corporate finance -session_5

Risk and ReturnCont…

Session 5

Portfolios

Rate of ReturnScenario Stock fund Bond fund Portfolio squared deviationRecession -7% 17% 5.0% 0.0016Normal 12% 7% 9.5% 0.0000Boom 28% -3% 12.5% 0.0012

Expected return 11.00% 7.00% 9.0%Variance 0.0205 0.0067 0.0010Standard Deviation 14.31% 8.16% 3.08%

The rate of return on the portfolio is a weighted average of the returns on the stocks and bonds in the portfolio:

SSBBP rwrwr

%)17(%50%)7(%50%5



Portfolios

The expected rate of return on the portfolio is a weighted average of the expected returns on the securities in the portfolio.

%)7(%50%)11(%50%9 )()()( SSBBP rEwrEwrE



Portfolios

The variance of the rate of return on the two risky assets portfolio is

BSSSBB2

SS2

BB2P )ρσ)(wσ2(w)σ(w)σ(wσ

where BS is the correlation coefficient between the returns on the stock and bond funds.



Portfolios

Observe the decrease in risk that diversification offers.

An equally weighted portfolio (50% in stocks and 50% in bonds) has less risk than either stocks or bonds held in isolation.

As long at the correlation coefficient is < 1, the standard deviation of a portfolio of two securities is less than the weighted average of the standard deviations of the individual securities.

In the above case: SD of portfolio= 3.08%Weighted average of SD = 14.31%*0.5 +

0.0816*0.5= 0.07155 + 0.0408 = 0.11235 = 11.235%This difference is due to the negative

correlation between the two securities.

Portfolo Risk and Return Combinations

5.0%

6.0%

7.0%

8.0%

9.0%

10.0%

11.0%

12.0%

0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%

Portfolio Risk (standard deviation)

Portf

olio

Ret

urn

% in stocks Risk Return0% 8.2% 7.0%5% 7.0% 7.2%10% 5.9% 7.4%15% 4.8% 7.6%20% 3.7% 7.8%25% 2.6% 8.0%30% 1.4% 8.2%35% 0.4% 8.4%40% 0.9% 8.6%45% 2.0% 8.8%

50.00% 3.08% 9.00%55% 4.2% 9.2%60% 5.3% 9.4%65% 6.4% 9.6%70% 7.6% 9.8%75% 8.7% 10.0%80% 9.8% 10.2%85% 10.9% 10.4%90% 12.1% 10.6%95% 13.2% 10.8%

100% 14.3% 11.0%

The Efficient Set for Two Assets

We can consider other portfolio weights besides 50% in stocks and 50% in bonds …

100% bonds

100% stocks

Portfolo Risk and Return Combinations

5.0%

6.0%

7.0%

8.0%

9.0%

10.0%

11.0%

12.0%

0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%

Portfolio Risk (standard deviation)Po

rtfol

io R

etur

n

% in stocks Risk Return0% 8.2% 7.0%5% 7.0% 7.2%10% 5.9% 7.4%15% 4.8% 7.6%20% 3.7% 7.8%25% 2.6% 8.0%30% 1.4% 8.2%35% 0.4% 8.4%40% 0.9% 8.6%45% 2.0% 8.8%50% 3.1% 9.0%55% 4.2% 9.2%60% 5.3% 9.4%65% 6.4% 9.6%70% 7.6% 9.8%75% 8.7% 10.0%80% 9.8% 10.2%85% 10.9% 10.4%90% 12.1% 10.6%95% 13.2% 10.8%

100% 14.3% 11.0%

The Efficient Set for Two Assets

100% stocks

100% bonds

Note that some portfolios are “better” than others. They have higher returns for the same level of risk or less.

Turn to page 349 of your books. Figure 10.3The point MV is called the Minimum Variance

portfolio

Portfolios with Various Correlations

Relationship depends on correlation coefficient-1.0 < r < +1.0

If r = +1.0, no risk reduction is possibleIf r = –1.0, complete risk reduction is possible

100% bonds

retu

rn

100% stocks

= 0.2

= 1.0

= -1.0Since any probable correlation of securities X and Y will range between – 1.0 and + 1.0, the triangle in the above figure specifies the limits to diversification. The risk-return curves for any correlations within the limits of – 1.0 and + 1.0, will fall within the triangle.

Portfolio Risk Depends onCorrelation between Assets

12

Investing wealth in more than one security reduces portfolio risk.

This is attributed to diversification effect. However, the extent of the benefits of portfolio

diversification depends on the correlation between returns on securities.

When correlation coefficient of the returns on individual securities is perfectly positive then there is no advantage of diversification. The weighted standard deviation of returns on individual securities is equal to the standard deviation of the portfolio.

Diversification always reduces risk provided the correlation coefficient is less than 1.

The Efficient Set for Many Securities

Consider a world with many risky assets; we can still identify the opportunity set of risk-return combinations of various portfolios.

retu

rn

P

Individual Assets

The Efficient Set for Many Securities

The section of the opportunity set above the minimum variance portfolio is the efficient frontier.

retu

rn

P

minimum variance portfolio

efficient frontier

Individual Assets

Investment Opportunity Set:The n-Asset Case

15

An efficient portfolio is one that has the highest expected returns for a given level of risk.

The efficient frontier is the frontier formed by the set of efficient portfolios.

All other portfolios, which lie outside the efficient frontier, are inefficient portfolios.

Efficient Portfolios of risky securities

16

An efficient portfolio is one that has the highest expected returns for a given level of risk. The efficient frontier is the frontier formed by the set of efficient portfolios. All other portfolios, which lie outside the efficient frontier, are inefficient portfolios.

Diversification and Portfolio Risk

Diversification can substantially reduce the variability of returns without an equivalent reduction in expected returns.

This reduction in risk arises because worse than expected returns from one asset are offset by better than expected returns from another.

However, there is a minimum level of risk that cannot be diversified away, and that is the systematic portion.

RISK DIVERSIFICATION:SYSTEMATIC AND UNSYSTEMATIC RISK

18

When more and more securities are included in a portfolio, the risk of individual securities in the portfolio is reduced.

This risk totally vanishes when the number of securities is very large.

But the risk represented by covariance remains.

Risk has two parts: 1. Diversifiable (unsystematic) 2. Non-diversifiable (systematic)

Systematic Risk19

Systematic risk arises on account of the economy-wide uncertainties and the tendency of individual securities to move together with changes in the market.

This part of risk cannot be reduced through diversification.

It is also known as market risk. Investors are exposed to market risk even when

they hold well-diversified portfolios of securities.Risk factors that affect a large number of assetsIncludes such things as changes in GDP, inflation,

interest rates, etc.

Examples of Systematic Risk20

Unsystematic Risk21

Unsystematic risk arises from the unique uncertainties of individual securities.

It is also called unique risk. These uncertainties are diversifiable if a

large numbers of securities are combined to form well-diversified portfolios.

Uncertainties of individual securities in a portfolio cancel out each other.

Unsystematic risk can be totally reduced through diversification.

Examples of Unsystematic Risk22

Total Risk23

Hence…Total Risk

Total risk = systematic risk + unsystematic risk

The standard deviation of returns is a measure of total risk.

For well-diversified portfolios, unsystematic risk is very small.

Consequently, the total risk for a diversified portfolio is essentially equivalent to the systematic risk.

Since the systematic risk can’t be diversified, the investor will require compensation for bearing this risk.

Diversified portfolios with no unsystematic risk, move with the market

Portfolio Risk and Number of Stocks, p355

Non diversifiable risk; Systematic Risk; Market Risk; COVAR

Diversifiable Risk; Nonsystematic Risk; Firm Specific Risk; Unique Risk, VARIANCE

n

In a large portfolio the variance terms are effectively diversified away, but the covariance terms are not.

Portfolio risk

Optimal Portfolio with a Risk-Free Asset

In addition to stocks and bonds, consider a world that also has risk-free securities like T-bills.

100% bonds

100% stocks

rf

retu

rn

Riskless Borrowing and Lending

Now investors can allocate their money across the T-bills and a balanced mutual fund.

100% bonds

100% stocks

rf

retu

rn

Balanced fund

CML

corporate finance -session_5

Economy & Finance