1. Investments Fall 2014 THE CAPM

2. F1F1 F0F0 E(F1)E(F1) F0F0 rfrf E(r)E(r) E(r)E(r) E(r)E(r) Riskfree investmentRisky investment INTRO I. – RETURN ON AN INVESTMENT

3. E ( F C ) σ ( r C ) E ( F D ) σ(r D ) F A F E(U) E(U*) E ( F B ) σ ( r B ) INTRO II. – REMEMBER THE UTILITY FUNCTION

4. E(U*) E(r)E(r) σ(r)σ(r) rArA σ(rB)σ(rB) E(rB)E(rB) σ(rC)σ(rC) E(rC)E(rC) σ(rD)σ(rD) E(rD)E(rD) INTRO III. – THE INDIFFERENCE CURVE(S)

5. If we assume that investors EFFICIENT PORTFOLIOS Harry Markowitz: 1952. Portfolio Selection –A portfolio is more than the group of its elements! –Markowitz constucts portfolios for investors, who „seek expected return, while avoid the deviation of return”. Avoid risk Maximize their utility Are perfectly rational

6. aiai a1a1 a2a2 a3a3 a4a4 a7a7 ajaj a6a6 a5a5 akak a8a8 E(ri)E(ri) E(r1)E(r1) E(r2)E(r2) E(r3)E(r3) E(r4)E(r4) E(rj)E(rj) E(r8)E(r8) E(rk)E(rk) E(r6)E(r6) E(r5)E(r5) E(r7)E(r7) E(rp)E(rp)

7. For two elements: UGLY FORMULAE I.

8. Correlation = 1 UGLY FORMULAE II.

9. Correlation = 0 –Deviations from mean cancelling out eachother –Depends on the number of elements UGLY FORMULAE III.

10. Negative correlation –„Cancel out” is faster –We do not need indefinitely many elements Correlations in between –Positive, but less than one Deviation decreases, but not to zero –Negative, but larger than minus one Still faster cancelling out, but not that fast Rule of thumb: –If correlation is not perfect, deviation decreases. The lower the pairwise correlations, the faster it approaches and the closer it gets to zero. –Example: from music to noise This is the very essence of modern portfolio theory!!! A LITTLE LESS UGLY FORM

11. SIMPLE EXAMPLE 37,5 37,5 25-50 25-50 37,5 50- 50% Sunglass business Raincoat business Sunny year 50 25 Rainy year 25 50 (correlation does not affect expected return!)

12. What happens to risk and return if we hold more than one security? Let’s begin with a two-security example. What kind of portfolios can we construct? LET’S HEAD BACK TO INVESTMENTS

13. k ij = -0,5 k ij = 0 k ij = 0,5 1 3 2 2,5 11,4 i 3,3 17,1 j k ij = 1k ij = -1 Securityij E(r) in %2.53.3 σ(r) in %11.417.1 TWO-SECURITY EXAMPLE

14. σ(r)σ(r) E(r)E(r) i j k FOR THREE SECURITIES

15. AND FOR A LOT... σ(r)σ(r) E(r)E(r)

16. The idea that we shouldn’t “put all of our eggs in one basket” has been around for a long time, but it was economist Harry Markowitz who formalized models to determine how best to diversify those “eggs” (our financial wealth) among different “baskets” (the various financial assets available) -> Diversification is good! If diversification is good and almost free, people will do it. „The hypothesis (or maxim) that the investor does (or should) maximize discounted return must be rejected. If we ignore market imperfections the foregoing rule never implies that there is a diversified portfolio which is preferable to all non- diversified portfolios. Diversification is both observed and sensible; a rule of behavior which does not imply the superiority of diversification must be rejected both as a hypothesis and as a maxim.” (Markowitz) LESSONS LEARNED

17. σ(r)σ(r) E(r)E(r) A B EFFICIENT PORTFOLIOS

18. σ2(r)σ2(r) Nr of portfolio elements (almost) efficient portfolio Idiosyncratic risk Market risk

19. σ(r)σ(r) E(r)E(r) A B1B1 B2B2

20. σ(r)σ(r) E(r)E(r) i j pl.: 0,4i + 0,6j pl.: -0,5i + 1,5j ADD THE RISKLESS LOAN/INVESTMENT

21. rfrf C2C2 C1C1 σ(r)σ(r) E(r)E(r) A M

22. Market portfolio σ(r)σ(r) E(r)E(r) M Capital market line E(rM)E(rM) σ(rM)σ(rM) rfrf

23. riri rMrM 1 1999. 03. 2000. 08. 2002. 11. 2000. 01. 2003. 10. 2001. 03. 2002. 02. βiβi εiεi BETA

24. 1 βiβi riri rMrM εiεi Beta is the slope of the characteristic line „Average” relationship, conditional expectation. The „epsilons” cancel out…

25. Total risk Market risk (Undiversifiable risk) (Systematic risk) (Aggregate risk) Idiosyncratic risk (Diversifiable risk) (Unsystematic risk) (Specific risk)

26. This is the Capital Asset Pricing Model = CAPM… Market portfolio Security Market line E ( r M ) 1 E ( r ) β r f

27. E(r)E(r) rfrf β

28. Past (average) behavior Future (expected) behavior 1 βiβi riri rMrM εiεi Market portfolio Capital Market line E ( r M ) 1 E ( r ) β r f MEASURE BETA

29. Then we can finally speak about performance In finance, Jensen's alpha is used to determine the abnormal return of a security or portfolio of securities over the theoretical expected return. 29 AND IF WE HAVE EXPECTED BEHAVIOR...