-free assets.

Rational investors maximize their utility by investing only in portfolio C and risk-free assets.

2. Why all types of risky assets are included in Portfolio C? As all investors will invest only portfolio C, they will

only buy those securities that are included in portfolio C.

If an asset is not included in portfolio C, no one will buy it; demand for that asset will drop and its share price will fall to the extent that it will become undervalued.

Being at an attractive price, the asset will gain interest among investors and will be included in portfolio C.

3. Why Portfolio C is completely diversified?

The benefit of diversification increases as we increase the number of securities in our portfolio,

In a large portfolio of assets, the chances of negative correlation between different assets increases.

As the negative correlation increases, portfolio risk decreases.

By including all risky assets in a portfolio, the portfolio becomes fully diversified.

Earning a (Even) Higher Return

If the investor wants to earn a return higher than that

of portfolio C, he has two options:

1. Invest in portfolio D.

2. Borrow at risk-free rate and invest in Portfolio C.

ExampleReturn on market portfolio = 20%

Risk of market portfolio (Portfolio C) = 14%

Risk-free rate = 8%

Investor’s Equity = Rs. 80000

Investor’s borrowing = Rs. 40000

Suppose, the investor Invests all his equity and the borrowed amount in portfolio C.

Calculate his risk and return on this new portfolio? Is there is a change in the risk and return as compared to the risk

and return before borrowing?

Weight of Risky assets = Investment in C / Investor’s Equity = 120,000 / 80,000 = 1.5

Weight of Risk-free assets = 1 - Weigh of C = 1 - 1.5 = - 0.5

Therefore,

SD of new portfolio = (1-WRf) σ Market

= (1- (- 0.5)14 = 1.5 x 14 = 21%

Return of the new portfolio = Rp = WaRa+WbRb

= -.5 (8) + 1.5 (20)

= -4% + 30%

= 26%

Options Without Borrowing With Borrowing

Risk 14% 21%

Return 20% 26%