Investments ch5 & 6 exam

Downside risk

Compares frequency of extreme negative outcomes to those in a normal distribution with the same mean and SD

Efficient Diversification

Construct portfolio that has lowest risk for a given expected return

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 a multiple of 1.05. 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

Ch5 Holding Period Return

Defined as dollars earned (=change in price + cash payments) per dollar invested. Applicable only for a single period setting.

Complete Portfolio

Defined as the overall portfolio composed of the risk-free asset and risky portfolio (P)

Risk Aversion (RA)

Degree to which an individual invests in risky assets (e.g. stocks) depends onhis/her risk aversion

Risk Premium (RP)

Difference between the expected HPR (of an asset, say i) and the risk-free rate(=rate you earn on a T-Bill) i.e. E(ri - rf )

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 the increase will result in a fall in the real estate of interest?

If inflation increases from 3% to 5%, according to the Fisher equation there will be aconcurrent increase in the nominal rate to offsets the increase in expected inflation.This gives investors an unchanged growth of purchasing power.

Nominal Rate

Interest rate in terms of nominal dollars

Kurtosis

Measure of the fatness of the tails of a probability distribution relative to that of a normal distribution. Indicates likelihood of extreme outcomes.

Mode

Most likely value

Fisher (1930)

Nominal rate ought to increase 1:1 with increases in expected inflation rate

Correlation coefficient

Ranges from -1 to +1

Inflation

Rate at which prices are rising, measured by the Consumer Price Index

Real Rate

Rate at which your purchasing power (PP) grows

Assume expected returns and standard deviations for all securities, as well as the risk- free rate for lending and borrowing, are known. Will investors arrive at the same optimal risky portfolio? Explain.

Since Stock A and Stock B are perfectly negatively correlated, a risk-free portfolio canbe created and the rate of return for this portfolio in equilibrium will always be the risk-free rate. To find the proportions of this portfolio [with w A invested in Stock A and w B= (1 -w A ) invested in Stock B], set the standard deviation equal to zero. With perfectnegative correlation, the portfolio standard deviation reduces to:sP = ABS[ w A sA -w B sB ]0 = 40 w A - 60(1 -w A ) Þ w A = .60The expected rate of return on this risk-free portfolio is:E(r) = ( .60 ́ .08) + ( .40 ́ .13) = 10.0%Therefore, the risk-free rate must also be 10.0%.

Your assistant gives you the following diagram as the efficient frontier of the group of stocks you asked him to analyze. The diagram looks a bit odd, but your assistant insists he double-checked his analysis. Would you trust him? Is it possible to get such a diagram?

Since these are annual rates and the risk-free rate was quite variable during the sampleperiod of the recent 20 years, the analysis has to be conducted with continuouslycompounded rates in excess of T-bill rates. Notice that to obtain cc rates we mustconvert percentage return to decimal. The decimal cc rate, ln(1 + percentage rate/100),can then be multiplied by 100 to return to percentage rates. Recall also that with ccrates, excess returns are just the difference between total returns and the risk-free (T-bill) rates. The bond portfolio is less risky as represented by its lower standard deviation. Yet, asthe portfolio table shows, mixing .87% of bonds with 13% stocks would have produceda portfolio less risky than bonds. In this sample of these 20 years, the average retterm-17urn onthe less risky portfolio of bonds was higher than that of the riskier portfolio of stocks.This is exactly what is meant by "risk." Expectation will not always be realized.

Geometric Average

Single per-period return that gives the same cumulative performance as the sequence of actual returns

In forming a portfolio of two risky assets, what must be true of the correlation coefficient between their returns if there are to be gains from diversification?

So long as the correlation coefficient is below 1.0, the portfolio will benefit from diversification because returns on component securities will not move in perfect lock step. The portfolio standard deviation will be less than a weighted average of thestandard deviations of the component securities

Naive Diversification

Split wealth evenly across all the securities i.e. hold an equally weighted portfolio

Arithmetic Average

Sum of returns in each period divided by number of periods

Suppose you've estimated that the fifth-percentile value at risk of a portfolio is −30%. Now you wish to estimate the portfolio's first-percentile VaR (the value below which lie 1% of the returns). Will the 1% VaR be greater or less than −30%?

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 asmaller VaR than a 5 percentile probability.

Dollar Weighted Return

The IRR of an investment

What is the sharpe ratio of the best feasible CAL?

The Sharpe ratio of the optimal CAL is:E(r P ) － r fs P= 12.88 － 5.523.34 = .3162

When adding a risky asset to a portfolio of many risky assets, which property of the asset has a greater influence on risk, its standard deviation or its covariance with the other assets? Explain.

The covariance with the other assets is more important. Diversification is accomplished via correlation with other assets. Covariance helps determine that number.

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

The excess return on the portfolio will be the same as long as you are consistent: youcan use either real rates for the returns on both the portfolio and the risk-free asset, ornominal rate for each. Just don't mix and match! So the average excess return, thenumerator of the Sharpe ratio, will be unaffected. Similarly, the standard deviation ofthe excess return also will be unaffected, again as long as you are consistent.

An investor ponders various allocations to the optimal risky portfolio and risk-free T-bills to construct his complete portfolio. How would the Sharpe ratio of the complete portfolio be affected by this choice?

The expected return of the portfolio will be impacted if the asset allocation is changed.Since the expected return of the portfolio is the first item in the numerator of the Sharpe ratio, the ratio will be changed.

Tabulate and draw the investment opportunity set of the two risky funds. Use investment proportions for the stock fund of 0% to 100% in increments of 20%. What expected return and standard deviation does your graph show for the minimum variance portfolio?

The parameters of the opportunity set are:E( r S ) = 15%, E( rB ) = 9%, sS = 32%, sB = 23%, r = 0.15, rf = 5.5% From the standard deviations and the correlation coefficient we generate the covariancematrix [note that Cov( rS , rB ) = rsS sB ]:Bonds StocksBonds 529.0 110.4Stocks 110.4 1024.0 The minimum-variance portfolio proportions are: w Min (S) = s B2 － Cov(r S, r B )s S2 + s B2 － 2Cov(r S, r B ) = 529 － 110.41,024 + 529 － (2 ×110.4) = .3142w Min (B) = 1 - .3142 = .6858The mean and standard deviation of the minimum variance portfolio are:E( rMin ) = ( .3142 ́ 15%) + ( .6858 ́ 9%) = 10.89%sMin = [ w %2s %2+ w &2s &2+ 2 w S w B Cov( rS , rB )] 1/2= [( .31422 ́ 1024) + ( .68582 ́ 529) + (2 ́ .3142 ́ .6858 ́ 110.4)] 1/2= 19.94%

What has been the historical average real rate of return on stocks, Treasury bonds, and treasury bills?

To answer this question with the data provided in the textbook, we look up thehistorical average for Treasury Bills, Treasury Bonds and stocks for 1926-2016 fromTable 5.3Arithmetic Average, Nominal ReturnsT-bills: 3.43%T-bonds: 5.51%Stocks: 11.91%To estimate the real rate of return, assume historical inflation of 2.99%:T-bills: 1.0343/1.0299 - 1 = .00427 = .43%T-bonds: 1.0551/1.0299 - 1 = .02446 = 2.45%Stocks: 1.1191/1.0299 - 1 = .08661 = 8.66%

The standard deviation of the market-index portfolio is 20%. Stock A has a beta of 1.5 and a residual standard deviation of 30%. a. What would make for a larger increase in the stock's variance: an increase of 0.15 in its beta or an increase of 3% (from 30% to 33%) in its residual standard deviation? b. An investor who currently holds the market-index portfolio decides to reduce the portfolio allocation to the market-index to 90% and to invest 10% in stock A. Which of the changes in (a) will have a greater impact on the portfolio's standard deviation?

Total variance = Systematic variance + Residual variance = β 2 Var( rM ) + Var(e)When β = 1.5 and σ(e) = .3, variance = 1.5 2 × .22 + .32 = .18. a. Both will have the same impact. Total variance will increase from .18 to .1989. b. Even though the increase in the total variability of the stock is the same in eitherscenario, the increase in residual risk will have less impact on portfoliovolatility. This is because residual risk is diversifiable. In contrast, the increasein beta increases systematic risk, which is perfectly correlated with the market-index portfolio and therefore has a greater impact on portfolio risk.

If you were to use only the two risky funds and still require an expected return of 12%, What would be the investment proportions of your portfolio? b. Compare its standard deviation to that of the optimal portfolio in the previous problem. What do you conclude?

Using only the stock and bond funds to achieve a mean of 12%, we solve:12 = 15w S + 9(1 -w S ) = 9 + 6w S Þ w S = .5Investing 50% in stocks and 50% in bonds yields a mean of 12% and standard deviationof: sP = [( .502 ́ 1,024) + ( .502 ́ 529) + (2 ́ .50 ́ .50 ́ 110.4)] 1/2 = 21.06%The efficient portfolio with a mean of 12% has a standard deviation of only 20.61%.Using the CAL reduces the standard deviation by 45 basis points.

A portfolios expected return is 12%, its standard deviation is 20%, and the risk free rate is 4%. Which of the following would make for the greatest increase in the portfolios Sharpe ratio?

a and b will have the same impact of increasing the Sharpe ratio from .40 to .45.

Skewness

a measure of the degree to which a distribution is asymmetrical

Suppose that many stocks are traded in the market and that it is possible to borrow at the risk free rate, rp. The characteristics of two of the stocks are as follows: Could the equilibrium rf be greater than 10%?

a. Although it appears that gold is dominated by stocks, gold can still be anattractive diversification asset. If the correlation between gold and stocks issufficiently low, gold will be held as a component in the optimal portfolio b. If gold had a perfectly positive correlation with stocks, gold would not be a partof efficient portfolios. The set of risk/return combinations of stocks and gold would plot as a straight line with a negative slope. (Refer to the above graphwhen correlation is 1.) The graph shows that when the correlation coefficient is 1, holding gold provides no benefit of diversification. The stock-only portfoliodominates any portfolio containing gold. This cannot be an equilibrium; theprice of gold must fall and its expected return must rise.

Suppose now that your portfolio must yield an expected return of 12% and be efficient, that is, on the best feasible CAL. a. What is the standard deviation of your portfolio? b. What is the proportion invested in the T-Bill fund and each of the two risky funds?

a. The equation for the CAL is:E( rC ) = rf + E(r P ) － r fs PsC = 5.5 + .3162sCSetting E( rC ) equal to 12% yields a standard deviation of 20.5566%.b. The mean of the complete portfolio as a function of the proportion invested inthe risky portfolio (y) is:E( rC ) = (l - y) rf + yE( rP ) = rf + y[E( rP ) - rf ] = 5.5 + y(12.88 - 5.5)Setting E( rC ) = 12% Þ y = .8808 (88.08% in the risky portfolio)1 - y = .1192 (11.92% in T-bills)To prevent rounding error, we use the spreadsheet with the calculation of theprevious parts of the problem to compute the proportion in each asset in the complete portfolio: b. If gold had a perfectly positive correlation with stocks, gold would not be a part of efficient portfolios. The set of risk/return combinations of stocks and golds(c) 0.205559955 = (12% - 5.5%)/Sharpe ratio(risky portfolio)W(risky portfolio) 0.880786822 = s(c)/s(risky portfolio)Proportion of stocks in complete portfolioW(s) = W(risky portfolio)*% in stock of the risky portfolio= 0.569541021 Proportion of bonds in complete portfolioW(b) = W(risky portfolio)*% in bonds of the risky portfolio= 0.311245801

Suppose that the returns on the stock fund presented in spreadsheet 6.1 were -40%, -14%, 17%, and 33% on the four scenarios. a. Would you expect the mean return and variance of the stock fund to be more than, less than, or equal to the values computed in spreadsheet 6.2? Why? b. Calculate the new values of mean return and variance for the stock fund using a format similar to spreadsheet 6.2. Confirm your intuition from part (a). c. Calculate the new value of the covariance between the stock and bond funds using a format similar to spreadsheet 6.4. Explain intuitively the change in the covariance.

a. Without doing any math, the severe recession is worse and the boom is better. Thus, there appears to be a higher variance, yet the mean is probably the samesince the spread is equally large on both the high and low side. The mean return, however, should be higher since there is higher probability given to the higher returns. b. see Eoc solutions c. Covariance has increased because the stock returns are more extreme in therecession and boom periods. This makes the tendency for stock returns to bepoor when bond returns are good (and vice versa) even more dramatic.

Mean

expected value i.e. the E(r) computed in earlier slides

Diversification

spread out your wealth across several different assets rather than invest all your wealth in one/few assets

Median

value above and below which we expect 50% of the observations