# Stock Market Return Risk Ratio

If you have a certain amount of money invested in the stock market for a moment of time, then there is an expected return on that investment (the stock market goes up on average), and a risk, a variance in that return (the stock market flucuates), both of which are proportional to the amount you have invested. As those moments of time are strung together, the expected returns for the different moments add, while the risks (since they are independent from one moment to the next) combine as square root of sum of squares. You have $10,000 to invest over one year. How should you allocate your investment over time to maximize your return/risk ratio?

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#### Solution Preview

Let there be n units of time.

For each unit of time let the expected return be E(r), (note that it is given in the problem that the returns can go up or down, but the expected return is positive, thus we deal with only expected returns).

Now, let us sum up the expected returns over n units of time, which is

n E(r). ...(1)

(As it is given in the problem that the expected returns for different moments add).

Now, we deal with the risk part.

The risk is the variance in the returns. Let us call it ...

#### Solution Summary

How should you allocate your investment over time to maximize your return/risk ratio is determined.

Stock Price, Risk Free Assets, Debt to Equity Ratio, Value

Company A and B are two identical companies with equal asset values of $50 million. Company A is financed by equity only and has 100,000 shares outstanding. Company B has perpetual risk-free debt in its capital structure with a market value of $20 million. Company B also has 100,000 shares outstanding. The risk-free rate is 1%.

(a) Assume that there are no taxes. What is A's share price? What is B's share price?

(b) Can you create a portfolio that mimics the risk-return profile of Company A (1 stock of Company A) and consists of the risk free asset and Company B's stock? If yes, describe the portfolio.

(c) Company B wants to achieve its long-term target Debt-to-Equity ratio of 0.5. How much debt or equity does Company B have to buy back? Calculate the value of the debt and equity after the buy-back.

(d) Let's consider that the corporate tax rate is 35% (assume that the levered value of Company B is still $50 million). Analyze the scenario in part (c) with corporate tax rates. What is the value of debt and equity after the buy-back in part (c).

(e) Why are the market values of debt and equity different in parts (c) and (d)?