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Chapter 06 - Efficient Diversification

Chapter 06

Efficient Diversification

1. 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 lockstep. The portfolio standard deviation will be less than a weighted average of the standard deviations of the component securities.

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

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

4. 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.

5. Total variance = Systematic variance + Residual variance = β2 Var(rM) + Var(e)

When β = 1.5 and σ(e) = .3, variance = 1.52 × .22 + .32 = .18. In the other scenarios:

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 either scenario, the increase in residual risk will have less impact on portfolio volatility. This is because residual risk is diversifiable. In contrast, the increase in beta increases systematic risk, which is perfectly correlated with the market-index portfolio and therefore has a greater impact on portfolio risk.

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 same since 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.

Calculation of mean return and variance for the stock fund:

c. Calculation of covariance:

Covariance has increased because the stock returns are more extreme in the recession and boom periods. This makes the tendency for stock returns to be poor when bond returns are good (and vice versa) even more dramatic.

a. One would expect variance to increase because the probabilities of the extreme outcomes are now higher.

Calculation of mean return and variance for the stock fund:

c. Calculation of covariance

Covariance has decreased because the probabilities of the more extreme returns in the recession and boom periods are now higher. This gives more weight to the extremes in the mean calculation, thus making their deviation from the mean less pronounced.

8. The parameters of the opportunity set are:

E(rS) = 15%, E(rB) = 9%, S = 32%, B = 23%,  = 0.15, rf = 5.5%

From the standard deviations and the correlation coefficient we generate the covariance matrix [note that Cov(rS, rB) = SB]:

	Bonds	Stocks
Bonds	529.0	110.4
Stocks	110.4	1024.0

The minimum-variance portfolio proportions are:

wMin(S) = = = .3142

wMin(B) = 1 – .3142 = .6858

The mean and standard deviation of the minimum variance portfolio are:

E(rMin) = ( .3142 15%) + ( .6858 9%)  10.89%

Min = [ + + 2 wS wB Cov(rS, rB)]1/2

= [( .31422 1024) + ( .68582 529) + (2 .3142 .6858 110.4)]1/2

= 19.94%

The graph approximates the points:

	E(r)
Minimum variance portfolio	10.89%	19.94%
Tangency portfolio	12.88%	23.3382%

10. The reward-to-variability ratio (Sharpe ratio) of the optimal CAL is:

= = .3162

11.

a. The equation for the CAL is:

E(rC) = rf + C = 5.5 + .3162C

Setting 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 in the 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% 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 the previous parts of the problem to compute the proportion in each asset in the complete portfolio:

12. Using only the stock and bond funds to achieve a mean of 12%, we solve:

12 = 15wS + 9(1 wS ) = 9 + 6wS wS = .5

Investing 50% in stocks and 50% in bonds yields a mean of 12% and standard deviation of: P = [( .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.

13.

a. Although it appears that gold is dominated by stocks, gold can still be an attractive diversification asset. If the correlation between gold and stocks is sufficiently 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 part of 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 graph when correlation is 1.) The graph shows that when the correlation coefficient is 1, holding gold provides no benefit of diversification. The stock-only portfolio dominates any portfolio containing gold. This cannot be an equilibrium; the price of gold must fall and its expected return must rise.

14. Since Stock A and Stock B are perfectly negatively correlated, a risk-free portfolio can be 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 wA invested in Stock A and wB = (1 –wA ) invested in Stock B], set the standard deviation equal to zero. With perfect negative correlation, the portfolio standard deviation reduces to:

P = ABS[wAAwBB]

0 = 40 wA  60(1 –wA) wA = .60

The 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%.

15. Since these are annual rates and the risk-free rate was quite variable during the sample period of the recent 20 years, the analysis has to be conducted with continuously compounded rates in excess of T-bill rates. Notice that to obtain cc rates we must convert 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 cc rates, 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, as the portfolio table shows, mixing .87% of bonds with 13% stocks would have produced a portfolio less risky than bonds. In this sample of these 20 years, the average return on the 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.

16. If the lending and borrowing rates are equal and there are no other constraints on portfolio choice, then the optimal risky portfolios of all investors will be identical. However, if the borrowing and lending rates are not equal, then borrowers (who are relatively risk averse) and lenders (who are relatively risk tolerant) will have different optimal risky portfolios.

17. No, it is not possible to get such a diagram. Even if the correlation between A and B were 1.0, the frontier would be a straight line connecting A and B.

18. In the special case that all assets are perfectly positively correlated, the portfolio standard deviation is equal to the weighted average of the component-asset standard deviations. Otherwise, as the formula for portfolio variance (Equation 6.6) shows, the portfolio standard deviation is less than the weighted average of the component-asset standard deviations. The portfolio variance is a weighted sum of the elements in the covariance matrix, with the products of the portfolio proportions as weights.

19. The probability distribution is:

Probability	Rate of Return
.7	100%
.3	-50%

Expected return = ( .7 1) + .3 ( .5) = 0.55 or 55%

Variance = [ .7 (1  0.55)2] + [ .3 (50  0.55)2] = 0.4725

Standard Deviation = = 0.6874 or 68.74%

20. The expected rate of return on the stock will change by beta times the unanticipated change in the market return: 1.2 ( .08 – .10) = –2.4%

Therefore, the expected rate of return on the stock should be revised to:

.12 – .024 = 9.6%

21.

a. The risk of the diversified portfolio consists primarily of systematic risk. Beta measures systematic risk, which is the slope of the security characteristic line (SCL). The two figures depict the stocks' SCLs. Stock B's SCL is steeper, and hence Stock B's systematic risk is greater. The slope of the SCL, and hence the systematic risk, of Stock A is lower. Thus, for this investor, stock B is the riskiest.

b. The undiversified investor is exposed primarily to firm-specific risk. Stock A has higher firm-specific risk because the deviations of the observations from the SCL are larger for Stock A than for Stock B. Deviations are measured by the vertical distance of each observation from the SCL. Stock A is therefore riskiest to this investor.

22. Using “Regression” command from Excel’s Data Analysis menu, we can run a regression of GM’s excess returns against those of S&P 500, and obtain the following data. The Beta of GM is .87.

23. A scatter plot results in the following diagram. The slope of the regression line is 2.0 and intercept is 1.0.

24.

a. Regression output produces the following:

alpha = 3.1792, beta = 1.3916, Residual St Dev = 11.5932

b. Sharpe Ratio of S&P = = – .6123/4.0316 = – .1519

c. Information Ratio = αG/(eG) = 3.1792/11.5932 = .2742

d. We use Equation 6.16 to compute

= = = –62.79%

Then, plug into Equation 6.17 to compute the optimal position of Google in the optimal risky portfolio and the weight in the market index:

= – .6279 / [1 + (– .6279 (1 – 1.3916)] = –50.40%

= 1– = 49.60%

e. SO = = = .3135

Sharpe ratio increases from – .1519 to .3135.

CFA 1

Answer:

E(rP) = ( .5 15) + ( .4 10) + ( .10 6) = 12.1%

CFA 2

Answer:

Fund D represents the single best addition to complement Stephenson's current portfolio, given his selection criteria. First, Fund D’s expected return (14.0 percent) has the potential to increase the portfolio’s return somewhat. Second, Fund D’s relatively low correlation with his current portfolio (+ .65) indicates that Fund D will provide greater diversification benefits than any of the other alternatives except Fund B. The result of adding Fund D should be a portfolio with approximately the same expected return and somewhat lower volatility compared to the original portfolio.

The other three funds have shortcomings in terms of either expected return enhancement or volatility reduction through diversification benefits. Fund A offers the potential for increasing the portfolio’s return, but is too highly correlated to provide substantial volatility reduction benefits through diversification. Fund B provides substantial volatility reduction through diversification benefits, but is expected to generate a return well below the current portfolio’s return. Fund C has the greatest potential to increase the portfolio’s return, but is too highly correlated to provide substantial volatility reduction benefits through diversification.

CFA 3

Answer:

a. Subscript OP refers to the original portfolio, ABC to the new stock, and NP to the new portfolio.

i. E(rNP) = wOP E(rOP ) + wABC E(rABC ) = ( .9 .67) + ( .1 1.25) = .7280%

ii. CovOP , ABC = CorrOP , ABC OP ABC = .40 2.37 2.95 = 2.7966

iii. NP = [wOP 2 OP 2 + wABC 2 ABC 2 + 2 wOP wABC (CovOP , ABC)]1/2

= [( .92 2.372) + ( .12 2.952) + (2 .9 .1 2.7966)]1/2

= 2.2672%

b. Subscript OP refers to the original portfolio, GS to government securities, and NP to the new portfolio.

i. E(rNP) = wOP E(rOP ) + wGS E(rGS ) = ( .9 .67) + ( .1 .42) = .6450%

ii. CovOP , GS = CorrOP , GS OP GS = 0 2.37 0 = 0

iii. NP = [wOP 2 OP 2 + wGS 2 GS 2 + 2 wOP wGS (CovOP , GS)]1/2

= [( .92 2.372) + ( .12 0) + (2 .9 .1 0)]1/2

= 2.1330%

c. Adding the risk-free government securities would result in a lower beta for the new portfolio. The new portfolio beta will be a weighted average of the individual security betas in the portfolio; the presence of the risk-free securities would lower that weighted average.

d. The comment is not correct. Although the respective standard deviations and expected returns for the two securities under consideration are identical, the correlation coefficients between each security and the original portfolio are unknown, making it impossible to draw the conclusion stated. For instance, if the correlation between the original portfolio and XYZ stock is smaller than that between the original portfolio and ABC stock, replacing ABC stocks with XYZ stocks would result in a lower standard deviation for the portfolio as a whole. In such a case, XYZ socks would be the preferred investment, assuming all other factors are equal.

e. Grace clearly expressed the sentiment that the risk of loss was more important to her than the opportunity for return. Using variance (or standard deviation) as a measure of risk in her case has a serious limitation because standard deviation does not distinguish between positive and negative price movements.

CFA 4

Answer:

a. Restricting the portfolio to 20 stocks, rather than 40 to 50, will very likely increase the risk of the portfolio, due to the reduction in diversification. Such an increase might be acceptable if the expected return is increased sufficiently.

b. Hennessy could contain the increase in risk by making sure that he maintains reasonable diversification among the 20 stocks that remain in his portfolio. This entails maintaining a low correlation among the remaining stocks. As a practical matter, this means that Hennessy would need to spread his portfolio among many industries, rather than concentrating in just a few.

CFA 5

Answer:

Risk reduction benefits from diversification are not a linear function of the number of issues in the portfolio. (See Figures 6.1 and 6.2 in the text.) Rather, the incremental benefits from additional diversification are most important when the portfolio is least diversified. Restricting Hennessy to 10 issues, instead of 20 issues, would increase the risk of his portfolio by a greater amount than reducing the size of the portfolio from 30 to 20 stocks.

CFA 6

Answer:

The point is well taken because the committee should be concerned with the volatility of the entire fund. Since Hennessy's portfolio is only one of six well-diversified portfolios, and is smaller than the average, the concentration in fewer issues might have a minimal effect on the diversification of the total fund. Hence, unleashing Hennessy to do stock picking may be advantageous.

CFA 7

Answer:

a. Systematic risk refers to fluctuations in asset prices caused by macroeconomic factors that are common to all risky assets; hence systematic risk is often referred to as market risk. Examples of systematic risk factors include the business cycle, inflation, monetary policy, and technological changes.

Firm-specific risk refers to fluctuations in asset prices caused by factors that are independent of the market, such as industry characteristics or firm characteristics. Examples of firm-specific risk factors include litigation, patents, management, and financial leverage.

b. Trudy should explain to the client that picking only the five best ideas would most likely result in the client holding a much more risky portfolio. The total risk of a portfolio, or portfolio variance, is the combination of systematic risk and firm-specific risk.

The systematic component depends on the sensitivity of the individual assets to market movements, as measured by beta. Assuming the portfolio is well-diversified, the number of assets will not affect the systematic risk component of portfolio variance. The portfolio beta depends on the individual security betas and the portfolio weights of those securities.

On the other hand, the components of firm-specific risk (sometimes called nonsystematic risk) are not perfectly positively correlated with each other and, as more assets are added to the portfolio, those additional assets tend to reduce portfolio risk. Hence, increasing the number of securities in a portfolio reduces firm-specific risk. For example, a patent expiration for one company would not affect the other securities in the portfolio. An increase in oil prices might hurt an airline stock but aid an energy stock. As the number of randomly selected securities increases, the total risk (variance) of the portfolio approaches its systematic variance.

Investment Opportunity Set

23 20.37 19.939999999999991 20.18 22.5 23.338218689827791 26.68 9 10.200000000000001 10.89 11.4 12.6 12.879765273325008 13.8

Standard Deviation (%)

Expected Return (%)

Investment Opportunity Set

23 20.37 19.939999999999987 20.18 22.5 23.338218689827787 26.68 9 10.200000000000001 10.89 11.4 12.6 12.879765273325015 13.8

Standard Deviation (%)

Expected Return (%)

Corr = -1 0.25 0.20500000000000004 0.16000000000000003 0.11499999999999998 7.0000000000000021E-2 2.5000000000000012E-2 8.4519768540102867E-8 2.000000000000015E-2 6.5000000000000002E-2 0.11000000000000006 0.1550000000000003 0.2 0.05 5.5000000000000014E-2 6.0000000000000032E-2 6.5000000000000002E-2 7.0000000000000021E-2 7.5000000000000011E-2 7.7777787167338269E-2 8.0000000000000043E-2 8.5000000000000048E-2 9.0000000000000024E-2 9.5000000000000043E-2 0.1 Corr = -0.5 0.25 0.21569654610123024 0.18330302779823371 0.15402921800749414 0.13 0.11456439237389601 0.11111110641635506 0.11135528725660059 0.12134661099511602 0.14177446878757829 0.16889345754054541 0.2 0.05 5.5000000000000014E-2 6.0000000000000032E-2 6.5000000000000002E-2 7.0000000000000021E-2 7.5000000000000011E-2 7.7777787167338269E-2 8.0000000000000043E-2 8.5000000000000048E-2 9.0000000000000024E-2 9.5000000000000043E-2 0.1 Corr = 0 0.25 0.22588713996153034 0.20396078054371167 0.18500000000000025 0.17 0.16007810593582122 0.15713483362426706 0.15620499351813358 0.15882380174268593 0.16763054614240239 0.18172781845386249 0.2 0.05 5.5000000000000014E-2 6.0000000000000032E-2 6.5000000000000002E-2 7.0000000000000021E-2 7.5000000000000011E-2 7.7777787167338269E-2 8.0000000000000043E-2 8.5000000000000048E-2 9.0000000000000024E-2 9.5000000000000043E-2 0.1 Corr = 0.5 0.25 0.23563743335896392 0.22271057451320089 0.21148285982556628 0.20223748416156745 0.19525624189766683 0.19245008159828239 0.19078784028338913 0.18901058171435828 0.19000000000000003 0.19371370627810516 0.2 0.05 5.5000000000000014E-2 6.0000000000000032E-2 6.5000000000000002E-2 7.0000000000000021E-2 7.5000000000000011E-2 7.7777787167338269E-2 8.0000000000000043E-2 8.5000000000000048E-2 9.0000000000000024E-2 9.5000000000000043E-2 0.1 Corr = 1 0.25 0.24500000000000025 0.24000000000000021 0.23500000000000001 0.23 0.22500000000000001 0.22222221283266191 0.22 0.21500000000000027 0.21000000000000021 0.20500000000000004 0.2 0.05 5.5000000000000014E-2 6.0000000000000032E-2 6.5000000000000002E-2 7.0000000000000021E-2 7.5000000000000011E-2 7.7777787167338269E-2 8.0000000000000043E-2 8.5000000000000048E-2 9.0000000000000024E-2 9.5000000000000043E-2 0.1

6-1

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Sheet1

sM	s(e)	b	TOTAL Variance	Corr Coeff
0.2	0.3	1.65	0.1989	0.7399
0.2	0.33	1.5	0.1989	0.6727
0.2	0.33	1.5	0.1989	0.6727

Sheet2

Sheet3

(A)(B)(C)(D)(E)(F)(G)

Col. BCol. B



Col. CCol. F

Severe recession0.05-40-2.0-51.22621.44131.07

Mild recession0.25-14-3.5-25.2635.04158.76

Normal growth0.40176.85.833.6413.46

Boom0.30339.921.8475.24142.57

11.2445.86

21.12

Scenario

Deviation

from

Expected

Return

Squared

Deviation

Expected Return =Variance =

Standard Deviation =

Rate of

ReturnProbability

Sheet1

2. Risk reduction benefits from diversification are not a linear function of the number of issues in the portfolio. (See Figures 6.1 and 6.2 in the text.) Rather, the incremental benefits from additional diversification are most important when the portfolio is least diversified. Restricting Hennessy to 10 issues, instead of 20 issues, would increase the risk of his portfolio by a greater amount than reducing the size of the portfolio from 30 to 20 stocks.

3. The point is well taken because the committee should be concerned with the volatility of the entire portfolio. Since Hennessy's portfolio is only one of six well-diversified portfolios, and is smaller than the average, the concentration in fewer issues might have a minimal effect on the diversification of the total fund. Hence, unleashing Hennessy to do stock picking may be advantageous.

4. In the regression of the excess return of Stock ABC on the market, the square of the correlation coefficient is 0.296, which indicates that 29.6% of the variance of the excess return of ABC is explained by the market (systematic risk).

Sheet2

(A)	(B)	(C)	(D)	(E)	(F)	(G)
Scenario	Probability	Rate of Return	Deviation from Expected Return	Squared Deviation
Col. B	Col. B
×	´
Col. C	Col. F
Severe recession	0.05	-40	-2.0	-51.2	2621.44	131.07
Mild recession	0.25	-14	-3.5	-25.2	635.04	158.76
Normal growth	0.40	17	6.8	5.8	33.64	13.46
Boom	0.30	33	9.9	21.8	475.24	142.57
Expected Return =	11.2	Variance =	445.86
Standard Deviation =	21.12

Sheet3

(A)(B)(C)(D)(E)(F)

Col. CCol. B

StockBond



FundFundCol. DCol. E

Severe recession0.05-51.2-14716.835.84

Mild recession0.25-25.210-252-63.00

Normal growth0.405.8317.46.96

Boom0.3021.8-10-218-65.40

Covariance =-85.6

Deviation from

Mean Return

ScenarioProbability

Sheet1

Sheet2

(A)	(B)	(C)	(D)	(E)	(F)	(G)	(A)	(B)	(C)	(D)	(E)	(F)
Scenario	Probability	Rate of Return	Col. B	Deviation from Expected Return	Squared Deviation	Col. B	Deviation from
´	´	Mean Return
Col. C	Col. F	Scenario	Probability	Col. C	Col. B
Severe recession	0.05	-40	-2	-51.2	2621.44	131.07	Stock	Bond	´	´
Mild recession	0.25	-14	-3.5	-25.2	635.04	158.76	Fund	Fund	Col. D	Col. E
Normal growth	0.4	17	6.8	5.8	33.64	13.46	Recession	0.3	-24	10	-240	-72
Boom	0.3	33	9.9	21.8	475.24	142.57	Normal	0.4	3	0	0	0
Expected Return =	11.2	Variance =	445.86	Boom	0.3	20	-10	-200	-60
Standard Deviation =	21.12	Covariance = -132
(A)	(B)	(C)	(D)	(E)	(F)
Deviation from
Mean Return
Scenario	Probability	Col. C	Col. B
Stock	Bond	´	´
Fund	Fund	Col. D	Col. E
Severe recession	0.05	-51.2	-14	716.8	35.84
Mild recession	0.25	-25.2	10	-252	-63.00
Normal growth	0.40	5.8	3	17.4	6.96
Boom	0.30	21.8	-10	-218	-65.40
Covariance =	-85.6
(A)	(B)	(C)	(D)	(E)	(F)	(G)
Scenario	Probability	Rate of Return	Col. B	Deviation from Expected Return	Squared Deviation	Col. B
´	´
Col. C	Col. F
Severe recession	0.05	-40	-2	-51.2	2621.44	131.07
Mild recession	0.25	-14	-3.5	-25.2	635.04	158.76
Normal growth	0.4	17	6.8	5.8	33.64	13.46
Boom	0.3	33	9.9	21.8	475.24	142.57
Expected Return =	11.2	Variance =	445.86
Standard Deviation =	21.12

Sheet3

Col. BCol. B



Col. CCol. F

Severe recession0.10-0.37-0.037-0.4650.21620.0216

Mild recession0.20-0.11-0.022-0.2050.04200.0084

Normal growth0.350.140.0490.0450.00200.0007

Boom0.350.300.1050.2050.04200.0147

0.0950.0454

0.2132

Expected Return =Variance =

Standard Deviation =

ScenarioProbability

Stock

Rate of

Return

Deviation

from

Expected

Return

Squared

Deviation

Sheet1

(A)	(B)	(C)	(D)	(E)	(F)	(G)
Scenario	Probability	Stock Rate of Return	Deviation from Expected Return	Squared Deviation
Col. B	Col. B
´	´
Col. C	Col. F
Severe recession	0.10	-0.37	-0.037	-0.465	0.2162	0.0216
Mild recession	0.20	-0.11	-0.022	-0.205	0.0420	0.0084
Normal growth	0.35	0.14	0.049	0.045	0.0020	0.0007
Boom	0.35	0.30	0.105	0.205	0.0420	0.0147
Expected Return =	0.095	Variance =	0.0454
Standard Deviation =	0.2132

Sheet2

(A)	(B)	(C)	(D)	(E)	(F)	(G)	(A)	(B)	(C)	(D)	(E)	(F)
Scenario	Probability	Rate of Return	Col. B	Deviation from Expected Return	Squared Deviation	Col. B	Deviation from
´	´	Mean Return
Col. C	Col. F	Scenario	Probability	Col. C	Col. B
Severe recession	0.05	-40	-2	-51.2	2621.44	131.07	Stock	Bond	´	´
Mild recession	0.25	-14	-3.5	-25.2	635.04	158.76	Fund	Fund	Col. D	Col. E
Normal growth	0.4	17	6.8	5.8	33.64	13.46	Recession	0.3	-24	10	-240	-72
Boom	0.3	33	9.9	21.8	475.24	142.57	Normal	0.4	3	0	0	0
Expected Return =	11.2	Variance =	445.86	Boom	0.3	20	-10	-200	-60
Standard Deviation =	21.12	Covariance = -132
(A)	(B)	(C)	(D)	(E)	(F)
Deviation from
Mean Return
Scenario	Probability	Col. C	Col. B
Stock	Bond	´	´
Fund	Fund	Col. D	Col. E
Severe recession	0.05	-51.2	-14	716.8	35.84
Mild recession	0.25	-25.2	10	-252	-63
Normal growth	0.4	5.8	3	17.4	6.96
Boom	0.3	21.8	-10	-218	-65.4
Covariance =	-85.6
(A)	(B)	(C)	(D)	(E)	(F)	(G)
Scenario	Probability	Rate of Return	Col. B	Deviation from Expected Return	Squared Deviation	Col. B
´	´
Col. C	Col. F
Severe recession	0.05	-40	-2	-51.2	2621.44	131.07
Mild recession	0.25	-14	-3.5	-25.2	635.04	158.76
Normal growth	0.4	17	6.8	5.8	33.64	13.46
Boom	0.3	33	9.9	21.8	475.24	142.57
Expected Return =	11.2	Variance =	445.86
Standard Deviation =	21.12

Sheet3

Col. CCol. B

StockBond



FundFundCol. DCol. E

Severe recession0.1-0.465-0.1220.056730.00567

Mild recession0.2-0.2050.119-0.024395-0.0049

Normal growth0.350.0450.0490.0022050.00077

Boom0.350.205-0.082-0.01681-0.0059

-0.036Covariance = -0.0043

Deviation from

Mean Return

ScenarioProbability

Expected return =

Sheet1

(A)	(B)	(C)	(D)	(E)	(F)	(G)
Scenario	Probability	Rate of Return	Col. B	Deviation from Expected Return	Squared Deviation	Col. B
´	´
Col. C	Col. F
Severe recession	0.1	-40	-2	-51.2	2621.44	131.07
Mild recession	0.2	-14	-3.5	-25.2	635.04	158.76
Normal growth	0.35	17	6.8	5.8	33.64	13.46
Boom	0.35	33	9.9	21.8	475.24	142.57
Expected Return =	11.2	Variance =	445.86
Standard Deviation =	21.12

Sheet2

(A)	(B)	(C)	(D)	(E)	(F)	(G)	(A)	(B)	(C)	(D)	(E)	(F)
Scenario	Probability	Rate of Return	Col. B	Deviation from Expected Return	Squared Deviation	Col. B	Deviation from
´	´	Mean Return
Col. C	Col. F	Scenario	Probability	Col. C	Col. B
Severe recession	0.1	-40	-2	-51.2	2621.44	131.07	Stock	Bond	´	´
Mild recession	0.2	-14	-3.5	-25.2	635.04	158.76	Fund	Fund	Col. D	Col. E
Normal growth	0.35	17	6.8	5.8	33.64	13.46	Recession	0.3	-24	10	-240	-72
Boom	0.35	33	9.9	21.8	475.24	142.57	Normal	0.4	3	0	0	0
Expected Return =	11.2	Variance =	445.86	Boom	0.3	20	-10	-200	-60
Standard Deviation =	21.12	Covariance = -132
(A)	(B)	(C)	(D)	(E)	(F)
Deviation from
Mean Return
Scenario	Probability	Col. C	Col. B
Stock	Bond	´	´
Fund	Fund	Col. D	Col. E
Severe recession	0.1	-0.465	-0.122	0.05673	0.005673
Mild recession	0.2	-0.205	0.119	-0.024395	-0.004879
Normal growth	0.35	0.045	0.049	0.002205	0.00077175
Boom	0.35	0.205	-0.082	-0.01681	-0.0058835
Expected return =	-0.036	Covariance =	-0.00431775
(A)	(B)	(C)	(D)	(E)	(F)	(G)
Scenario	Probability	Rate of Return	Col. B	Deviation from Expected Return	Squared Deviation	Col. B
´	´
Col. C	Col. F
Severe recession	0.05	-40	-2	-51.2	2621.44	131.07
Mild recession	0.25	-14	-3.5	-25.2	635.04	158.76
Normal growth	0.4	17	6.8	5.8	33.64	13.46
Boom	0.3	33	9.9	21.8	475.24	142.57
Expected Return =	11.2	Variance =	445.86
Standard Deviation =	21.12

Sheet3

% in stocks% in bondsExp. ReturnStd dev.Sharpe Ratio

0.001.000.090.230.15

0.200.800.100.200.23

0.31420.68580.10890.19940.2701Minimum Variance Portfolio

0.400.600.110.200.29

0.600.400.130.230.32

0.64660.35340.12880.2333820.3162Tangency Portfolio

0.800.200.140.270.31

1.000.000.150.320.30

Sheet1

E(rs)	15%
E(rb)	9%	% in stocks	% in bonds	Exp. Return	Std dev.	Sharpe Ratio
Sigma(s)	32%	0.00	1.00	0.09	0.23	0.15
Sigma(b)	23%	0.20	0.80	0.10	0.20	0.23
Corr(s,b)	0.15	0.3142	0.6858	0.1089	0.1994	0.2701	Minimum Variance Portfolio
Rf	5.50%	0.40	0.60	0.11	0.20	0.29
0.60	0.40	0.13	0.23	0.32
0.6466	0.3534	0.1288	0.233382	0.3162	Tangency Portfolio
0.80	0.20	0.14	0.27	0.31
1.00	0.00	0.15	0.32	0.30

W(risky portfolio)0.880786822=

(c)/

(risky portfolio)

Proportion of stocks in complete portfolio

W(s) = W(risky portfolio)*% in stock of the risky portfolio

=0.569541021

Proportion of bonds in complete portfolio

W(b) = W(risky portfolio)*% in bonds of the risky portfolio

=0.311245801

Sheet1

E(rs)	15%
E(rb)	9%	% in stocks	% in bonds	Exp. Return	Std dev.	Sharpe Ratio
Sigma(s)	32%	0.00	1.00	0.09	0.23	0.15
Sigma(b)	23%	0.20	0.80	0.10	0.20	0.23
Corr(s,b)	0.15	0.3142	0.6858	0.1089	0.1994	0.2701	Minimum Variance Portfolio
Rf	5.50%	0.40	0.60	0.11	0.20	0.29
0.60	0.40	0.13	0.23	0.32
0.6466	0.3534	0.1288	0.2334	0.3162	Tangent Line Portfolio
0.80	0.20	0.14	0.27	0.31
1.00	0.00	0.15	0.32	0.30
11.b
s(c)	0.2055599546	= (12% - 5.5%)/Sharpe ratio(risky portfolio)
W(risky portfolio)	0.8807868217	= s(c)/s(risky portfolio)
Proportion of stocks in complete portfolio
W(s) = W(risky portfolio)*% in stock of the risky portfolio
=	0.5695410207
Proportion of bonds in complete portfolio
W(b) = W(risky portfolio)*% in bonds of the risky portfolio
=	0.311245801

Annual returns from Table 2Continuously compounded ratesExcess returns

Year

Large

Stock

Long-

Term T-

BondsT-Bills

Large

Stock

Long-

Term T-

BondsT-Bills

Large

Stock

Long-

Term T-

Bonds

198931.3419.498.3827.2617.818.0519.219.76

1990-3.207.137.84-3.256.897.55-10.80-0.66

199130.6618.395.6026.7416.885.4521.2911.43

19927.717.793.507.437.503.443.994.06

19939.8715.482.909.4114.392.866.5511.53

19941.29-7.183.911.28-7.453.84-2.55-11.29

199537.7131.675.6032.0027.515.4526.5522.06

199623.07-0.815.2020.76-0.815.0715.69-5.88

199733.1715.085.2528.6514.055.1223.538.93

199828.5813.524.8525.1412.684.7420.407.94

199921.04-8.744.6919.10-9.154.5814.51-13.73

2000-9.1020.275.88-9.5418.465.71-15.2512.74

2001-11.894.213.86-12.664.123.79-16.450.34

2002-22.1016.791.63-24.9715.521.62-26.5913.90

200328.692.381.0225.222.351.0124.211.34

200410.887.711.1910.337.431.189.146.24

20054.916.502.984.796.302.941.863.36

200611.78-1.214.8111.14-1.224.706.44-5.92

20073.5310.254.673.479.764.56-1.105.20

2008-38.491.341.55-48.601.331.54-50.14-0.21

Average3.534.06

SD19.648.88

Corr(stocks,bonds)0.13

StocksBondsMeanSD

0.014.068.88

0.10.94.018.47

0.20.83.958.55

0.30.73.909.10

0.40.63.8510.05

0.50.53.7911.29

0.60.43.7412.74

0.70.33.6914.34

0.80.23.6316.04

0.90.13.5817.81

1.003.5319.64

Min-Var0.13380.86623.998.44

Weights inPortfolio

Sheet1

Sheet2

Annual returns from Table 2	Continuously compounded rates	Excess returns
Year	Large Stock	Long-Term T-Bonds	T-Bills	Large Stock	Long-Term T-Bonds	T-Bills	Large Stock	Long-Term T-Bonds
1989	31.34	19.49	8.38	27.26	17.81	8.05	19.21	9.76
1990	-3.20	7.13	7.84	-3.25	6.89	7.55	-10.80	-0.66
1991	30.66	18.39	5.60	26.74	16.88	5.45	21.29	11.43
1992	7.71	7.79	3.50	7.43	7.50	3.44	3.99	4.06
1993	9.87	15.48	2.90	9.41	14.39	2.86	6.55	11.53
1994	1.29	-7.18	3.91	1.28	-7.45	3.84	-2.55	-11.29
1995	37.71	31.67	5.60	32.00	27.51	5.45	26.55	22.06
1996	23.07	-0.81	5.20	20.76	-0.81	5.07	15.69	-5.88
1997	33.17	15.08	5.25	28.65	14.05	5.12	23.53	8.93
1998	28.58	13.52	4.85	25.14	12.68	4.74	20.40	7.94
1999	21.04	-8.74	4.69	19.10	-9.15	4.58	14.51	-13.73
2000	-9.10	20.27	5.88	-9.54	18.46	5.71	-15.25	12.74
2001	-11.89	4.21	3.86	-12.66	4.12	3.79	-16.45	0.34
2002	-22.10	16.79	1.63	-24.97	15.52	1.62	-26.59	13.90
2003	28.69	2.38	1.02	25.22	2.35	1.01	24.21	1.34
2004	10.88	7.71	1.19	10.33	7.43	1.18	9.14	6.24
2005	4.91	6.50	2.98	4.79	6.30	2.94	1.86	3.36
2006	11.78	-1.21	4.81	11.14	-1.22	4.70	6.44	-5.92
2007	3.53	10.25	4.67	3.47	9.76	4.56	-1.10	5.20
2008	-38.49	1.34	1.55	-48.60	1.33	1.54	-50.14	-0.21
Average	3.53	4.06
SD	19.64	8.88
Corr(stocks,bonds)	0.13
Weights in	Portfolio
Stocks	Bonds	Mean	SD
0.0	1	4.06	8.88
0.1	0.9	4.01	8.47
0.2	0.8	3.95	8.55
0.3	0.7	3.90	9.10
0.4	0.6	3.85	10.05
0.5	0.5	3.79	11.29
0.6	0.4	3.74	12.74
0.7	0.3	3.69	14.34
0.8	0.2	3.63	16.04
0.9	0.1	3.58	17.81
1.0	0	3.53	19.64
Min-Var	0.1338	0.8662	3.99	8.44

Sheet3

SUMMARY OUTPUT

Regression Statistics

Multiple R0.37

R Square0.14

Adjusted R Square0.12

Standard Error8.32

Observations60.00

Coefficients

Standard

Errort StatP-value

Intercept(1.65) 1.08 (1.52) 0.13

S&P 5000.87 0.28 3.05 0.00

Sheet1

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.37
R Square	0.14
Adjusted R Square	0.12
Standard Error	8.32
Observations	60.00
Coefficients	Standard Error	t Stat	P-value
Intercept	(1.65)	1.08	(1.52)	0.13
S&P 500	0.87	0.28	3.05	0.00

Sheet2

Sheet3

y = 1.0 + 2.0 x

-2

-1

-0.5

0.5

Market Return, Percent

Generic

Return,

Percent

Sheet: Sheet1

Sheet: Sheet2

Sheet: Sheet3

Sheet: Sheet4

Sheet: Sheet5

Sheet: Sheet6

Sheet: Sheet7

Sheet: Sheet8

Sheet: Sheet9

Sheet: Sheet10

Sheet: Sheet11

Sheet: Sheet12

Sheet: Sheet13

Sheet: Sheet14

Sheet: Sheet15

Sheet: Sheet16

Market Return

Generic Return

Int.

1.0

Slope

2.0

AlphaBetaE(r) - rfVARSD

S&P-0.612316.25414.0316

Google3.17921.39162.3271163.244312.7767

regression

摘要輸出

迴歸統計
R 的倍數	0.4391106443
R 平方	0.192818158
調整的 R 平方	0.1766745211
標準誤	11.5932403983
觀察值個數	52

ANOVA
	自由度	SS	MS	F	顯著值
迴歸	1	1605.3001022264	1605.3001022264	11.9439107723	0.0011274181
殘差	50	6720.1611466516	134.403222933
總和	51	8325.461248878

	係數	標準誤	t 統計	P-值	下限 95%	上限 95%	下限 95.0%	上限 95.0%
截距	3.1791979619	1.6264881358	1.9546395033	0.0562285545	-0.087699539	6.4460954627	-0.087699539	6.4460954627
X 變數 1	1.3915903302	0.4026596643	3.4559963502	0.0011274181	0.5828246085	2.200356052	0.5828246085	2.200356052

Problem 6.24

6.24
Note: Google monthly returns available only as of September 2004; 52 observations up to December 2008.

	Return (from Yahoo)		from French	Excess Returns
Date	Google	S&P	T-bills	Google	S&P	E^2		S&P	G
Aug-00	47.10	1.29	0.11	46.99	1.18	1778.56			3.1791979619	Alpha
Sep-00	-4.54	4.46	0.11	-4.65	4.35	192.75			1.3915903302	Beta
Oct-00	5.94	3.01	0.15	5.79	2.86	1.86		16.25	163.2443382133	Variance
Nov-00	1.47	-2.24	0.16	1.31	-2.40	2.17		4.03	12.7767107744	Standard Deviation
Dec-00	-3.90	2.09	0.16	-4.06	1.93	98.56		4.03	12.7767107744
Jan-01	-3.98	-1.83	0.16	-4.14	-1.99	20.66		-0.61	2.33
Feb-01	21.88	-1.87	0.21	21.67	-2.08	456.99		-0.1518738742	0.1821381405	Sharpe
Mar-01	26.03	3.22	0.21	25.82	3.01	340.39			11.5932403983	Standard Error
Apr-01	6.09	0.15	0.24	5.85	-0.09	7.84
May-01	-2.17	3.82	0.23	-2.40	3.59	112.02			0.2742285895	Information Ratio
Jun-01	-0.61	-0.93	0.24	-0.85	-1.17	5.75
Jul-01	10.65	0.80	0.3	10.35	0.50	41.90
Aug-01	17.59	-2.36	0.29	17.30	-2.65	317.33
Sep-01	8.81	4.39	0.27	8.54	4.12	0.14		Alpha	Beta	E(r) - rf	VAR	SD	Sharpe Ratio	SD(e)
Oct-01	2.46	-0.19	0.31	2.15	-0.50	0.11	S&P			-0.6123	16.2541	4.0316	-0.1519
Nov-01	4.29	2.41	0.32	3.97	2.09	4.46	Google	3.1792	1.3916	2.3271	163.2443	12.7767	0.1821	11.59
Dec-01	-16.19	0.57	0.35	-16.54	0.22	400.98
Jan-02	7.55	1.65	0.34	7.21	1.31	4.86
Feb-02	7.16	1.26	0.37	6.79	0.89	5.67
Mar-02	-11.04	-3.01	0.36	-11.40	-3.37	97.78
Apr-02	12.78	0.26	0.43	12.35	-0.17	88.48
May-02	-7.81	0.45	0.4	-8.21	0.05	131.20
Jun-02	-2.09	2.18	0.4	-2.49	1.78	66.38
Jul-02	6.17	2.70	0.42	5.75	2.28	0.35
Aug-02	18.53	3.16	0.41	18.12	2.75	123.76
Sep-02	1.77	1.99	0.41	1.36	1.58	16.11
Oct-02	-5.02	0.77	0.42	-5.44	0.35	83.00
Nov-02	8.91	1.51	0.4	8.51	1.11	14.35
Dec-02	-10.38	-1.96	0.44	-10.82	-2.40	113.54
Jan-03	1.94	1.16	0.38	1.56	0.78	7.34
Feb-03	2.89	4.42	0.43	2.46	3.99	39.45
Mar-03	5.63	3.39	0.44	5.19	2.95	4.40
Apr-03	4.98	-1.46	0.41	4.57	-1.87	15.96
May-03	-2.43	-3.13	0.4	-2.83	-3.53	1.21
Jun-03	1.03	1.28	0.4	0.63	0.88	14.24
Jul-03	10.10	3.88	0.42	9.68	3.46	2.85
Aug-03	24.63	1.35	0.32	24.31	1.03	387.90
Sep-03	-1.98	-3.87	0.32	-2.30	-4.19	0.12
Oct-03	-0.22	-1.13	0.34	-0.56	-1.47	2.87
Nov-03	-18.39	-6.04	0.27	-18.66	-6.31	170.43
Dec-03	-16.50	-2.58	0.21	-16.71	-2.79	256.17
Jan-04	-6.52	-0.90	0.13	-6.65	-1.03	70.47
Feb-04	30.38	4.77	0.17	30.21	4.60	425.58
Mar-04	2.00	1.51	0.17	1.83	1.34	10.28
Apr-04	-10.14	-8.35	0.17	-10.31	-8.52	2.64
May-04	-10.01	-0.90	0.17	-10.18	-1.07	140.83
Jun-04	-2.21	1.55	0.15	-2.36	1.40	55.96
Jul-04	-13.55	-9.42	0.12	-13.67	-9.54	12.79
Aug-04	-10.28	-16.52	0.15	-10.43	-16.67	92.08
Sep-04	-18.48	-6.96	0.08	-18.56	-7.04	142.56
Oct-04	5.01	0.99	0.02	4.99	0.97	0.22
Nov-04	10.04	-8.22	0.09	9.95	-8.31	335.84
						134.40
						11.59

Comparison The Fama-French market factor (Mkt) is better diversified than the SP 500, containing about 10 times as many stocks. However, the additional stocks in Mkt are relatively small, and hence the performance of the value weighted portfolios is expected to be quite similar, and the correlation of the excess returns very high. Notice also that a sample of 81 annual returns with as large a SD as we have here is still quite small.l comparing the continuously compounded excess returns we see that the difference between the two portfolios is indeed quite small, and the correlation coefficient between their returns is 0.99, almost perfect. The SD of the two portfolios is indistinguishable. Both deviate from the normal distribution as seen from the negative skew and positive kurtosis (that is, they have fat tails -- although not large.) Accordingly, the VaR (5% percentile) of the two is smaller than what is expected from a normal with same mean and SD. This is also indicated by the lower minimum excess return for the period. The serial correlation is also small and indistinguishable across the portfolios. As a result of all this, we expect the risk premium of the two portfolios to be similar, as we find from the sample. Notice that the SD of the 81-year average for the SP500 is somewhat higher (because of the small positive serial correlation) than 20.46/sqrt(81) = 2.27, and almost identical for the Mkt. This SD is far larger than the difference in the average of the two portfolios (0.02%). Notice also that the excess return of both portfolios have a small negative correlation with the risk-free rate. Since we expect the risk-free rate to be highly correlated with the rate of inflation, this suggests that equities are not a perfect hedge against inflation. More rigorous analysis of this point is important, but beyond the scope of this question.

Analysis The bond portfolio is less risky, its SD is less than half that of the stock portfolio. Yet, as the portfolio table shows, mixing 0.87% of bonds with 13% (riskier) stocks would have produced a portfolio less risky than bonds. In the particular sample of these 20 years, the average return on the less risky portfolio (bonds) was higher than that of the riskier portfolio (stocks). This is exactly what is meant by 'risk,' expectation are not going to always be realized.

Comment The last 5 years have marked the end of GM as we know it, ending in bankruptcy in 2009. This development is reflected in the negative intercept (–1.65% per month). Obviously, this development was not widely expected, or the stock price would have plunged far earlier.

Analysis (1) Five-year betas. The beta of Ford has been quite aggressive, representing the sensitivity of sales to economic conditions. The beta of GM is much lower than that of Ford, representing the large part of the finance arm of the company. The Beta of Toyota is defensive for U.S, investors, reflecting the large part of Toyota's business abroad. (2) Five-year alpha estimates. The large negative alpha values of Ford and GM reflect the greater than expected deterioration of these companies. Toyota's positive alpha is due to the pre-crisis years, when its performance was stellar, better than expected. (3) Beta over two-year (first and last) subperiods. As a first pass we note that large standard deviation of the beta estimates. Non of the subperiod estimates deviates from the overall period estimate by more than two standard deviation. That is, the t-statistic of deviation from the overall period is not significant for any of the companies subperiod beta estimates. Looking beyond the aforementioned observation, the differences can be attributed to different alpha values during the subperiods. The case of Toyota is most revealing: The alpha estimate for the first two years is positive and for the last two years negative (both large). Following a good performance in the "normal" years prior to the crisis, Toyota surprised investors with a negative perofrmance, beyond what could be expected from the index. This suggests that a beta of around 0.5 is more reliable. The shift of the intercepts from positive to negative when the index moved to largely negative returns, explains why the line is steeper when estimated for the overall period. Draw a line in the positive qudrant for the index with a slope of 0.5 and positive intercept. Then draw a line with similar slope in the negative quadrant of the index with a negative intercept. You can see that a line that reconciles the observations for both quadrants will be steeper. The same logic explains part of the behavior of subperiod betas for Ford and GM.

Problem 6.24

SP 500 index and 26-week moving average

Sheet1

Relative strength FSRBX vs. SP 500

Analysis It appears that FSRBX (fund investing in large financial firms) is consistently falling relative to the SP 500, with some semblence of recovery toward the end of 2008. However, this recovery doesn't yet show a clear trend.

SUMMARY OUTPUT: Regression of Google on S&P 500 (excess returns)

Regression Statistics

Multiple R0.4391

R Square0.1928

Adjusted R Square0.1767

Standard Error11.5932

Observations52.0000

Coefficients

Standard

Error t Stat P-value

Intercept3.1792 1.6265 1.9546 0.0562

S&P 5001.3916 0.4027 3.4560 0.0011

GoogleSpy

Google1.00

S&P 5000.44 1.00

Sheet1

SUMMARY OUTPUT: Regression of Google on S&P 500 (excess returns)
Regression Statistics
Multiple R	0.4391
R Square	0.1928
Adjusted R Square	0.1767
Standard Error	11.5932
Observations	52.0000
Coefficients	Standard Error	t Stat	P-value
Intercept	3.1792	1.6265	1.9546	0.0562
S&P 500	1.3916	0.4027	3.4560	0.0011
Google	Spy
Google	1.00
S&P 500	0.44	1.00

Sheet2

Sheet3

s(e)b

TOTAL

VarianceCorr Coeff

0.20.31.650.19890.7399

0.20.331.50.19890.6727