Risk Aversion and Capital Allocation to Risky Assets

Transcription

1 Risk Aversion and Capital Allocation to Risky Assets

2 Allocation to Risky Assets Assumption : Investors are rational -Investors will avoid risk unless there is a reward In another word, investor will try to maximize their return, given a level of risk Different investor have different risk aversion. The optimal allocation between a risky portfolio and a risk-free asset varies according to investor s risk aversion.

3 Risk Aversion Investors are willing to consider: risk-free assets risky assets with positive risk premiums Each investor has different risk aversion therefore the risk preference are different Portfolio attractiveness increases with expected return and decreases with risk. Hence the need for portfolio management : An important aspect of portfolio management is Asset allocation

4 Asset Allocation The process to choose asset classes How much should you own in stock? How much should you own in bond? How much cash reserve? Asset Allocation provide the mean to control for risk.

5 The Risk-Free Asset The closest market instrument to risk-free is Gov. Bonds Only the government can issue default-free bonds. - Being default-free is not enough to make them riskfree in real terms Because. 1) inflation 2) the mismatch between maturity and investor s desired holding period Even though, T-bills usually viewed as the risk free asset Many money market instruments also considered risk-free in practice.

6 A Portfolios of One Risky Asset and a Risk-Free Asset It s possible to create a complete portfolio by splitting investment funds between safe and risky assets. Let y = portion allocated to the risky portfolio, P (1-y) = portion to be invested in risk-free asset, F.

7 Example y = the fraction of portfolio allocated in risky portfolio (P) Thus, the portfolio risk premium = 15%-7% = 8%

8 Example (ctd.) The risk premium of the complete portfolio, C, equals the risk premium of the risky asset times the fraction of the portfolio invested in risky asset (y)

9 Example (ctd.) The standard deviation of the complete portfolio equals the standard deviation of the risky asset times the fraction of the portfolio invested in risky asset (y) Since the portfolio of risky asset has the SD of 22%

10 Example (ctd.) and Risk premium 8%, r f =7%, SD p = 22%

11 Measuring portfolio performance The reward-to-volatility ratio (Sharpe Ratio) Sharpe ratio : measure the risk premium per level of per unit of standard deviation in an investment asset or a trading strategy Ex. A portfolio with SD = 20%, have portfolio return of 12%. Given the risk-free return = 4%, Sharpe ratio =? = (12%-4%) / 20% = 0.4 What is sharp ratio of a risk-free asset?

12 Capital Allocation Line

13 Ex. Capital Allocation Line Consider two portfolio P (y=1) and Portfolio C (y=0.5) Expected Return Risk Premium Standard Deviation (σ) Sharp Ratio Port P (y=1) 15% 8% 22% 8/22 = 0.36 Port C (y=0.5) 7%+0.5*8% = 11% 0.5*8% = 4% 0.5*22% = 11% 4/11 = 0.36

14 CAL : Borrowing To summarize : the risk premium and the standard deviation of the complete portfolio increase in proportion to the investment in the risky portfolio (y) All the points in CAL have the same slope = The sharp ratio = 8/22 = 0.36 To archive higher return than the return from 100% risky portfolio P (15%), the investor could leverage his/her position by borrowing to increase E(r) c > 15% This mean y > 1 and borrowing portion = y-1

15 Example : Leveraged CAL Suppose the investment budget is $300,000, and you borrows $120,000 more. You invest $420,000 in risky asset. What is your portfolio s return? Sol. In this case y = 420,000 / 300,000 = 1.4 Thus, E(r c ) = 7% (8%) = 18.2%

16 Example : Leveraged CAL Another way to find the portfolio returns is : You expect to earn 15% of 420,000 63,000 But you need to pay 7% of 120,000 8,400 Thus, the net income ,600 You initial investment was 300,000 Thus, return 54,600 / 300, % Your portfolio still exhibit the same sharp ratio (all points in CAL) σ c = 1.4* 22 = 30.8% S = (E(r c )-r f ) / σ c = 0.36

17 Risk Aversion and Capital Allocation The optimal allocation of an investor will depend on his/her risk aversion The more risk-averse investors will choose to invest less in risky asset and more of the risk-free asset. To find the optimal allocation, we need to know the investor s risk-aversion This can be measured by the degree of risk aversion (A)

18 Coefficient of Risk Aversion Investor s degree of Risk Aversion (A) : measure the risk premium per level of per unit of variance. Aka : price of risk Ex. An investor who hold a portfolio with annual risk premium of 10%, variance (SD 16%), we would infer the investor s degree of risk aversion = 0.10 / = 3.91

19 Risk Aversion and Capital Allocation We can find the investor's optimum allocation, y, by dividing the risky portfolio s price of risk by the investor s required price of risk Ex. What is the optimum allocation for an investor A=3.91, who considering investing in an index fund which give annual return of 12%, r f =4%, and SD of 20%. What is the expected return of the optimum portfolio?

20 Risk Aversion and Capital Allocation Risky Portfolio s risk premium to variance ratio = ( )/(0.2*0.2) = 2 y = 2/3.91 = 0.51 Thus, optimum allocation is 51% in risky portfolio and the other 49% in the risk-free asset E(r) = 0.51*12%+0.49*4% = 8.08%

21 Passive strategies & CML The capital allocation line provided by one-month T-bills and a broad index of common stocks are called Capital Market Line (CML) Passive strategies based on the assumption that securities are always fairly priced Thus, there is no need to undertake security analysis. Investor can just choose a portfolio of all stocks in a broad market index such as the S&P500. This strategies called indexing

22 Passive strategies Main benefit of passive strategies is to avoid search cost (cost of acquiring information about securities) Flip side of passive strategies is active strategies No doubt a really good fund manager can beat the market but Can you find him/her?, Even if you do, what about his/her fees? (Active -> cost of finding a good fund managers that consistently beat the market)