This document discusses concepts related to portfolio optimization and the capital asset pricing model (CAPM). It provides information on calculating expected portfolio returns and volatility based on the returns and correlations of individual stocks. It also discusses how adding risk-free assets such as Treasury bills affects the efficient frontier and introduces the capital allocation line. The Sharpe ratio is presented as a measure of portfolio return relative to risk. The separation theorem and using the tangent portfolio to maximize returns for a given risk level are also summarized.

1. Optimal Portfolio Choice

2. 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?

3. Combining Risks
Portfolio Returns
Year Stock Returns
North 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%

5. Year Deviation from Mean
(RN – RˉN) (Rw – Rˉw) (Rt –Rˉt)
North West & West
Air (RN – RˉN) (Rw – Rˉw)
West air & Tex
oil (Rw – Rˉw) (Rt –
Rˉt)
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

6. 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.

7. Interpreting Correlation

8. 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

9. 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?

10. 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?

11. 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:

12. 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?

13. 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%

14. 0% 40%20% 30%10%
0%
25%
20%
15%
10%
5%
60%50%
30%
Coca-Cola
Intel
Inefficient
Portfolios
Efficient
Portfolios
(0 , 1)
(0.2 , 0.8)
(0.4 , 0.6)
(0.6 , 0.4)
(0.8 , 0.2)
(1 , 0)
Volatility (Standard Deviation)
ExpectedReturn

15. 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?

16. Effect of Correlation

17. 0% 40%20% 30%10%
0%
25%
20%
15%
10%
5%
60%50%
30%
Coca-Cola
Intel
Volatility (Standard Deviation)
ExpectedReturn
Correlation = -1
Correlation = +1
Effect of Correlation on Volatility and Expected Return

18. 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

19. 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?

20. Risk-Free Saving and Borrowing
Consider 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)

21. 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).

22. 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 – Free
Investment
Investing in P and the
Risk-free environment
Efficient Frontier of
Risky Investments
Volatility (standard deviation)
ExpectedReturn
Buying Stocks on Margin

23. Margin Investing
Suppose 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?

24. F
ExpectedReturn
D B
σS σm
Changing efficient frontier with risk-free asset
Capital Allocation Line
P
C
A
M
Rs
Rm

25. 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?

26. 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.

27. P
Risk-Free
Investment
Efficient Frontier Including
Risk-Free Investment
Tangent or
Efficient Portfolio
Efficient Frontier of
Risky Investments
Volatility (standard deviation)
ExpectedReturn

28. Sharpe Ratio
The 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.

29. 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.

30. Optimal Portfolio Choice
Your 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.

31. 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.

32. 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