1. Fi8000 Valuation of Financial Assets
Spring Semester 2010
Dr. Isabel Tkatch
Assistant Professor of Finance

2. Investment Strategies
Lending vs. Borrowing (risk-free asset)
Lending: a positive proportion is invested in the risk-free asset (cash outflow in the present: CF0 < 0, and cash inflow in the future: CF1 > 0)
Borrowing: a negative proportion is invested in the risk-free asset (cash inflow in the present: CF0 > 0, and cash outflow in the future: CF1 < 0)

3. Lending vs. Borrowing A A
Lend B
Borrow C rf rf

4. Investment Strategies
A Long vs. Short position in the risky asset
Long:
A positive proportion is invested in the risky asset (cash outflow in the present: CF0 < 0, and cash inflow in the future: CF1 > 0)
Short:
A negative proportion is invested in the risky asset (cash inflow in the present: CF0 > 0, and cash outflow in the future: CF1 < 0)

5. Long vs. Short E(R) A B STD(R) Long A and Short B Long A and Long B
Short A and Long B B
STD(R)

6. Investment Strategies
Passive risk reduction:
The risk of the portfolio is reduced if we invest a larger proportion in the risk-free asset relative to the risky one
The perfect hedge:
The risk of asset A is offset (can be reduced to zero) by forming a portfolio with a risky asset B, such that ρAB=(-1)
Diversification:
The risk is reduced if we form a portfolio of at least two risky assets A and B, such that ρAB<(+1)
The risk is reduced if we add more risky assets to our portfolio, such that ρij<(+1)

7. One Risky Fund and one Risk-free Asset: Passive Risk Reduction
Reduction in portfolio risk B
Increase of portfolio Risk C rf rf

8. Two Risky Assets with ρAB=(-1): The Perfect Hedge
Minimum Variance is zero
Pmin B
STD(R)

9. The Perfect Hedge – an Example
What is the minimum variance portfolio if we assume that
μA=10%; μB=5%; σA=12%; σB=6% and ρAB=(-1)?

10. The Perfect Hedge – Continued
What is the expected return μmin and the standard deviation of the return σmin of that portfolio?

11. Diversification: the Correlation Coefficient and the Frontier
ρAB=(-1)
-1<ρAB<1
ρAB=+1 B
STD(R)

12. Diversification: the Number of Risky assets and the Frontier
STD(R)

13. Diversification: the Number of Risky assets and the Frontier
STD(R)

14. Diversification: the Number of Risky assets and the Frontier
STD(R)

15. Diversification: the Number of Risky assets and the Frontier
STD(R)

16. Capital Allocation: n Risky Assets
State all the possible investments – how many possible investments are there?
Assuming you can use the Mean-Variance (M-V) rule, which investments are M-V efficient?
Present your results in the μ-σ (mean – standard-deviation) plane.

17. The Expected Return and the Variance of the Return of the Portfolio
wi = the proportion invested in the risky asset i (i=1,…n)
p = the portfolio of n risky assets (wi invested in asset i)
Rp = the return of portfolio p
μp = the expected return of portfolio p
σ2p = the variance of the return of portfolio p

18. The Set of Possible Portfolios in the μ-σ Plane
E(R)
The Frontier i
STD(R)

19. The Set of Efficient Portfolios in the μ-σ Plane
E(R)
The Efficient Frontier i
STD(R)

20. Capital Allocation: n Risky Assets
The investment opportunity set:
{all the portfolios {w1, … wn} where Σwi=1}
The Mean-Variance (M-V or μ-σ ) efficient investment set:
{only portfolios on the efficient frontier}

21. The case of n Risky Assets: Finding a Portfolio on the Frontier
Optimization:
Find the minimum variance portfolio for a given expected return
Constraints:
A given expected return;
The budget constraint.

22. The case of n Risky Assets: Finding a Portfolio on the Frontier

23. Capital Allocation: n Risky Assets and a Risk-free Asset
State all the possible investments – how many possible investments are there?
Assuming you can use the Mean-Variance (M-V) rule, which investments are M-V efficient?
Present your results in the μ-σ (mean – standard-deviation) plane.

24. The Expected Return and the Variance of the Return of the Possible Portfolios
wi = the proportion invested in the risky asset i (i=1,…n)
p = the portfolio of n risky assets (wi invested in asset i)
Rp = the return of portfolio p
μp = the expected return of portfolio p
σ2p = the variance of the return of portfolio p

25. The Set of Possible Portfolios in the μ-σ Plane (only n risky assets)
E(R)
The Frontier i
STD(R)

26. The Set of Possible Portfolios in the μ-σ Plane (risk free asset included)rf
STD(R)

27. The Set of Efficient Portfolios in the μ-σ Plane
The Capital Market Line: μp= rf + [(μm-rf) / σm]·σp μ m i rf σ

28. The Separation Theorem
The asset allocation process of the risk-averse investors can be separated into two stages:
1.Choose the optimal portfolio of risky assets m
(The allocation between risky securities is identical for all the
investors)
2.Choose the optimal allocation of funds between the risky
portfolio m and the risk-free asset rf – choose a portfolio on
the CML (The allocation between the risky portfolio and the
risk free asset is personal and depends on the risk
preferences of each investor)

29. Capital Allocation: n Risky Assets and a Risk-free Asset
The investment opportunity set:
{all the portfolios {w0, w1, … wn} where Σwi=1}
The Mean-Variance (M-V or μ-σ ) efficient investment set:
{all the portfolios on the Capital Market Line - CML}

30. n Risky Assets and One Risk-free Asset: Finding the Market Portfolio
Optimization:
Find the minimum variance portfolio for a given expected return
Constraints:
A given expected return;
The budget constraint.

31. n Risky Assets and One Risk-free Asset: Finding the Market Portfolio

32. n Risky Assets and One Risk-free Asset: Finding the Market Portfolio

33. Example
Find the market portfolio if there are only two risky assets, A and B, and a risk-free asset rf.
μA=10%; μB=5%; σA=12%; σB=6%; ρAB=(-0.5) and rf=4%

34. Example Continued

35. Example Continued

36. Practice Problems BKM 7th Ed. Ch. 7: 1-13, 17-22, 25-26
BKM 8th Ed. Ch. 7: 4-19 CFA: 4-6, 10-11
Mathematics of Portfolio Theory:
Read and practice parts 11-13