Investments chapter 5

Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be either $50,000 or $150,000, with equal probabilities of 0.5. The alternative riskless investment in T-bills pays 5%. A.) If you require a risk premium of 10%, how much will you be willing to pay for the portfolio? B.) Suppose that the portfolio can be purchased for the amount you found in (a). What will be the expected rate of return on the portfolio? C.) Now suppose that you require a risk premium of 15%. What is the price that you will be willing to pay?

A.) The expected cash flow is: (0.5 × $50,000) + (0.5 × 150,000) = $77,500 With a risk premium of 10% over the risk-free rate of 5%, the required rate of return is 15%. Therefore, the present value of the portfolio is:$77,500/1.15 = $67,391 B.) If the portfolio is purchased for $67,391 and provides an expected cash inflow of $77,500, then the expected rate of return [E(r)] is as follows:$67,391 × [1 + E(r)] = $77,500 Therefore, E(r) = 15%. The portfolio price is set to equate the expected rate of return with the required rate of return. C) If the risk premium over T-bills is now 15%, then the required return is:5% + 15% = 20% The present value of the portfolio is now:$77,500/1.20 = $64,583.33

XYZ stock price and dividend stock history are as follows: Year Beginning-of-year Price Dividend Paid at Year-End 2015 $100 $4 2016 110 4 2017 90 4 2018 95 4 An investor buys four shares of XYZ at the beginning of 2010, buys another three shares at the beginning of 2011, sells one share at the beginning of 2012, and sells all six remaining shares at the beginning of 2013. a. What are the arithmetic and geometric average time-weighted rates of return for the investor? b-1. Prepare a chart of cash flows for the four dates corresponding to the turns of the year for January 1, 2010, to January 1, 2013. b-2. What is the dollar-weighted rate of return? (Hint: If your calculator cannot calculate internal rate of return, you will have to use a spreadsheet or trial and error.)

Arithmetic mean: 2.25% Geometric mean: 2.22% 1/1/2010: -432 1/1/2011: -320 1/1/2012: 126 1/1/2013: 642 rate of return: 0.8771%

You've just decided upon your capital allocation for the next year, when you realize that you've underestimated both the expected return and the standard deviation of your risky portfolio by 4%. Will you increase, decrease, or leave unchanged your allocation to risk-free T-bills?

Decrease. Typically, standard deviation exceeds return. Thus, an underestimation of 4% in each will artificially decrease the return per unit of risk. To return to the proper risk return relationship the portfolio will need to decrease the amount of risk free investments.

When estimating a sharpe ratio, would it make sense to use the average excess real return that accounts for inflation?

No, since all numbers are presented in nominal figures, should use nominal data

The real interest rate approximately equals the nominal rate minus the inflation rate. Suppose the inflation rate increases from 3% to 5%. Does the Fisher equation imply that this increase will result in a fall in the real rate of interest? Explain.

The Fisher equation predicts that the nominal rate will equal the equilibrium real rate plus the expected inflation rate. Hence, if the inflation rate increases from 3% to 5% while there is no change in the real rate, then the nominal rate will increase by 2%. On the other hand, it is possible that an increase in the expected inflation rate would be accompanied by a change in the real rate of interest. While it is conceivable that the nominal interest rate could remain constant as the inflation rate increased, implying that the real rate decreased as inflation increased, this is not a likely scenario

The stock of Business Adventures sells for $40 a share. Its likely dividend payout and end-of-year price depend on the state of the economy by the end of the year as follows: Dividend Stock Price Boom $2.00 $50 Normal economy 1.00 43 Recession 0.50 34 a. Calculate the expected holding-period return and standard deviation of the holding-period return. All three scenarios are equally likely. b. Calculate the expected return and standard deviation of a portfolio invested half in Business Adventures and half in Treasury Bills is 4%.

a. boom (50-40+2)/40= 0.3 (30%)normal (43-40+1)/40= 0.1 (10%recession (34-40+0.5)/40 = -0.1375 (-13.75%)E(r)=1/3(30%) + 1/3(10%) + 1/3(-13/75%)E(r)=0.875 (8.75%)Var = sum p(s)[r(s)-E(r)^2]Var = 319.79St. Dev. = SqRt 319.79 = 17.88% b. E(rp) = wi[E(r)]+w2[rf]wi= proportion of the portfolio invested in stock of business adventuresrf= return on treasury billsw2= proportion of portfolio invested in treasury billsE(rp)=6.375%St. Dev.= wi x variance=0.5x17.88%=8.94%