1. Portfolio Theory & Related Topics

2. THE CAPM (Capital Asset Pricing Model) developed by Sharpe, Treynor
later contributions by Mossin, Lintner, Black
all done in the sixties
William Sharpe, Nobel Prize, Economics, 1990

3. Assumptions
Individual investors are price takers (cannot affect prices)
Single-period investment horizon
Investments are limited to traded financial assets
No taxes, regulations, short-selling constraints and transaction costs

4. Assumptions (cont’d)
Information is costless and available to all investors
Investors are rational mean-variance optimizers (risk averse individuals)
Homogeneous expectations (all investors believe the same about r’s and σ’s)
Investors can borrow/ lend at RF

5. Return μ
M = T RF
Capital Market Line (CML) σ

6. Important (actually, critical) observation…
Each investor puts a fraction of his/her wealth in (i) the risk free asset; and (ii) the tangent portfolio
But, all investors MUST hold the risky asset in the same proportion
Thus, they MUST hold the same risky (tangent) portfolio..
THE TANGENT PORTFOLIO IS THE MARKET PORTFOLIO

7. The market portfolio (M) is an efficient portfolio
Market portfolio = Tangent portfolio
The CML shows different combinations of the risk-free asset and the market (M) portfolio; investors select a position along the CML line according to risk preferences. (Passive investment strategy)
CML equation

8. For a given risky (i) security, we have that the return of that security can be written as (CAPM Model, security market line)

9. Proof
The idea is to match the slope of the CML equation at point M, with the slope of the efficient frontier
See Notes

10. VAR(ri) = βi2 x VAR (rM) + VAR (εi)
Non-systematic (“error”) component
Alternatively, and more formally
Systematic risk
TOTAL RISK = Systematic RISK non-systematic RISK
VAR(ri) = βi2 x VAR (rM) VAR (εi)

11. CAPM, Issues and Observations
Anomalies
Roll’s critique

12. APPENDIX

13. Sharpe Ratio The Sharpe Ratio is:
Fact: The tangent portfolio has the maximum Sharp Ratio of any portfolio or stock in the market

14. Sharpe’s Insight Two facts
Everybody’s portfolio is a combination of the tangent portfolio and the riskless asset.
The sum of everybody’s portfolio of risky assets is the portfolio of all risky assets --- the market portfolio
This implies that everybody must be holding the market portfolio because
Everybody holds the same portfolio
Sum of everybody’s portfolio is the market

15. Security Market Line and Capital Market Line

16. Estimación del beta (II)
Ejemplo: Beta con respecto a índice de mercado IPSA de Copec
* Retornos mensuales, estimación a septiembre 2008
IN56A – Primavera Gonzalo Maturana F

17. Single factor Model
Single factor model equation defines a linear fit to data
We have several independent observations of the rate of return and factor
Error is measured by the vertical distance from a point to the line