1. Investment

2. An Investor’s Perspective An investor has two choices in investment. Risk free asset and risky asset For simplicity, the return on risk free asset is zero. The return on risky asset is 1+d for probability p, 1-d for probability 1-p Investor want to maximize his long term return. How should he allocate resources between risk free and risky assets?

3. Solution Suppose the investor will allocate portion x into risky asset and portion 1-x into riskless asset. The expected rate of return for him is

4. To determine the value of x at which the portfolio will have the maximal rate of return, we differentiate the above formula with respect to x.

5. The above differentiation equals zero when At this value of x, the portfolio obtains the highest expected geometric return.

6. Some numerical examples Assume d = 25%, which is roughly equivalent to standard deviation of 25% for a stock. We set p = 0.55, 0.575, 0.60, 0.625

7. Example Suppose p = 0.575, d = 0.25 for the risky asset and the risk free rate is 0. What are the expected geometric returns if the portion of the risky asset is 30%, 60% and 90%? What are the expected arithmetic returns if the portion of the risky asset is 30%, 60% and 90%?

8. Answers expected geometric return0.0084870.0113570.008397 expected arithmetic return0.011250.02250.03375

9. Geometric and arithmetic returns In practice, arithmetic means are used to measure performance. Geometric means provide more relevant measure for investors Example: First year 100% return, second year -50%. What is the average return from two years?

10. Higher Risk, Higher Return? True up to a certain level. If p = 0.6, up to the level of 80% of risky asset, higher risk, higher return. Over 80% limit, higher risk, lower return. In the past, high equity return. Putting all assets in equity may provide high return. If equity premium is lower in the future, as in the past ten years in most of the stock markets, return and risk pictures could be different.

11. A comparison with standard theory In standard theory, there is always a risk return tradeoff. In our theory, there is a highest possible return at a certain point. The standard theory is a two parameter theory. Ours is a one parameter theory.

12. Utility based or return based? A deeper sense of difference is whether economic theory should be utility based on return based Measurement of company performances is return based Human decisions are utility based. It is often claimed that human beings have free will. However, company managers are human beings and have free will as well.

13. Both companies and individuals are subject to the requirement of positive returns. If a person makes a bad investment decision, his wealth will shrink and his impact on market will decline.

14. Difference on investment decisions In CAPM, there is a capital market line. Investors can pick any point on capital market line based on his utility function. In a geometric return based theory, the ones choose higher rates of return will gradually holding higher shares of total wealth than the ones choose lower rates of return.

15. Capital Market Line