1. Choosing an Investment Portfolio
Chapter 12
Choosing an Investment Portfolio

2. Objectives
To understand the process of personal portfolio selection in theory and in practice
To build a quantitative model of the trade-off between risk and reward

3. Contents The Process of Personal Portfolio Selection
The Trade-Off between Expected Return and Risk
Efficient Diversification with Many Risky Assets

4. Portfolio Selection
A process of trading off risk and expected return to find the best portfolio of assets and liabilities

5. Portfolio Selection
The Life Cycle
Time Horizons
Risk Tolerance

6. The Life Cycle In portfolio selection the best strategy
depends on an individual ‘s personal
circumstances:
Family status
Occupation
Income
Wealth

7. Time Horizons
Planning Horizon: The total length of time for which one plans
Decision Horizon: The length of time between decisions to revise the portfolio
Trading Horizon: The minimum time interval over which investors can revise their portfolios.

8. Risk Tolerance
The characteristic of a person who is more willing than the average person to take on additional risk to achieve a higher expected return

9. The Trade-Off between Expected Return and Risk
Objective:
To find the portfolio that offers
investors the highest expected
rate of return for any degree of
risk they are willing to tolerate

10. Portfolio Optimization
Find the optimal combination of risky assets
Mix this optimal risky-asset portfolio with the riskless asset.

11. Riskless Asset
A security that offers a perfectly predictable rate of return in terms of the unit of account selected for the analysis and the length of the investor’s decision horizon

12. Combining a Riskless Asset and a Single Risky Asset

13. Combining the Riskless Asset and a Single Risky Asset
The expected return of the portfolio is the weighted average of the component returns
mp = W1*m1 + W2*m2
mp = W1*m1 + (1- W1)*m2

14. Combining the Riskless Asset and a Single Risky Asset
The volatility of the portfolio is not quite as simple:
sp = ((W1* s1) W1* s1* W2* s2 + (W2* s2)2)1/2

15. Combining the Riskless Asset and a Single Risky Asset
We know something special about the portfolio, namely that security 2 is riskless, so s2 = 0, and sp becomes:
sp = ((W1* s1)2 + 2W1* s1* W2* 0 + (W2* 0)2)1/2
sp = |W1| * s1

16. Combining the Riskless Asset and a Single Risky Asset
In summary
sp = |W1| * s1, And:
mp = W1*m1 + (1- W1)*rf , So:
If W1<0,
mp = [(rf -m1)/ s1]*sp + rf , Else mp = [(m1-rf )/ s1]*sp + rf

18. 100% Risky 100% Risk-less Long both risky and risk-free
Long risky and short risk-free
100% Risky
Long both risky and risk-free
100% Risk-less

19. To obtain a 20% Return
You settle on a 20% return, and decide not to pursue on the computational issue
Recall: mp = W1*m1 + (1- W1)*rf
Your portfolio: s = 20%, m = 15%, rf = 5%
So: W1 = (mp - rf)/(m1 - rf)
= ( )/( ) = 150%

20. To obtain a 20% Return Assume that you manage a $50,000,000 portfolio
A W1 of 1.5 or 150% means you invest (go long) $75,000,000, and borrow (short) $25,000,000 to finance the difference

21. sp = |W1| * s1 = 1.5 * 0.20 = 0.30 To obtain a 20% Return
How risky is this strategy?
sp = |W1| * s1 = 1.5 * 0.20 = 0.30
The portfolio has a volatility of 30%

22. Portfolio of Two Risky Assets
Recall from statistics, that two random variables, such as two security returns, may be combined to form a new random variable
A reasonable assumption for returns on different securities is the linear model:

23. Equations for Two Shares
The sum of the weights w1 and w2 being 1 is not necessary for the validity of the following equations, for portfolios it happens to be true
The expected return on the portfolio is the sum of its weighted expectations

24. Equations for Two Shares
Ideally, we would like to have a similar result for risk
Later we discover a measure of risk with this property, but for standard deviation:

25. Correlated Common Stock
The next slide shows statistics of two common stock with these statistics:
mean return 1 = 0.15
mean return 2 = 0.10
standard deviation 1 = 0.20
standard deviation 2 = 0.25
correlation of returns = 0.90
initial price 1 = $57.25
Initial price 2 = $72.625

29. Formulae for Minimum Variance Portfolio

30. Formulae for Tangent Portfolio

31. Example: What’s the Best Return given a 10% SD?

32. Achieving the Target Expected Return (2): Weights
Assume that the investment criterion is to generate a 30% return
This is the weight of the risky portfolio on the CML

33. Achieving the Target Expected Return (2):Volatility
Now determine the volatility associated with this portfolio
This is the volatility of the portfolio we seek