1. The CAPM
Investments
Fall 2016

2. Intro I. – Return on an investmentF1 F0
E(F1) rf
E(r)
Riskfree investment
Risky investment

3. Intro II. – Remember the utility function( F B ) σ r E ( F C ) σ r F A
E(U)
E(U*) E ( F D ) σ r

4. Intro III. – The indifference curve(s)
E(U*)
E(r)
σ(r)
σ(rD)
E(rD)
σ(rC)
E(rC)
σ(rB)
E(rB) rA

5. Efficient portfolios If we assume that investors
Avoid risk
Are perfectly rational
Maximize their utility
Harry Markowitz: 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”.

6. a2 a1 a3 a4 a6 a5 a7 a8 aj ak ai E(rp) E(r2) E(r1) E(r3) E(r4) E(r6)
E(rj)
E(r8)
E(rk)
E(r6)
E(r5)
E(r7) ai a1 a2 a3 a4 a7 aj a6 a5 ak a8
E(ri)
E(rp)

7. Ugly formulae I.
For two elements:

8. Ugly formulae II.
Correlation = 1

9. Ugly formulae III. Correlation = 0
Deviations from mean cancelling out eachother
Depends on the number of elements

10. A little less ugly form Negative correlation Correlations in between
„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!!!

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

12. Let’s head back to investments
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?

13. Two-security example Security i j E(r) in % 2.5 3.3 σ(r) in % 11.4
kij= -1
kij= -0,5
kij= 0
kij= 1
kij= 0,5
Security i j
E(r) in %
2.5
3.3
σ(r) in %
11.4
17.1
3,3
17,1 j
2,5
11,4 i

14. For three securities
σ(r)
E(r) i j k

15. And for a lot...
σ(r)
E(r)

16. Lessons learned
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)

17. Efficient portfolios
σ(r)
E(r) B A

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

19. σ(r)
E(r) A B1 B2

20. Add the riskless loan/investmentj
pl.: -0,5i + 1,5j
pl.: 0,4i + 0,6j

21. σ(r)
E(r) A C1 M C2 rf

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

23. Beta 1 ri βi εi rM

24. ri εi Beta is the slope of the characteristic line
„Average” relationship, conditional expectation. ri βi εi 1
The „epsilons” cancel out… rM

25. (Undiversifiable risk)
Risk decomposition
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…( r )
Security Market line
This is the Capital Asset Pricing Model = CAPM…
Market portfolio E ( r ) M r f 1 β

27. E(r) rf β

28. Past (average) behavior Future (expected) behavior
Measure beta
Capital Market line 1 βi ri rM εi E ( r )
Market portfolio E ( r ) M r f 1 β
Past (average) behavior
Future (expected) behavior

29. And if we have expected behavior...
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.