Investment

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Investment
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An Investors 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 1d 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?

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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

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• 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.

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• The above differentiation equals zero when

At this value of x, the portfolio obtains the
highest expected geometric return.
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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

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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?

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Answers
expected geometric return 0.008487 0.011357 0.008397
expected arithmetic return 0.01125 0.0225 0.03375
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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?

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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.

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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.

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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.

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• 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.

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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.

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Capital Market Line