1. Chapter 12: Choosing an Investment Portfolio
Objective
To understand the theory of personal
portfolio selection in theory
and in practice

2. Chapter 12: Contents The process of personal portfolio selection
The trade-off between expected return and risk
Efficient diversification with many risky assets

3. The Concept of ‘Portfolio’
A person’s wealth portfolio includes
Assets: stocks, bonds, shares in unincorporated business, houses or apartments, pensions benefits, insurance policies, etc.
Liabilities: student loans, auto loans, home mortgages, etc.

4. Portfolio Selection
A study of how people should invest their wealth optimally
A process of trading off risk and expected return to find the best portfolio of assets and liabilities
Narrow and broad definitions:
How much to invest in stocks, bonds, and other securities
Whether to buy or rent one’s house
What types and amounts of insurance to purchase
How to manage one’s liabilities
How much to invest in one’s human capital

5. Portfolio Selection
Although there are some general rules for portfolio selection that apply to virtually everyone, there is no single portfolio or portfolio strategy that is best for everyone.

6. The Life Cycle
In portfolio selection, the best strategy depends on an individual’s personal circumstances (family status, occupation, income, wealth).
Illustrations
Young couple: buy a house and take out a mortagage loan / older couple: sell house and invest in assets provding a steady stream of income.
Investing in stock market: Chang (30, a security analyst) / Obi (30, an English teacher).
Buying insurance policies: Miriam (a parent with dependent children) / Sanjiv (a single person with no dependents).

7. Time Horizon
In formulating a plan for portfolio selection, you begin by determining your goals and 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 / its determination and impacts
Investment strategy & trading horizon: portfolio insurance or dynamic portfolio strategy.

8. Risk Tolerance A major determinant of portfolio choices
It is influenced by such characteristics as
age, family status, job status, wealth, and
other attributes that affect a person’s ability to maintain his standard of living in the face of adverse movements in the market value of his investment portfolio

9. Professional Asset Managers
Investment advisors & “finished products” from a financial intermediary
Specialization, information and cost advantages

10. The Trade-off between Expected Return and Risk
The objective is to find the portfolio which offers investors the highest expected rate of return for the degree of risk they are willing to tolerate.
Two step process:
find the optimal combination of risky assets.
mix this optimal risk-asset 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.
For example, if the U.S dollars is taken as the unit of account and the decision horizon is half a year, the riskless rate is the interest rate on U.S Treasury bills maturing after half a year.

12. Rates of Return on Risky Assets
Required return depends on the risk of the investment.
Greater the risk, greater the return
Risk premium

17. Measuring Portfolio Return
Portfolio of n risky assets
Ii : the initial investment in asset i (if Ii <0, short selling)
wi: the proportion of the portfolio investing in asset I
ri : the rate of return on asset I
rp: the rate of return on the portfolio

18. Short Selling
Ik < 0 : short selling (borrowing) asset k

19. Mean and Variance of Portfolio Return
: the expected value of ri
: the standard deviation of ri
: the correlation between ri and rj

20. Variance with 2 Securities

21. An Example: A Portfolio of BM and FM
Suppose you invest $6000 in Bristol-Myers at an expected return of 15%, and $4000 in Ford Motor at an expected return of 21%.
The standard deviation of the return on BM’s stock is 18.6%, while the standard deviation of the return on FM is 28%.
The correlation between the returns is 0.4.

22. Portfolios of BM and FM Expected Return (%) Ford Motor 40% F M 60% BM
Bristol-Myers
Expected Return (%) ●
Ford Motor
40% F M
60% BM
Standard Deviation (%)

23. Portfolios of Two Correlated Common Stock
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

25. Is one “better”? Portfolio of Two Securities Security 1 Security 2
0.00
0.05
0.10
0.15
0.20
0.25
0.17
0.19
0.21
0.23
0.27
0.29
Standard Deviation
Expected Return
Efficient Portfolio
Is one “better”?
Minimum Variance Portfolio
Sub-optimal Portfolio

26. Formula for Minimum Variance Portfolio

27. Portfolio Selection with n Risky Assets
s.t.
Harry Markowitz (1952): Portfolio Selection, Journal of Finance 10

28. Solution：

29. where 10

30. minimum-variance portfolio
Portfolio of many risky assets
Efficient frontier: the set of portfolios offering the highest expected return for any given standard deviation.
Standard Deviation (%)
Expected Return (%)
efficient frontier
minimum-variance portfolio 10

31. Combining the Riskless Asset and a Single Risky Asset: An illustration
Let’s suppose that you have $100,000 to invest.
You are choosing between a riskless asset with a interest of 6% per year and a risky asset with an expected rate of return of 14% per year and a standard deviation of 20%.
How much of your $100,000 should you invest in the risky asset?

32. Mean and Standard Deviation

33. The Risk-Return Trade-off Line
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.05
0.15
0.2
0.25
0.3
Standard Deviation
Expected Return S J H F G R
inefficient

34. Combining the Riskless Asset and a Single Risky Asset
We know something special about the portfolio, namely that security 2 is riskless, so σ2 = 0, and σp becomes
where

36. CML Long risky and short risk-free Long both risky and risk-free
100% Risk-less

37. Risk Premium
Sharpe Ratio
The slope measure the extra expected return the market offers for each extra risk a investor is willing to bear

38. Achieving a Target Expected Return
To find the portfolio corresponding to an expected rate of return of 0.11 per year, we substitute 0.11 for E(rp) and solve for w1.
Thus, the portfolio mix is 62.5% risky asset and 37.5% riskless asset.

39. Portfolios of the Riskless Security and Two Risky Securities
The riskless security and two risky securities with the following statistics:
riskless rate of return rf = 0.06
mean return 1 = 0.14
mean return 2 = 0.08
standard deviation 1 = 0.20
standard deviation 2 = 0.15
correlation of returns = 0

40. The Optimal Combination of the Three Securities
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.05
0.15
0.2
0.25
0.3
Standard Deviation
Expected Return S R T E ◆
Tangent Portfolio

41. Formula for Tangent Portfolio
12154 . ) ( E = T r
14595 s

42. Efficient Trade-off Line
New efficient trade-off line:
Compare the old trade-off line connecting points F and S.
Clearly the investor is better off.

43. Achieving a Target Expected Return
The investment criterion is to generate a 10% expected rate of return.
Thus, the portfolio mix is 35% riskless asset and 65% tangent portfolio, namely 45% risky security 1 and 20% risky security 2.

44. Selecting the Preferred Portfolio
It is important to note that in finding the optimal combination of risky assets, we do not need to know anything about investor preferences.
There is always a particular optimal portfolio of risky assets that all risk-averse investors who share the same forecasts of rates of return will combine with the riskless asset to reach their most-preferred portfolio.

45. The Rationale for Portfolio Selection
Return
Risk
Low Risk
High Return
High Risk
Low Return

46. Portfolio of many risky assets and the riskless asset
Standard Deviation (%)
Expected Return (%)
Short sell rf
Efficient frontier
Tangent Portfolio 10

47. Two-Fund Separation Theorem (Tobin, 1958)
Efficient Frontier
The jelly fish shape contains all possible combinations of risk and return: The feasible set.
The red line constitutes the efficient frontier of portfolios of risky assets: Highest return for given risk.
The tangent portfolio T is the optimal portfolio of risky assets that all risk-averse investors will combine with the riskless asset.
Standard Deviation
Expected Return T
Two-Fund Separation Theorem (Tobin, 1958)

48. Theory & Practice
The static mean-variance model & elementary theory of mutual fund financial intermediation.
Dynamic versions integrating intertemporal optimization of the life-cycle consumption-saving decisions with the allocation of those savings among alternative investments & a richer theory for the role of securities and financial intermediation.
Optimal combination of assets & optimal hedging portfolio more tailored to the needs of different clienteles.