Homework. Due Monday 11/2/2009 at beginning of class Chapter 6: 2 Additional Homework: Download data for

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1 Data Code Go to User: SUNYSB Password: STONYBROOK11794 Download software for WorldwatchInsight and Marketlink and corresponding manuals Login using your personal login name and password

2 Homework Due Monday 11//009 at beginning of class Chapter 6: Additional Homework: Download data for stocks on the Yahoo!Financeand compute performance statistics. Each person will receive an with the stocks and the performance characteristics Wednesday 11/4/009: XuDong substituting and projects due by midnight

3 Mean-Variance Portfolio Theory Typical investments have uncertain returns Single Period Investing: Money invested at the initial time and payoff attained at period end, e.g. zero coupon bond Ways of treating investment uncertainty: Mean-Variance Analysis, Utility Function Analysis, Arbitrage or Comparison Analysis Mean-Variance uses Probability Theory Leads to CAPM

4 Asset Return When Buying Asset: Investment instrument that may be bought and sold If you buy an asset at time t 0 for amount X 0 and sell it at time t 1 for X 0, the rate of return, r, for the asset is r X X X 0 X X 0 1

5 Example: Google Stock You buy Google at the closing price on 1/31/08 and sell it at the closing price on 9/30/09. What is your rate of return? 1/31/08 closing price: $ /30/09 closing price: $ r $ $ $ $ %

6 Asset Returning When Shorting Shorting: The act of selling an asset that you do not own Arrange to borrow the asset from someone who owns it and then sell it with the intent of buying it at a later date, presumably at a lower price, and repaying the lender. Only profitable if the asset declines in price Theoretically losses are unlimited risk of shorting higher than risk of going long

7 Examples of Shorting Speculation: Betting on an asset declining As a hedge: Remove undesirable exposures. A copper producer wants to lock in the price of copper that is being produced for the year; copper futures can be sold now, guaranteeing a price. Statistical arbitrage: Only way to neutralize unwanted factors in a portfolio

8 Assumptions for Shorting In reality, shorting has a cost, a borrow rate for theoretical work assume zero Borrow rate is usually tied to lending rates Stocks and bonds for shorting may be sourced by your broker Return computation is the same as for buying, but treat the initial outlay as negative r X 1 X 0 X 0

9 Profits on Shorting Assume you shorted 1 share of Google instead of buying $ and buy it back at $ Since the price of Google went up, lost $188. r $ $ $ % $ *.6117 $188.

10 Portfolio Return Suppose nassets are available for a portfolio If we have a total amount Xto invest in n assets such that n Xi i 1 For i 1,, we invest an amount X i in the ith asset Each asset has a weight in the portfolio, w i, which is the fraction of the total investment st n i 1 w i 1

11 Portfolio Return w i > 0 is a long, w i <0 is a short Let R i denote the total return of asset i Total amount of money returned by the portfolio at the end of the period is R n RwX i i i 1 X n i 1 wr i i

12 Portfolio Return Both the total return and the rate of return of a portfolio of assets are equal to the weighted sum of the corresponding individual asset returns, with the weight of an asset being its relative weight (in purchase cost) in the portfolio R n wr r i i i 1 i 1 n w i r i

13 Portfolio Return Example Compute the total return and rate of return of the following portfolio purchased on 9/30/09 and sold on 10/3/09: 10 shares AAPL, 100 shares GE, 10 shares IBM Compute the return of the portfolio if you are short 10 shares of GE

14 Random Variable Future asset prices are generally uncertain at the time of purchase Describe asset prices as random variables with a probability distribution provides some way to put Important Values: Expected Return, Variance, Probability Density Distribution, Covariance Stock prices and returns

15 Random Variables Suppose xis a random quantity that can take on any one of a finite number of specific values, x 1, x,..., x m Assume that with each possible x i, there is a probability p i that represents the relative chance of an occurrence of x i. The p i s satisfy n i 1 p i 1 and x quantified in this way is a random variable p i 0 for each i

16 Example: Roll of a Die What are the possible outcomes of roll of a die? What are the probabilities associated with each outcome? Show that the number of dots on the face of a die when it is rolled is a random variable

17 Expected Value The expected value or mean of a random variable x is the average value obtained by regarding the probabilities as frequencies E( x) n i1 xi pi x

18 Properties of Expected Value Certain Value: If y is a known value (not random), then E(y) y Linearity: If yand zare random, then E( ay+ bz) ae( y) + for any real values of be( z) a andb Nonnegativity: If x is random and greater than 0 E( x) 0

19 Variance A measure of the degree of possible deviation from the mean, usually denoted σ y ] ) [( ) var( y y E y y σ is called the standard deviation and is another measure of how variable yis from its mean ] ) ( y y E y σ σ y ] [ ] ) [( ) var( x y E y y E y

20 Example: Roll of a Die What is the expected value of a roll of a die? What is the variance of a roll of die? What is the standard deviation of a roll of a die? What do these numbers mean?

21 Several Random Variable Two random variables: To describe them, must have probabilities for all combinations of the two values that are possible, and assign probabilities for each combination Two die that are rolled simultaneously Price of the S&P 500 and IBM Special Case: Independent random variables the outcome of one does not depend on other

22 Covariance x Let x 1 and be two random variables with expected values x1 and x The covariance of these two r.v. sis defined to be cov( x cov( x 1 1,, x x ) ) E[( x1 x1 )( x x)] σ1, E( x 1 x ) x 1 x The covariance of two r.v. sdescribes their mutual dependence

23 Covariance and Correlation If two r.v. sx 1 and x have covariance 0, they are independent If two r.v. sx 1 and x have positive covariance, they are positively correlated If two r.v. s have negative covariance, they are negatively correlated

24 Yuan-EUR and Yuan-YEN CNY/YEN CNY/EUR Area of Interest

25 Correlation Coefficient Covariance Bound: the covariance of two random variables satisfies σ σ 1 1σ Correlation Coefficient: A useful summary statistic σ ρ σσ ρ 1 1

26 Variance of a Sum The variance of a sum of r.v. sxand Y Var ( x+ y) E[( x x+ y y) ] σ x + σ σ xy+ y If X and Y are independent Var ( x+ y) σ σ x + y