1. Lecture 4
Portfolio Tools

2. Portfolio Weights
Financial Markets and Corporate Strategy, David Hillier

3. The Two-Stock Portfolio
A portfolio consists of £1 million in Vodafone equity and £3 million in British Airways equity. What are the portfolio weights of the two equities? Answer: The portfolio has a total value of £4 million. The weight on Vodafone is £1,000,000/£4,000,000  .25 or 25 per cent, and the weight on British Airways is £3,000,000/£4,000,000 = .75 or 75 per cent.
Financial Markets and Corporate Strategy, David Hillier

4. Short Sales and Portfolio Weights
To sell short shares or bonds, the investor must borrow the securities from someone who owns them. This is known as taking a short position in a security. To close out the short position, the investor buys the investment back and returns it to the original owner.
Short Position: Negative Weight
Long Position: Positive Weight
Financial Markets and Corporate Strategy, David Hillier

5. The weights of component assets must sum to one
Feasible Portfolios
The weights of component assets must sum to one
Weights can be negative (short selling or borrowing) or positive (long position)
Financial Markets and Corporate Strategy, David Hillier

6. Computing Portfolio Weights for a Portfolio of many Securities
Describe the weights of a €40,000 portfolio invested in four securities. The amounts invested in each security are as follows:
Financial Markets and Corporate Strategy, David Hillier

7. The Portfolio Weighted Average Method
Portfolio Returns
The Ratio Method
The Portfolio Weighted Average Method
Financial Markets and Corporate Strategy, David Hillier

8. Computing Portfolio Returns for a Two Security Portfolio
A £4,000,000 portfolio consists of £1,000,000 of Vodafone equity and £3,000,000 of British Airways equity. If Vodafone shares have a return of 10 per cent and British Airways shares have a return of 5 per cent, determine the portfolio return using both (1) the ratio method and (2) the portfolio-weighted average method.
Financial Markets and Corporate Strategy, David Hillier

9. The Portfolio Return Formula
Financial Markets and Corporate Strategy, David Hillier

10. Expected Portfolio Returns
The FTSE 100 stock index portfolio had returns of 7.69 per cent in 2006, per cent in 2007, per cent in 2008, and per cent in If the returns between 2006 and 2009 are unbiased estimates of future returns, what is your estimate of the expected return of the FTSE 100?
Financial Markets and Corporate Strategy, David Hillier

11. Expected Return Properties
Financial Markets and Corporate Strategy, David Hillier

12. Expected Portfolio Returns and Arbitrary Weights
The Palisades Quant Fund has a portfolio weight of x in the Eurostoxx 50 index, which has an expected return of 11 per cent. The fund’s investment in Treasury bills, with a portfolio weight of 1 – x, earns 5 per cent. What is the expected return of the portfolio? Answer:
Financial Markets and Corporate Strategy, David Hillier

13. Result 4.1
The expected portfolio return is the portfolio-weighted average of the expected returns of the individual stocks in the portfolio:
Financial Markets and Corporate Strategy, David Hillier

14. Variance
Financial Markets and Corporate Strategy, David Hillier

15. Computing Variances
Compute the variance of the return of a €400,000 investment in the hypothetical company SINTEL. Assume that over the next period SINTEL earns 20 per cent 8/10 of the time, loses 10 per cent 1/10 of the time, and loses 40 per cent 1/10 of the time.
Financial Markets and Corporate Strategy, David Hillier

16. Estimating Variances with Historical Data
Estimate the variance of the return of the FTSE 100. Recall from Example 4.5 that the annual returns of the FTSE 100 from 2006 to 2009 were 7.69 per cent, per cent, per cent, and per cent, respectively, and that the average of these four numbers was 1.40 per cent.
Financial Markets and Corporate Strategy, David Hillier

17. Standard Deviation
Financial Markets and Corporate Strategy, David Hillier

18. Covariance and Correlation
Financial Markets and Corporate Strategy, David Hillier

19. Variance of a two-asset portfolio
Financial Markets and Corporate Strategy, David Hillier

20. Correlation, Diversification and Portfolio Variances
Financial Markets and Corporate Strategy, David Hillier

21. Result 4.2
Given positive portfolio weights on two assets, the lower the correlation, the lower the variance of the portfolio.
Financial Markets and Corporate Strategy, David Hillier

22. Result 4.3
The standard deviation of either (1) a portfolio of two investments where one of the investments is riskless, or (2) a portfolio of two investments that are perfectly positively correlated, is the absolute value of the portfolio-weighted average of the standard deviations of the two investments.
Financial Markets and Corporate Strategy, David Hillier

23. Result 4.4
The formula for the variance of a portfolio return is given by: where ij  covariance between the returns of assets i and j.
Financial Markets and Corporate Strategy, David Hillier

24. Computing the Variance of a Portfolio of Three Assets
Consider three assets on the Irish Stock Exchange, Allied Irish Banks (AIB) plc, CRH plc, and Ryanair Holdings plc, denoted respectively as assets 1, 2, 3. In the first eleven months of 2010, the covariance between the daily returns of AIB and CRH (assets 1 and 2) was ; between CRH and Ryanair (2 and 3), ; and between AIB and Ryanair (1 and 3), The variances of the three equities are respectively , and , and the weights on AIB, CRH and Ryanair are respectively x1 1/3; x2 1/6; and x3 1/2. Calculate the variance of the portfolio.
Financial Markets and Corporate Strategy, David Hillier

25. Variance of a Portfolio
Financial Markets and Corporate Strategy, David Hillier

26. Result 4.5
For any asset, indexed by k, the covariance of the return of a portfolio with the return of asset k is the portfolio-weighted average of the covariances of the returns of the investments in the portfolio with asset k’s return – that is:
Financial Markets and Corporate Strategy, David Hillier

27. The Mean-Standard Deviation Diagram
The mean-standard deviation diagram, plots the means (Y-axis) and the standard deviations (X-axis) of all feasible portfolios in order to develop an understanding of the feasible means and standard deviations that portfolios generate.
Financial Markets and Corporate Strategy, David Hillier

28. Result 4.6
Whenever the portfolio mean and standard deviations are portfolio-weighted averages of the means and standard deviations of two investments, the portfolio mean-standard deviation outcomes are graphed as a straight line connecting the two investments in the mean-standard deviation diagram.
Financial Markets and Corporate Strategy, David Hillier

29. Exhibit 4.3
Financial Markets and Corporate Strategy, David Hillier

30. When Both Portfolio Weights are Positive
Financial Markets and Corporate Strategy, David Hillier

31. When the Risky Investment has a Negative Portfolio Weight
Financial Markets and Corporate Strategy, David Hillier

32. Result 4.7
When investors employ risk-free borrowing (that is, leverage) to increase their holdings in a risky investment, the risk of the portfolio increases.
Financial Markets and Corporate Strategy, David Hillier

33. Portfolios of Two Perfectly Positively Correlated Assets
Financial Markets and Corporate Strategy, David Hillier

34. Forming a Riskless Portfolio with Two Perfectly Negatively Correlated Assets
Over extremely short time intervals, the return of IBM is perfectly negatively correlated with the return of a put option on the company. The standard deviation of the return on IBM equity is 18 per cent per year, while the standard deviation of the return on the option is 54 per cent per year. What portfolio weights on IBM and its put option create a riskless investment over short time intervals?
Financial Markets and Corporate Strategy, David Hillier

35. Portfolios of Two Assets with Arbitrary Correlation
Financial Markets and Corporate Strategy, David Hillier

36. Result 4.8
The covariance of an asset return with the return of a portfolio is proportional to the variance added to the portfolio return when the asset’s portfolio weight is increased by a small amount, keeping the weighting of other assets fixed by financing the additional holdings of the asset with another investment that has zero covariance with the portfolio.
Financial Markets and Corporate Strategy, David Hillier

37. How to Use Covariances Alone to Reduce Portfolio Variance
Vodafone and British Airways shares have respective covariances of .001 and .002 with a portfolio. Vodafone and British Airways are already in the portfolio. Now, change the portfolio’s composition slightly by holding a few more shares of Vodafone and reducing the holding of British Airways by an equivalent sterling amount. Does this increase or decrease the variance of the overall portfolio?
Financial Markets and Corporate Strategy, David Hillier

38. Result 4.9
If the difference between the covariances of the returns of assets A and B with the return of a portfolio is positive, slightly increasing the portfolio’s holding in asset A and reducing the position in asset B by the same amount increases the portfolio return variance. If the difference is negative, the change will decrease the portfolio return variance.
Financial Markets and Corporate Strategy, David Hillier

39. Result 4.10
The portfolio of a group of assets that minimizes return variance is the portfolio with a return that has an equal covariance with every asset return.
Financial Markets and Corporate Strategy, David Hillier

40. Forming a Minimum Variance Portfolio for Asset Allocation
Historically, the return of the S&P 500 Index (S&P) has had a correlation of .8 with the return of the Dimensional Fund Advisors small cap fund, which is a portfolio of small equities that trade mostly on NASDAQ. S&P has a standard deviation of 20 per cent per year – that is, S&P  .2. The DFA small cap fund return has a standard deviation of 39 per cent per year – that is, DFA  .39. What portfolio allocation between these two investments minimizes variance?
Financial Markets and Corporate Strategy, David Hillier

41. Finding the Minimum Variance Portfolio
Suntharee Beers, a Thai firm, wants to branch out internationally. Recognizing that the markets in Japan and Europe are difficult to break into, it is contemplating capital investments to open franchises in India (investment 1), Russia (investment 2) and China (investment 3). Given that Suntharee has a fixed amount of capital to invest in foreign franchising, and recognizing that such investment is risky, Suntharee wants to find the minimum variance investment proportions for these three countries. Solve Suntharee’s portfolio problem. Assume that the returns of the franchise investments in the three countries have covariances as given in the next slide:
Financial Markets and Corporate Strategy, David Hillier

42. Financial Markets and Corporate Strategy, David Hillier

43. Thank You