# Risk and Return: Past and Prologue

**Pages:**11 (1986 words)

**Published:**December 3, 2013

RISK AND RETURN: PAST AND PROLOGUE

1.The 1% VaR will be less than –30%. As percentile or probability of a return declines so does the magnitude of that return. Thus, a 1 percentile probability will produce a smaller VaR than a 5 percentile probability.

2.The geometric return represents a compounding growth number and will artificially inflate the annual performance of the portfolio.

3.No. Since all items are presented in nominal figures, the input should also use nominal data.

4.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.

5.Using Equation 5.6, we can calculate the mean of the HPR as: E(r) = = (0.3 0.44) + (0.4 0.14) + [0.3 (–0.16)] = 0.14 or 14% Using Equation 5.7, we can calculate the variance as:

Var(r) = 2 = – E(r)]2

= [0.3 (0.44 – 0.14)2] + [0.4 (0.14 – 0.14)2] + [0.3 (–0.16 – 0.14)2] = 0.054

Taking the square root of the variance, we get SD(r) = = = = 0.2324 or 23.24%

6.We use the below equation to calculate the holding period return of each scenario: HPR =

a.The holding period returns for the three scenarios are:

Boom:(50 – 40 + 2)/40 = 0.30 = 30%

Normal:(43 – 40 + 1)/40 = 0.10 = 10%

Recession:(34 – 40 + 0.50)/40 = –0.1375 = –13.75%

E(HPR) =

= [(1/3) 0.30] + [(1/3) 0.10] + [(1/3) (–0.1375)] = 0.0875 or 8.75%

Var(HPR) = – E(r)]2

= [(1/3) (0.30 – 0.0875)2] + [(1/3) (0.10 – 0.0875)2] + [(1/3) (–0.1375 – 0.0875)2]

= 0.031979

SD(r) = = = = 0.1788 or 17.88%

b.E(r) = (0.5 8.75%) + (0.5 4%) = 6.375%

= 0.5 17.88% = 8.94%

7.

a.Time-weighted average returns are based on year-by-year rates of return.

YearReturn = [(Capital gains + Dividend)/Price]

2010-2011 (110 – 100 + 4)/100 = 0.14 or 14.00%

2011-2012 (90 – 110 + 4)/110 = –0.1455 or –14.55% 2012-2013(95 – 90 + 4)/90 = 0.10 or 10.00%

Arithmetic mean: [0.14 + (–0.1455) + 0.10]/3 = 0.0315 or 3.15% Geometric mean: – 1

= 0.0233 or 2.33%

b.

Date

1/1/20101/1/20111/1/20121/1/2013

Net Cash Flow–300–208110396

TimeNet Cash flowExplanation

0–300Purchase of three shares at $100 per share

1–208Purchase of two shares at $110,

plus dividend income on three shares held

2 110Dividends on five shares,

plus sale of one share at $90

3 396Dividends on four shares,

plus sale of four shares at $95 per share

The dollar-weighted return is the internal rate of return that sets the sum of the present value of each net cash flow to zero: 0 = –$300 + + +

Dollar-weighted return = Internal rate of return = –0.1661%

8.

a.Given that A = 4 and the projected standard deviation of the market return = 20%, we can use the below equation to solve for the expected market risk premium: A = 4 = =

E(rM) – rf = AM2 = 4 (0.20) = 0.16 or 16%

b.Solve E(rM) – rf = 0.09 = AM2 = A (0.20) , we can get A = 0.09/0.04 = 2.25

c.Increased risk tolerance means decreased risk aversion (A), which results in a decline in risk premiums.

9.From Table 5.4, we find that for the period 1926 – 2010, the mean excess return for S&P 500 over T-bills is 7.98%.

E(r) = Risk-free rate + Risk premium = 5% + 7.98% = 12.98%

10.To answer this question with the data provided in the textbook, we look up the real returns of the large stocks, small stocks, and Treasury Bonds for 1926-2010 from Table 5.2, and the real rate of return of T-Bills in the same period from Table 5.3: Total Real Return – Geometric Average

Large Stocks: 6.43%

Small Stocks: 8.54%

Long-Term T-Bonds: 2.06%

Total Real Return – Arithmetic Average

Large Stocks: 8.00%

Small Stocks: 13.91%

Long-Term T-Bonds: 1.76%

T-Bills: 0.68% (Table 5.3)

11.

a.The expected cash...

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