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```									Ch. 10, Portfolios of Stocks

Risky Assets Only

Econ 134A

Page - 1

Spring ‘05

Previously…
For an individual stock, its risk is measured by the variance (or standard deviation) of its returns. That is, the higher the variance, the higher the risk of a particular stock. This lecture: How do we measure the risk and return of a portfolio of multiple stocks?

Econ 134A

Page - 2

Spring ‘05

Expected return of a two stock portfolio

Rportfolio  XA RA  XB RB
XA = the proportion of Stock A in the portfolio. XB = the proportion of Stock B in the portfolio.
XA + XB = 1.

So, the expected return on a portfolio is a weighted average of the expected returns on the individual stocks.
Econ 134A Page - 3 Spring ‘05

Why is that the case?
Suppose we have two stocks (and to make things easy, two states). XA=75%
RA State 1 State 2 AVG 12% 8% 10% RB 20% 4% 12% Rport 14% 7% 10.5%

Rport  Rport1  prob1  Rport 2  prob2   XARA1  XB1RB   prob1  ...  XARA1  prob1  XBRB1  prob1  ...  XA RA  XB RB
Econ 134A Page - 4 Spring ‘05

From last lecture: RA=17.5%, RB=5.5%
XA 0% 25% 50% 75% 100% 125% XB 100% 75% 50% 25% 0% -25% E[R] 5.5% 8.5% 11.5% 14.5% 17.5% 20.5%

Rportfolio  XA RA  XB RB  25 % 17 .5%  75 %  5.5%  8.5%
Econ 134A Page - 5 Spring ‘05

Variance and SD of two-stock portfolios

 2 portfolio  XA 2A  2 XAXBAB  XB B
2 2 2

2

 2 portfolio  XA 2A  2 XAXBABAB  XB B
2

2

portfolio   2 portfolio
The expected return of the portfolio is a weighted average of the two stock’s returns. Is the same thing true for variance and standard deviation? No—There is a cross term with the covariance (or correlation coefficient).
Econ 134A Page - 6 Spring ‘05

Variance and SD example:
Recall: 2A=668.75, 2B=132.25, A,B= -48.75 XA 0% 25% 50% XB 100% 75% 50% 2portfolio 132.25 97.91 175.86 portfolio 11.5 9.89 13.26

75% 100%

25% 0%

336.16 668.75

19.14 25.86

 2 por tfolio  .252  2 A  2  .25  .75   2 AB  .752  2 B
 .252 668.75  .375 48.75  .752 132.25  97.91
Econ 134A Page - 7 Spring ‘05

Summary of results
Average return is the correct measure of return for the portfolio, while the standard deviation is the correct measure of risk for a portfolio. XA 0% XB 100% E[R] 5.5 SD 11.5

25% 50% 75% 100%

75% 50% 25% 0%

8.5 11.5 14.5 17.5

9.89 13.26 19.14 25.86

Econ 134A

Page - 8

Spring ‘05

Graphing;

Econ 134A

Page - 9

Spring ‘05

Diversification Effects
(Or, which stock to pick, finally!) If you were strongly risk averse, you might think you would be best off selecting the less risky asset, Stock B. However, you can make yourself unambiguously better off by adding some of the more risky asset, Stock A, to your portfolio (until you reach the Minimum Variance portfolio) Why?
Econ 134A Page - 10 Spring ‘05

Efficient Frontier and MV portfolio

Econ 134A

Page - 11

Spring ‘05

Definitions from the previous graph


MV portfolio:
– The point on the feasible set with the lowest variance (or standard deviation)



Feasible Set (or Opportunity set):
– The points on the standard deviation-expected return plane that are attainable



Efficient Frontier:
– Those points in the feasible set that are not dominated.

Econ 134A

Page - 12

Spring ‘05

Diversification Effect Take II

The SD of the portfolio is less than the weighted average of the SD of the individual securities.

This happens whenever the correlation between the securities is less than one.
Econ 134A Page - 13 Spring ‘05

Diversification Effect for various ’s

Econ 134A

Page - 14

Spring ‘05

Multiple security case:
Expected return of a portfolio is just the weighted average of the securities in the portfolio (regardless of number) Variance is not! Pretty complicated. We’ll do an example of how to calculate it on the next page. Another result: variance of the return of the portfolio is more dependent on the covariance between the securities than on the variances

Econ 134A

Page - 15

Spring ‘05

Variance of multiple stock portfolios
Stock A Stock A Var(A) Stock B Stock C Stock D Stock E

Stock B
Stock C Stock D Stock E

Cov(BA) Var(B)

Cov(BC) Cov(BD) Cov(BE)
Cov(CD) Cov(CE) Cov(DE)

Cov(CA) Cov(CB) Var(C)

Cov(DA) Cov(DB) Cov(DC) Var(D)

Cov(EA) Cov(EB) Cov(EC) Cov(ED) Var(E)

1 Each entry is weighted by Var( port )  n Var  n 2  n  Cov  2 n the share of each stock in 1  1 the portfolio and then  Var  1  Cov n  n summed. If we let all as n  , variances and covariances  Cov be equal and have an equal weighting of stocks, we get:
Econ 134A Page - 16 Spring ‘05

Multiple Stock Graph
Same idea, but now the entire shaded region is feasible. There still will be a MV portfolio, and the efficient frontier is the upper edge of the region

Econ 134A

Page - 17

Spring ‘05

between the forward rate and the spot rate (forward premium or discount) is approximately equal to the difference in interest rates between the two countries. Arbitrage in the forward and spot markets helps to hold this relationship in place.

Links changes in exchange rates with differences in inflation rates and the purchasing power of each nation’s currency.  In the long run, exchange rates adjust so that the purchasing power of each currency tends to be the same.  Exchange rate changes tend to reflect international differences in inflation rates.  Countries with high inflation tend to experience currency devaluation.

The Law of One Price
In competitive markets where there are no transportation costs or barriers to trade, the same goods sold in different countries sell for the same price if all the different prices are expressed in terms of the same currency.  This proposition underlies the PPP relationship.  Arbitrage allows the law of one price to hold for commodities that can be shipped to other countries and resold.

Exchange Rate Risk
Translation exposure - foreign currency assets and liabilities that, for accounting purposes, are translated into domestic currency using the exchange rate, are exposed to exchange rate risk.  However, if markets are efficient, investors know that any translation losses are “paper” losses and are unrealized.


Exchange Rate Risk
exposure - refers to transactions in which the monetary value is fixed before the transaction actually takes place.  Ex: your firm buys foreign goods to be received and paid for at a later date. The exchange rate can change, which can affect the price actually paid.
 Transaction

Multinational Working-Capital Management
Leading and Lagging  Lead: dispose of a net asset position in a weak currency.  Pay a net liability position in a weak currency.  Lag: Delay collection of a net asset position in a strong currency. Delay payment of a net liability position in a weak currency.

Direct Foreign Investment
Risks  Business Risk - firms must be aware of the business climate in both the US and the foreign country.  Financial Risk - not much difference between financial risks of foreign operations and those of domestic operations.

Direct Foreign Investment
Risks  Political Risk - firms must be aware that many foreign governments are not as stable as the U.S.  Exchange Rate Risk - exchange rate changes can affect sales, costs of goods sold, etc. as well as the firm’s profit in dollars.

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