Optimal Portfolio Choice

Portfolio Expected Return

Suppose you invest Rs.10,000 in Ford and

Rs.30,000 in Johnson and Johnson stock. You

expect a return of 10% for Ford and 16% for J &

J. What is the expected return for your portfolio?

Combining Risks Portfolio Returns

Year Stock ReturnsNorth Air West Air Tex oil

½ RN + ½ RW ½ RW + ½ RT

1998 21% 9% -2% 15% 3.5%

1999 30% 21% -5% 25.5% 8.0%

2000 7% 7% 9% 7.0% 8.0%

2001 -5% -2% 21% -3.5% 9.5%

2002 -2% -5% 30% -3.5% 12.5%

2003 9% 30% 7% 19.5% 18.5%

Average Returns 10.0% 10.0% 10.0% 10.0% 10.0%

Volatility 13.4% 13.4% 13.4% 12.1% 5.1%

Year Deviation from Mean(RN – RˉN) (Rw – R wˉ ) (Rt –Rˉt)

North West & West Air (RN – R N) (Rw – R w) ˉ ˉ

West air & Tex oil (Rw – R w) (Rt –ˉR t)ˉ

1998 11% -1% -12% -0.0011 0.0012

1999 20% 11% -15% 0.0220 -0.0165

2000 -3% -3% -1% 0.0009 0.0003

2001 -15% -12% 11% 0.0180 -0.0132

2002 -12% -15% 20% 0.0180 -0.0300

2003 -1% 20% -3% -0.0020 -0.0060

Covariance 0.0112 -0.0128

Correlation 0.624 -0.713

Computing Covariance and Correlation between Pairs of Stocks

Interpreting Covariance

• Positive covariance signifies the fact that the

stocks move together in the same direction.

• If the stocks move in opposite directions, the

covariance is negative.

• Correlation quantifies the relationship between

the stocks and is between +1 and -1.

Interpreting Correlation

Historical Annual Volatilities and Correlation (based on monthly returns)

Microsoft Dell Delta Air Lines

American Airline

General Motors

Ford Motors

Anheuser - Busch

Volatility 42% 54% 50% 72% 33% 37% 18%

Correlation with

Microsoft 1.00 0.65 0.27 0.19 0.22 0.26 -0.07

Dell 0.65 1.00 0.19 0.18 0.32 0.32 0.10

Delta Air 0.27 0.19 1.00 0.69 0.31 0.38 0.19

American 0.19 0.18 0.69 1.00 0.35 0.58 0.11

General Motors

0.22 0.32 0.31 0.35 1.00 0.64 0.11

Ford 0.26 0.32 0.38 0.58 0.64 1.00 0.10

Anheuser - Busch

-0.07 0.10 0.19 0.11 0.11 0.10 1.00

Calculating the Portfolio returns and Portfolio Volatility

• Let’s say you have 2 stocks: I-flex and HCL. Assume that I-flex’s average return over the last 5 years has been 20% per year and that of HCL has been 25%. Also assume that the standard deviations of those returns were 30% and 40% respectively.

• If the correlation coefficient for these two stocks is 0.8, what would be the standard deviation of a portfolio invested 40% in I-fled and 60% in HCL?

• If the correlation coefficient were 0.5 instead, would the portfolio standard deviation be greater than or less than in (a)? Why?

Calculation of Expected Return and Portfolio Volatility

The common stocks of Bajaj and Colgate have expected

returns of 15% and 20% respectively, while their deviations

from the expected returns are 20% and 40%. The expected

correlation coefficient between the stocks is 0.36.1. What is the expected value of return and the portfolio volatility of a portfolio consisting of (a) 40% Bajaj and 60% Colgate? (b) 40% Colgate and 60% Bajaj? 2.How should the correlation coefficient move to bring the portfolio risk still lower?

Efficient Portfolio with Two Stocks

Consider Intel Corporation and the Coca-Cola

Company. The returns of these two companies

were uncorrelated.

Suppose an investor wants to invest in these

two portfolios for which the details are given as

under:

Choosing a Portfolio

Correlation with

Stock Expected Return

Volatility Intel Coca-Cola

Intel 26% 50% 1.0 0.0

Coca-cola 6% 25% 0.0 1.0

How should the investor choose a portfolio of these two stocks? Calculate a Portfolio for its returns and volatility if you invest 40% in Intel and 60% in Coca-Cola?

Expected Returns and Volatility for Different Portfolios of Two Stocks

Portfolio Weights Expected Return(%) Volatility (%)

X(Intel) X(Coca-cola) Expected return of Portfolio SD [R p]

1.00 0.00 26% 50%

0.80 0.20 22% 40.3%

0.60 0.40 18% 31.6%

0.40 0.60 14% 25%

0.20 0.80 10% 22.3%

0.00 1.00 6% 25%

0% 40%20% 30%10%

0%

25%

20%

15%

10%

5%

60%50%

30%

Coca-Cola

Intel

InefficientPortfolios

EfficientPortfolios

(0 , 1)

(0.2 , 0.8)

(0.4 , 0.6)

(0.6 , 0.4)

(0.8 , 0.2)

(1 , 0)

Volatility (Standard Deviation)

Expe

cted

Ret

urn

Improving Returns with an Efficient Portfolio

Sally has invested 100% of her money in Coca-

Cola and is seeking investment advice. She

would like to earn the highest expected return

possible without increasing her volatility. Which

portfolio would you recommend?

Effect of Correlation

0% 40%20% 30%10%

0%

25%

20%

15%

10%

5%

60%50%

30%

Coca-Cola

Intel

Volatility (Standard Deviation)

Expe

cted

Ret

urn

Correlation = -1

Correlation = +1

Effect of Correlation on Volatility and Expected Return

Suppose you have Rs 20,000 in cash to invest. You decided to short sell Rs 10,000 worth of Coca-Cola stock and invest the proceeds from your short sale, plus your Rs 20,000 in Intel. At the end of the year, you decide to liquidate your portfolio. If the two stocks have the following realized returns, what is the return on your portfolio?

PO Div1 + P1 Return

Intel 25.00 31.50 26%Coca - Cola 40.00 42.40 6%

Calculation of Returns from Short Selling

Volatility with Short Sales

Suppose Intel stock has a volatility of 50%, Coca-

Cola stock has a volatility of 25% and the stocks

are uncorrelated.

• What is the volatility of a portfolio that is short

Rs.10,000 of Coca-Cola and long Rs.30,000 of

Intel?

Risk-Free Saving and BorrowingConsider an arbitrary risky portfolio with returns Rp. Lets look at the effect on risk and return of putting a fraction x of our money in the portfolio, while leaving the remaining fraction (1-x) in risk-free Treasury bills with a yield of rf .

E[Rxp] = rf + x(E[Rp] – rf)

Volatility of the Portfolio

• Next let’s compute the volatility. The volatility

of the risk-free investment is zero.

• Therefore the volatility of this investment is

only equal to the fraction of the volatility of

the portfolio. That is it is equal to xSD(Rp).

25%

20%

15%

10%

5%

0%0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20%

P Buying P on Margin

X = 200%

X = 150%

X = 100%

X = 50%

Risk – FreeInvestment

Investing in P and the Risk-free environment

Efficient Frontier of Risky Investments

Volatility (standard deviation)

Expe

cted

Ret

urn

Buying Stocks on Margin

Margin InvestingSuppose you have Rs.10,000 in cash and you decide to borrow another Rs.10,000 at a 5% interest rate to invest in the stock market. You invest the entire Rs.20,000 in Portfolio Q with a 10% expected return and a 20% volatility. 1.What is the expected return and volatility of your investment? 2.What is your realized return if Q goes up 30% over the course of the year?3. What return do you realize if Q falls by 10% over the course of the year?

F

Expe

cted

Ret

urn

D B

σS σm

Changing efficient frontier with risk-free asset Capital Allocation Line

P

C

A

M

Rs

Rm

Capital Allocation Line Mr.Xing is considering an investment of Rs.4,00,000. He wants to allocate a part of the funds to a portfolio of risk-free securities providing a return of 8% and the remaining part in a share portfolio, that has an expected return and the standard deviation of 24% each.a.If Mr.Xing is prepared to assume a maximum risk of 15%, what is the best return he can hope for?b.In case Mr.Xing needs a return of 32%, what minimum risk would he have to assume?

Identifying the Tangent Portfolio

• By forming a portfolio out of risk- free asset and a portfolio which is effectively on the efficient portfolio at a level higher than the Portfolio P, we get a line tangent to the efficient portfolio which is steeper than the line that is through P.

• If the line is steeper, then for any level of volatility, we will earn a higher expected return.

P

Risk-FreeInvestment

Efficient Frontier IncludingRisk-Free Investment

Tangent orEfficient Portfolio

Efficient Frontier ofRisky Investments

Volatility (standard deviation)

Expe

cted

Ret

urn

Sharpe RatioThe slope of the return through a given portfolio P is often referred to as the Sharpe ratio of the portfolio. Sharpe Ratio = Portfolio Excess Return Portfolio Volatility = E[Rp] – rf

σ p

The Sharpe ratio measures the ratio of reward-to-volatility provided by a portfolio.

Separation Theorem• The portfolio managers offer the same set of

assets on the efficient portfolio to all their clients.

• Generally the don’t conduct separate research for identifying suitable portfolios for each individual client, thereby saving cost, time and effort.

• To accommodate individual preferences of the clients all they need to do is to adjust the proportion of investment in the two assets.

Optimal Portfolio ChoiceYour uncle calls and asks for investment advice. Currently, he has Rs.1,00,000 invested in a portfolio P as shown in the above graph. This portfolio has an expected return of 10.5% and a volatility of 8%. Suppose the risk-free rate is 5% and the tangent portfolio has an expected return of 18.5% and a volatility of 13%.a.Recommend a portfolio to maximize his return without increasing the risk.b.Recommend a portfolio to minimize his risk but maintaining the same expected return.

Cost of Capital for Investment i

• The cost of capital of investment (i) is equal to

the expected return of the best available

portfolio in the market with the same

sensitivity to systematic risk.

• And the best available portfolio is the Tangent

Portfolio or the portfolio that has the highest

Sharpe ratio of any portfolio in the economy.

Cost of Capital for Investment i

• Because all other risk is diversifiable, it is an investment’s beta with respect to the efficient portfolio that measures its sensitivity to systematic risk and therefore its cost of capital.

rі = rf + β