# Finance calculations

Ch 7

5. For all values of r1,2:

E(Rport) = (.6 x .10) + (.4 x .15) = .12

5(a).

5(b).

5(c).

5(d).

5(e).

5(f).

5(g).

6(a). E(Rp) = (1.00 x .12) + (.00 x .16) = .12

6(b). E(Rp) = (.75 x .12) + (.25 x .16) = .13

6(c). E(Rp) = (.50 x .12) + (.50 x .16) = .14

6(d). E(Rp) = (.25 x .12) + (.75 x .16) = .15

6(e). E(Rp) = (.05 x .12) + (.95 x .16) = .158

7. DJIA S&P Russell Nikkei

Month (R1) (R2) (R3) (R4) R1-E(R1) R2-E(R2) R3-E(R3) R4-E(R4)

1 .03 .02 .04 .04 .01667 .00333 .01333 .00833

2 .07 .06 .10 -.02 .05667 .04333 .07333 -.05167

3 -.02 -.01 -.04 .07 -.03333 -.02667 -.06667 .03883

4 .01 .03 .03 .02 -.00333 .01333 .00333 -.01167

5 .05 .04 .11 .02 .03667 .02333 .08333 -.01167

-.06 -.04 -.08 .06 -.07333 -.05667 -.10667 .02833

## Sum .08 .10 .16 .19

7(a).

7(b). 1 = (.01667)2+ (.05667)2+ (-.03333)2+ (-.00333)2+ (.03667)2 + (-.07333)2

= .00028 + .00321 + .00111 + .00001 + .00134 + .00538 = .01133

1 = (.00226)1/2 = .0476

2 = (-.00333)2 + (.04333)2 + (-.02667)2 + (.01333)2 + (.02333)2 + (-.05667)2

= .00001 + .00188 + .00071 + .00018 + .00054 + .00321 = .00653

2 = (.01306)1/2 = .0361

3 = (.01333)2 + (.07333)2 + (-.06667)2 + (.00333)2 + (.08333)2 + (-.106672)2

= .00018 + .00538 + .00444 + .00001 + .00694 + .01138 = .02833

3 = (.00567) 1/2 = .0753

4 = (.00833)2+(-.05167)2+ (.03833)2+ (-.01167)2+(-.01167)2 + (.02833)2

= .00007 + .00267 + .00147 + .00014 + .00014 .00080 = .00529

4 = (.001058)1/2 = .0325

7(c).

7(d). Correlation equals the covariance divided by each standard deviation.

Correlation (DJIA, S&P) = 0.001678/ [(0.0476)(0.0361)] = .9765

Correlation (S&P, R2000) = 0.002604/ [(0.0361)(0.0753)] = .9579

Correlation (S&P, Nikkei) = -0.001054/ [(0.0361)(0.0325)] = -0.8984

Correlation (R2000, Nikkei) = -0.002054/ [(0.0753)(0.0325)] =-0.8393

7(e).

The resulting correlation coefficients suggest a strong positive correlation in returns for the S&P 500 and the Russell 2000 combinations (.96), preventing any meaningful reduction in risk (.05518) when they are combined. Since the S&P 500 and Nikkei have a negative correlation (-.90), their combination results in a lower standard deviation (.009875).

Ch 8

2. E(Ri) = RFR + i(RM – RFR)

= .10 + i(.14 – .10)

= .10 + .04i

2a.

Stock Beta (Required Return) E(Ri) = .10 + .04i

### N 1.25 .10 + .04(1.25) = .10 + .05 = .150

D -.20 .10 + .04(-.20) = .10 – .008 = .092

2b.

#### Stock

Current Price Expected Price Expected Dividend

#### U

22

24

0.75

N

48

51

2.00

D

37

40

1.25

Stock Beta Required Estimated Evaluation

### U .85 .134 .1250 Overvalued

N 1.25 .150 .1042 Overvalued

D -.20 .092 .1149 Undervalued

If you believe the appropriateness of these estimated returns, you would buy stocks D and sell stocks U and N.

E(R)

N

14% U

*U’

*D’ * N’

D

-0.5 -0.2 0.5 .085 1.0 1.25

3a. Q: 4.8%/10.5% = 0.4571

R: 7%/14% = 0.5000

S: 1.6%.5% = 0.3200

T: 8.7%/18.5% = 0.4703

U: 3.2%/7.5% = 0.4267

3b. The CML slope, [E(RMKT ) – RFR ]/ σMKT , is the ratio of risk premium per unit of risk. Portfolio R has the highest ratio, 0.5000, of these five portfolios so it is most likely the market portfolio. Thus, the slope of the CML is 0.5; its intercept is 3%, the risk-free rate.

3c. The CML equation, based on the above analysis, is E(Rportfolio ) = 3% + (0.50) σportfolio . If the desired standard deviation is 7.0% the expected portfolio return is 6.5%:

E(Rportfolio ) = 3% + (0.50) (7%) = 6.5% . The answer is no, it is not possible to earn an expected return of 7% with a portfolio whose standard deviation is 7%.

3d. Using the CML equation, we set the expected portfolio return equal to 7% and solve for the standard deviation:

E(Rportfolio ) = 7% = 3% + (0.50) σportfolio 4% = (0.50) σportfolio σ = 4%/0.50 = 8%.

Thus, 8% is the standard deviation consistent with an expected return of 7%.

To find the portfolio weights with result in a risk of 8% and expected return of 7%, recall that the covariance between the risk-free asset and the market portfolio is zero. Thus, the portfolio standard deviation calculation simplifies to: σportfolio = wMKT (σMKT ) and the weight of the risk-free asset is 1 – wMKT .

Doing this, we have σportfolio = 8% = wMKT (14.0% ), so wMKT = 8%/14.0% = 0.5714 and wrisk-free asset = 1 – 0.5714 = 0.4286. As a check, the weighted average expected return should equal 7%:

0.5714 (10%) + 0.4286(3%) = 7.0% which it does. Remember to use the expected return of the market portfolio, 10%, in this calculation.

3e. To find the portfolio weights with result in a risk of 18.2%, recall that the covariance between the risk-free asset and the market portfolio is zero. Thus, the portfolio standard deviation calculation simplifies to: σportfolio = wMKT (σMKT ) and the weight of the risk-free asset is 1 – wMKT .

Doing this, we have σportfolio = 18.2% = wMKT (14.0% ), so wMKT = 18.2%/14.0% = 1.30; wrisk-free asset = 1 – (1.3) = -0.30. This portfolio is a borrowing portfolio; 30% of the funds will be borrowed (we will use margin) and 130% of the initial funds are invested in the market portfolio.

The expected return will be the weighted average of the risk-free and market portfolio returns:

1.30 (10%) + (-0.30) (3%) = 12.1% .

We can also use the CML equation to find the expected return:

E(Rportfolio ) = 3% + (0.50) σportfolio = 3% + (0.50)(18.2%) = 12.1%. Thus, both methods agree, as they should, on the expected portfolio return.

4. With a risk premium of 5% and risk-free rate of 4.5%, the security market line is:

E(return) = 4.5% + (5%)β. Information about the level of diversification of the portfolios is not given, nor is information about the market portfolio. But a portfolio’s beta is the weighted average of the betas of the securities held in the portfolio so the SML can be used to evaluate managers Y and Z.

4a. Expected return (Y) = 4.5% + (5%)β = 4.5% + (5%)(1.20) = 10.50%.

Expected return (Z) = 4.5% + (5%)β = 4.5% + (5%)(0.80) = 8.50%.

4b. Alpha is the difference between the actual return and the expected return based on portfolio risk:

Alpha of manager Y = actual return – expected return = 10.20% – 10.50% = -0.30%

Alpha of manager Z = actual return – expected return = 8.80% – 8.50% = 0.30%

4c. A positive alpha means the portfolio outperformed the market on a risk-adjusted basis; it would plot above the SML. A negative alpha means the opposite, that the portfolio underperformed the market on a risk-adjusted basis; it would plot below the SML.

In this case, manager Z outperformed the market portfolio on a risk-adjusted basis by 30 basis points (0.30%). Manager Y underperformed, returning 30 basis points less (-0.30%) than expected based upon the risk of Y’s portfolio.

5(a).

then COVi,m = (ri,m)(i)( m)

For Intel:

COV i,m = (.72)(.1210)(.0550) = .00479

For Ford:

COV i,m = (.33)(.1460)(.0550) = .00265

For Anheuser Busch:

COV i,m = (.55)(.0760)(.0550) = .00230

For Merck:

COV i,m = (.60)(.1020)(.0550) = .00337

5(b). E(Ri) = RFR + Bi(RM – RFR)

= .08 + Bi(.15 – .08)

= .08 + .07Bi

Stock Beta E(Ri) = .08 + .07Bi

Intel 1.583 0.1908

Ford .876 0.1413

Anheuser Busch .760 0.1332

Merck 1.114 0.1580

5(c). .20 *Intel

*AB

RM = .15 *Ford

.10 *Merck

RFR=.08

1.0 Beta

Intel, Ford, and Anheuser all have estimated return (given in part c) exceeding their expected returns (computed in part b); they are undervalued and are potential “buy” candidates. Merck is overvalued as its estimated return (10%) is less than the return required by the SML (15.8%); it is a potential candidate for selling.

6. Chelle General (R1 – E(R1) x

Year (R1) Index (RM) R1 – E(R1) RM – E(RM) RM – E(RM)

1 37 15 27.33 6 163.98

2 9 13 -.67 4 -2.68

3 -11 14 -20.67 5 -103.35

4 8 -9 -1.67 -18 30.06

5 11 12 1.33 3 3.99

4 9 -5.67 0 0.00

= 58 = 54 = 92.00

E(R1) = 9.67 E(M) = 9

6(a). The correlation coefficient can be computed as follows:

6(b). The standard deviations are: 15.5649% for Chelle Computer and 9.0554% for index, respectively.

6(c). Beta for Chelle Computer is computed as follows:

Ch 9

1(a). In general for the APT, E(Rq) = 0 + 1bq1 + 2bq2

For security J:

E(RJ) = 0.05 + 0.02×0.80 + 0.04×1.40

= 0.05 + 0.016 + 0.056

= .1220 or 12.20%

For Security L:

E(RL) = 0.05 +0.02×1.60 + 0.04×2.25

= 0.05 + 0.032 + 0.09

= .172 or 17.20%

1(b). Total return = dividend yield + capital gain yield

For security J, the dividend yield is \$0.75/\$22.50 = 0.033 or 3.33%

For security J, the expected capital gain is therefore 12.20%– 3.33% = 8.87%

Therefore:

The expected price for security J is \$22.50x(1.0887) = \$25.50

For security L, the dividend yield is \$0.75/\$15.00 = 0.05 or 5%

For security L the expected capital gain is therefore 17.20% – 5.00% = 12.20%

Therefore:

The expected price for security L is \$15.00x(1.1220) = \$16.83

The answer can be found using the holding period return:

For security J: [(P1 – \$22.50) + 0.75] / \$22.50 = 0.1220; solving for P1 , we obtain \$25.50.

For security l: [(P1 – \$15) + 0.75] / \$15 = 0.1720; solving for P1 , we obtain \$16.83.

2.

2(a).

 Coefficients Factor Risk Premia 3 -factor MKT SMB HML 1980-2009 1927-09 MSFT 0.966 -0.018 -0.388 MKT 7.11% 7.92% CSX 1.042 -0.043 0.370 SMB 1.50% 3.61% XRX 1.178 0.526 0.517 HML 5.28% 5.02% Expected excess returns: Using factor premium from: 1980-2009 1927-09 MSFT 4.79% 5.64% CSX 9.30% 9.95% XRX 11.89% 13.82%

Sample calculation:

MSFT, using 1980-2009 premia:

(0.966)(7.11%) + (-0.018)(1.50%) + (-0.388)(5.28%) = 4.79%

2b.

 coefficients Factor Risk Premia 4-factor MKT SMB HML MOM 1980-2009 1927-09 MSFT 0.777 -0.655 -0.186 -0.074 MKT 7.11% 7.92% CSX 0.593 -0.123 0.084 0.042 SMB 1.50% 3.61% XRX 0.804 0.100 0.496 -0.158 HML 5.28% 5.02% MOM 7.99% 9.79% Expected excess returns: Using factor premium from: 1980-2009 1927-09 MSFT 2.97% 2.13% CSX 4.81% 5.09% XRX 7.22% 7.67%

2(c). The excess returns for all the stocks for both periods seem moderately large. This is partly due to the fact that we are using out-of-sample numbers to do the estimating. That is, the regressions were estimated using 2005-2009 data, but the estimates were made using data from much longer periods.

2(d). No, we wouldn’t expect the factor betas to remain constant over time. The sensitivity of a particular company’s return to a specific factor will change as the character of the firm changes. For instance, growth companies don’t remain growth companies forever, but tend to mature. Thus their factor betas would change to reflect the slowdown in growth but increased stability of earnings.

3(a). RQRS = 4.5 +7.5×1.24

= 4.5 + 6.825

= 13.8%

RTUV = 4.5 + 7.5×0.91

= 4.5 + 6.825

= 11.325%

RWXY = 4.5 + 7.5×1.03

= 4.5 +7.725

= 12.225%

3(b). RQRS = 4.5 + 7.5×1.24 + (-0.3)x(-0.42) + 0.6×0.00

= 4.5 + 9.30 + 0.126 +0.00

= 13.926%

RTUV = 4.5 + 7.5×0.91 + (-0.3)x(0.54) + 0.6×0.23

= 4.5 + 6.825 – 0.162 + 0.138

= 11.301%

RWXY = 4.5 + 7.5×1.03 + (-0.3)x(-0.09) + 0.6×0.00

= 4.5 + 7.725 + 0.027 + 0.00

= 12.252%

3(c). Assuming that the factor loadings are significant the three factor model should be more useful to the extent that the non-market factors pick up movements in returns not captured by the market return. To be practical, however, the differences in the expected returns are small.

3(d). Because the factor loadings on MACRO2 are zero for two of the stocks, it appears that MACRO2 is not a systematic factor, i.e., one that generally affects all stocks. It may represent industry- or firm-specific factors.

4(a). E(RD) = 5.0 + 1.21 + 3.42 = 13.1%

E(RE) = 5.0 + 2. + 2.

Solving the second equation for 1 in terms of 2, we get:

1 = (10.4 – 2.62)/2.6 = (4.0 – 1.02)

Substituting that into the first equation:

1.2(4 – 2 )+ 3.42 = 8.1

Solving for 2, we find 2 = 1.5. Using this value we can determine from either equation that 1 is equal to 2.5.

4(b). Because neither stock pays a dividend, the total return is all due to price appreciation. Therefore for stock D:

P0x(1.131) = \$55

P0 = \$55/1.131

= \$48.63

And for stock E:

P0x(1.154) = \$36

P0 = \$36/1.154

= \$31.20

4(c). From part (a), the risk premium for factor 1 was 2.5%. The new risk factor is thus 2.5% + 0.25%, or 2.75%. The new expected returns are:

E(RD) = 5.0 + (1.2×2.75) +(3.4×1.5)

= 5.0 + 3.3 + 5.1

= 13.4%

E(RE) = 5.0 + (2.6×2.75) + (2.6×1.5)

= 5.0 + 7.15 + 3.9

= 16.05%

4(d). D: PD0(1 + 0.134) = \$55

PD0 = \$55/1.134

PD0 = \$48.50

E: PE0(1 + .1605) = \$36

PE0 = \$36/(1.1605)

PE0 = \$31.02

5(a). Because no stock pays a dividend, all return is due to price appreciation.

E(RA) = 1.1×0.04 + 0.8×0.02

= 0.044 + 0.016

= 0.06 or 6%

E(Price A) = \$30(1.06) = \$31.80

E(RB) = 0.7×0.04 + 0.6×0.02

= 0.28 + .012

= 0.04 or 4%

E(Price B) = \$30(1.04) = \$31.20

E(RC) = 0.3×0.04 + 0.4×0.02

= 0.12 + 0.008

= 0.02 or 2%

E(Price C) = \$30(1.02) = \$30.60

5(b). In order to create a riskless arbitrage investment, an investor would short 1 share of A and one share of C, and buy 2 shares of B. The weights of this portfolio are WA = -0.5, WB = +1.0, and WC = -0.5. The net investment is:

Short 1 share A = +\$30

Buy 2 shares B = – \$60

Short 1 share C = +\$30

Net investment = \$ 0

The risk exposure is:

Risk Exposure Factor 1 Factor 2

A (-0.5)x1.1 (-0.5)x0.8

B (+1.0)x0.7 (+1.0)x0.6

(-0.5)x0.3 (-0.5)x0.4

Net Risk Exposure 0 0

At the end of the period the profit is given by:

Profit = (\$30 – \$31.50) + 2x(\$35 – 30) + (\$30 – \$30.50)

= -\$1.50 + \$10 – \$0.50

= \$8

Ch 14

9.

Note: the lease sale price is \$100,000 at the end of year 3.

The analyst would recommend investing in the lease obligation since, for each dollar invested, the present value of the future cash flows generated by the lease is greater than the present value of the cash flow that will be realized when the stock is sold.

Present Value of Common Stock

Year Cash Flow PV @ 10% \$ PV

1 \$0 .909 \$ 0

2 \$0 .826 0

3 \$100,000 x (1.1)3 = \$133,100 .751 \$100,000

\$100,000

Present Value of Lease Cash Flow

Year Cash Flow PV @ 10% \$ PV

1 \$0 .909 \$ 0

2 \$15,000 .826 12,390

3 \$100,000 + \$25,000 = \$125,000 .751 93,875

\$106,265

Ratio of common stock PV for \$ 1 invested = 1.00

Ratio of lease PV for \$1 invested (\$106,265/\$100,000) = 1.06

11(a). Security Market Line

i. Fair-value plot. The following template shows, using the CAPM, the expected return, ER, of Stock A and Stock B on the SML. The points are consistent with the following equations:

ER on stock = Risk-free rate + Beta x (Market return – Risk-free rate)

ER for A = 4.5% + 1.2(14.5% – 4.5%)

= 16.5%

ER for B = 4.5% + 0.8(14.5% – 4.5%)

= 12.5%

ii. Analyst estimate plot. Using the analyst’s estimates, Stock A plots below the SML and Stock B, above the SML.

Expected Return

*Stock A

14.5% *Stock B

4.5%

0.8 1.2 Beta

11(b). Over vs. Undervalue

Stock A is overvalued because it should provide a 16.5% return according to the CAPM whereas the analyst has estimated only a 16.0% return.

Stock B is undervalued because it should provide a 12.5% return according to the CAPM whereas the analyst has estimated a 14% return.

12(a)(i) Return on Equity (ROE) = Profit Margin X Asset Turnover X Financial Leverage

OR

ROE = (Net Income/Revenue) X (Revenue/Assets) X (Assets/Equity)

(ii) ROE= 510/5140 X 5140/3100 X 3100/2200

9.9% X 1.66 X 1.41 = 23.16%

Acceptable variations:

Balance sheet averages may be used for assets and equity. The calculations become

ROE= 510/5140 X 5140/3025 X 3025/2150

9.9% X 1.70 X 1.41 = 23.73%

Beginning of period equity may also be used in calculating leverage as shown below:

ROE= 510/5140 X 5140/3100 X 3100/2100

9.9% X 1.66 X 1.48 = 24.32%

(iii) Sustainable Growth = ROE X Retention Ratio (RR)

where RR= 1 – Dividend payout ratio = 1 – .60/1.96 = .694

23.16% X .694 =16.07% (using end of period balances)

or 23.73% X .694 =16.47% (using average assets and equity)

or 24.32% X .694 =16.88% (using beginning equity)

12(b). If the problem were temporary, management could simply accumulate resources in anticipation of future growth. However, assuming this trend continues longer-term as the question stated, there are four alternative courses of action that management can take when actual growth falls below sustainable growth:

1. Acquire, merge with, or invest in another company (buy growth). Investing internally is acceptable if the core concepts of increasing earnings growth and using excess cash flow are discussed.

2. Return cash to shareholders by increasing the dividend or the dividend payout ratio

3. Return cash to shareholders by buying back stock

4. Reduce liabilities (decrease leverage or pay off debt)

Ch 15

3(a). Portfolio turnover is the dollar value of securities sold in a year divided by the average value of the assets:

Fund W: 37.2/289.4 = .1285 or 12.85%

Fund X: 569.3/653.7 = 0.8709 or 87.09%

Fund Y: 1,453.8/1,298.4 = 1.1197 or 111.97%

Fund Z: 437.1/5,567.3 = 0.0785 or 7.85%

(b) Passively managed funds will have low portfolio turnover ratios and should have low expenses ratios. On this basis, Funds W and Z are the most likely passively managed portfolios; X and Y are most likely to be actively managed.

(c) The tax cost ratio is compute as [1 – (1 + TAR)/(1+PTR)] x 100 where TAR represents tax-adjusted return and PTR is the pre-tax return. Our calculations are as follows:

Fund W: [1 – (1 + 0.0943)/(1+0.0998)] x 100 = 0.50%

Fund X: [1 – (1 + 0.0887)/(1+0.1065)] x 100 = 1.61%

Fund Y: [1 – (1 + 0.0934)/(1+0.1012)] x 100 = 0.71%

Fund Z: [1 – (1 + 0.0954)/(1+0.0983)] x 100 = 0.26%

The tax cost ratio represents the percentage of an investor’s assets that are lost to taxes on a yearly basis due to the trading strategy employed by the fund manager. Funds Z and W are the most tax-efficient (least assets lost to taxes) and Funds X and Y were the least tax-efficient.

6(a). EUpk = ERp – (p2/RTk)

Portfolios Ms. A Mr. B

1. 8 – (5/8) = 7.38 8 – (5/27) = 7.81

1. 9 – (10/8) = 7.75 9 – (10/27) = 8.63

1. 10 – (16/8) = 8.00 10 – (16/27) = 9.41

1. 11 – (25/8) = 7.88 11 – (25/27) = 10.07

6(b). The optimal portfolio is the one with the highest expected utility. Thus, portfolio 3 represents the optimal strategic allocation for Ms. A, while Portfolio 4 is the optimal allocation for Mr. B. Since Mr. B has a higher risk tolerance, he is able to pursue more volatile portfolios with higher expected returns.

6(c). For Ms. A: Portfolio 1 = Portfolio 2

8 – (5/RT) = 9 – (10/RT)

RT = 5

In other words, a risk tolerance factor of 5 would leave Ms. A indifferent between having Portfolio 1 or Portfolio 2 as her strategic allocation.

8 a) The table below shows that Manager A’s average return is less than the index while Manager B’s average exceeded that of the index. But performing several t-tests show that neither manager’s performance differed significantly from that of the index.

b) The table below shows the difference between Manager A’s performance and the index, as well as the difference between Manager B’s performance and the index. Manager A did the better job of limiting the client’s exposure to unsystematic risk as the difference between A’s returns and those of the index has a smaller standard deviation than that of the difference between B’s returns and those of the index.

 Period Manager A Manager B Index A minus Index B minus Index 1 12.80% 13.90% 11.80% 1.00% 2.10% 2 -2.10% -4.20% -2.20% 0.10% -2.00% 3 15.60% 13.50% 18.90% -3.30% -5.40% 4 0.80% 2.90% -0.50% 1.30% 3.40% 5 -7.90% -5.90% -3.90% -4.00% -2.00% 6 23.20% 26.30% 21.70% 1.50% 4.60% 7 -10.40% -11.20% -13.20% 2.80% 2.00% 8 5.60% 5.50% 5.30% 0.30% 0.20% 9 2.30% 4.20% 2.40% -0.10% 1.80% 10 19.00% 18.80% 19.70% -0.70% -0.90% Average 5.89% 6.38% 6.00% -0.11% 0.38% Std Dev 11.41% 11.77% 11.66% Std dev = tracking error 2.11% 3.00%

Ch 24

3(a). Using the above data, the arithmetic average return per year is 3.65%. On an annual compounded (geometric average) basis, the average annual return is 3.42%.

This latter answer is the same as if the annual return is computed using only the end points, shares were worth \$11.44 at the end of year 4 and were purchased for \$10, giving a compounded return of (\$11.44 / \$10).25 -1 = 3.42%

3(b). (12.30 / 10.00) (1/4) – 1 = 5.31%

3(c). Ignoring commission, shares were purchased \$10.69 and sold at 10.08, a return of -5.7%

3(d). Change is NAV is \$9.85 – \$11.25 = \$-1.40; the percentage change is \$-1.40 / 11.25

= -12.44%

4(a). Client 1 Client 2

.0100 x 5,000,000 = 50,000 .0100 x 5,000,000 = 50,000

.0075 x 5,000,000 = 37,500 .0075 x 5,000,000 = 37,500

.0060 x 10,000,000 = 60,000 .0060 x 10,000,000 = 60,000

.0040 x 7,000,000 = 28,000 .0040 x 77,000,000 = 308,000

27,000,000 175,500 97,000,000 455,500

4(b). 175,500/27,000,000 = .0065 or 0.65%

455,000/97,000,000 = .004696 = 0.47%

4(c). Costs of management do not increase at the same rate as the managed assets because substantial economies of scale exist in managing assets.

6(a). Beginning value = \$27.15 x 257.876 = \$7,001.33

#### Ending value = \$30.34 x 257.876 = 7,823.96

(\$7,823.96 – \$7,001.33) + 288.82

Return = = 15.87%

\$7,001.33

which can be computed on a per-share basis:

[(\$30.34 – 27.15) + 1.12] / \$27.15 = 0.1587 or 15.87%

6(b). Only the dividend and capital gain distribution is taxable; the shares are not yet sold so the change in NAV does not represent a taxable (realized) gain or loss:

[(\$30.34 – 27.15) + (1.12) (1 – 0.30)] / \$27.15

= 3.97 / \$27.15 = 0.1464 or 14.64%

6(c). The investor received a distribution of \$1.12 per share which, at the year-end NAV, purchases \$1.12/\$30.34 = 0.036915 shares. Since the investor owned 257.876 shares, he can purchase 9.519 shares (if there were no tax). If the distribution were taxalb,e the after-tax distribution of \$1.12 (1-.30) = \$0.784 could purchase 0.0258405 shares—for a total of 6.664 shares.

7.

 Year 1 Year 2 Stock Shares (000) price MV (000) Shares (000) price MV (000) A 100 \$45.25 \$4,525.00 100 \$48.75 \$4,875.00 B 225 \$25.38 \$5,710.50 225 \$24.75 \$5,568.75 C 375 \$14.50 \$5,437.50 375 \$12.38 \$4,642.50 D 115 \$87.13 \$10,019.95 115 \$98.50 \$11,327.50 E 154 \$56.50 \$8,701.00 154 \$62.50 \$9,625.00 F 175 \$63.00 \$11,025.00 175 \$77.00 \$13,475.00 G 212 \$32.00 \$6,784.00 212 \$38.63 \$8,189.56 H 275 \$15.25 \$4,193.75 275 \$8.75 \$2,406.25 I 450 \$9.63 \$4,333.50 450 \$27.45 \$12,352.50 J 90 \$71.25 \$6,412.50 90 \$75.38 \$6,784.20 K 87 \$42.13 \$3,665.31 87 \$49.63 \$4,317.81 L 137 \$19.88 \$2,723.56 0 \$27.88 \$0.00 M 0 \$17.75 \$0.00 150 \$19.75 \$2,962.50 Cash \$3,542.00 \$2,873.00 Total \$77,073.57 \$89,399.57 Expenses \$730,000.00 \$830,000.00 a. NAV = Sum of market values + cash divided by 5,430,000 shares number in 000 \$77,073.57 divided by 5,430 = \$14.19 Note: NAV is (market value of assets – liabilities) /# shares. Expenses are not included in this calculation b. NAV = Sum of market values + cash divided by 5,430,000 shares number in 000 \$89,399.57 divided by 5,430 = \$16.46 Percent change: 15.99% c. # shares = cash account / year 2 NAV = 174.502 shares d. Year 2 Dollars to Number of Shares (000) price MV (000) % holding be sold shares sold A 100 \$48.75 \$4,875.00 5.63% \$297,593.56 6104.48 B 225 \$24.75 \$5,568.75 6.44% \$339,943.41 13735.09 C 375 \$12.38 \$4,642.50 5.37% \$283,400.64 22891.81 D 115 \$98.50 \$11,327.50 13.09% \$691,485.34 7020.16 E 154 \$62.50 \$9,625.00 11.12% \$587,556.52 9400.90 F 175 \$77.00 \$13,475.00 15.57% \$822,579.12 10682.85 G 212 \$38.63 \$8,189.56 9.46% \$499,930.32 12941.50 H 275 \$8.75 \$2,406.25 2.78% \$146,889.13 16787.33 I 450 \$27.45 \$12,352.50 14.28% \$754,056.30 27470.17 J 90 \$75.38 \$6,784.20 7.84% \$414,140.35 5494.03 K 87 \$49.63 \$4,317.81 4.99% \$263,579.99 5310.90 L 0 \$27.88 \$0.00 0.00% \$0.00 0.00 M 150 \$19.75 \$2,962.50 3.42% \$180,845.32 9156.72 Total value, shares only \$86,526.57 100.0% \$5,282,000.00 Amount to liquidate: \$16.31 x 500,000 = \$8,155,000 Less cash: 8155000 minus 2873000 = \$5,282,000

Ch 25

5(a). Overall performance (Fund 1) = 26.40% – 6.20% = 20.20%

Overall performance (Fund 2) = 13.22% – 6.20% = 7.02%

5(b). E(Ri) = 6.20 + (15.71 – 6.20)

= 6.20 + (9.51)

Total return (Fund 1) = 6.20 + (1.351)(9.51) = 6.20 + 12.85 = 19.05%

where 12.85% is the required return for risk

Total return (Fund 2) = 6.20 + (0.905)(9.51) = 6.20 + 8.61 = 14.81%

where 8.61% is the required return for risk

5(c)(i). Selectivity1 = 20.2% – 12.85% = 7.35%

Selectivity2 = 7.02% – 8.61% = -1.59%

5(c)(ii).Ratio of total risk1 = 1/m = 20.67/13.25 = 1.56

Ratio of total risk2 = 2/m = 14.20/13.25 = 1.07

R1 = 6.20 + 1.56 (9.51) = 6.20 + 14.8356 = 21.04%

R2 = 6.20 + 1.07 (9.51) = 6.20 + 10.1757 = 16.38%

Diversification1 = 21.04% – 19.05% = 1.99%

Diversification2 = 16.38% – 14.81% = 1.57%

5(c)(iii). Net Selectivity = Selectivity – Diversification

Net Selectivity1 = 7.35% – 1.99% = 5.36%

Net Selectivity2 = -1.59% – 1.57% = -3.16%

5(d). Even accounting for the added cost of incomplete diversification, Fund 1’s performance was above the market line (best performance), while Fund 2 fall below the line.

6.

 a. Year Mgr X Return Mgr Y Return 1 -1.5 -6.5 2 -1.5 -3.5 3 -1.5 -1.5 4 -1.0 3.5 5 0.0 4.5 6 4.5 6.5 7 6.5 7.5 8 8.5 8.5 9 13.5 12.5 10 17.5 13.5 Average 4.5 4.5 Std Dev 6.90 6.63 Semi-dev 0.65 4.20 Semi-deviation considers only the returns that are below the average. b. Sharpe ratio: (average return minus risk-free rate) / standard deviation Mgr X: 0.435 Mgr Y: 0.452 Best performer c. Sortino ratio: (average return minus minimum acceptable return)/semi-deviation Mgr X: 4.602 Best performer Mgr Y: 0.714

d. The Sharpe and Sortino measures should provide the same performance ranking when the return distributions are symmetrical for the funds or managers under consideration. Asymmetric distributions, such as when one manager is hedging risk exposure or using a portfolio insurance strategy, the performance rankings should differ.

7.

7(a)(i). .6(-5) + .3(-3.5) + .1(0.3) = -4.02%

7(a)(ii). .5(-4) + .2(-2.5) + .3(0.3) = -2.41%

7(a)(iii). .3(-5) + .4(-3.5) + .3(0.3) = -2.81%

Manager A outperformed the benchmark fund by 161 basis points while Manager B beat the benchmark fund by 121 basis points.

7(b)(i). [.5(-4 + 5) + .2(-2.5 + 3.5) + .3(.3 -.3)] = 0.70%

7(b)(ii). [(.3 – .6) (-5 + 4.02) + (.4 – .3) (-3.5 + 4.02) + (.3 -.1)(.3 + 4.02)] = 1.21%

Manager A added value through her selection skills (70 of 161 basis points) and her allocation skills (71 of 161 basis points). Manager B added value totally through his allocation skills (121 of 121 basis points).

10(i). Dollar-Weighted Return

Manager L:

500,000 = -12,000/(1+r) – 7,500/(1+r)2- 13,500/(1+r)3 – 6,500/(1+r)4- 10,000/(1+r)5+

625,000/(1+r)5

###### Solving for r, the internal rate of return or DWRR is 2.75%

Manager M:

700,000 = 35,000/(1+r) + 35,000/(1+r)2+35,000/(1+r)3+35,000/(1+r)4+35,000/(1+r)5 +

625,000/(1+r)5

###### Solving for r, the internal rate of return or DWRR is 2.98%.

10(ii). Time-weighted return

Manager L:

Periods HPR

1 [(527,000 – 500,000) – 12,000]/500,000 = .03

2 [(530,000 – 527,000) – 7,500]/527,000 = -.0085

3 [(555,000 – 530,000) – 13,500]/530,000 = .0217

4 [(580,000 – 555,000) – 6,500]/555,000 = .0333

5 [(625,000 – 580,000) – 10,000]/580,000 = .0603

TWRR = [(1 + .03)(1 – .0085)(1 + .0217)(1 + .0333)(1 + .0603)]1/5 – 1

= (1.143) 1/5 – 1= 1.02712 – 1 = .02712 = 2.71%

Manager M:

Periods HPR

1 [(692,000 – 700,000) + 35,000]/700,000 = .03857

2 [(663,000 – 692,000) + 35,000]/692,000 = .00867

3 [(621,000 – 663,000) + 35,000]/663,000 = -.01056

4 [(612,000 – 621,000) + 35,000]/621,000 = .04187

5 [(625,000 – 612,000) + 35,000]/612,000 = .0784

TWRR = [(1 + .03857)(1 + .00867)(1 – .01056)(1 + .04187)(1 + .0784)]1/5 – 1

= (1.1646) 1/5 – 1= 1.03094 – 1 = .03094 = 3.094%

EV – (1 – DW)(Contribution)

10(iii). Dietz approximation method = – 1

BV + (DW)(Contribution)

In this case, DW = (91 – 45.5)/91 = 0.50

Manager L:

Periods HPY

1 [(527,000 – (1 -.50)(12,000)]/[500,000 + (.50)(12,000)] – 1

= (527,000 –6,000/(500,000 + 6,000) – 1 = 521,000/506,000 – 1 = .0296

2 (530,000 – (1 -.50)(7,500)]/[527,000 + (.50)(7,500)] – 1

= 526,250/530,750 – 1 = -.0085

3 (555,000 – (1 -.50)(13,500)]/[530,000 + (.50)(13,500)] – 1

= 548,250/536,750 – 1 = .0214

4 (580,000 – (1 -.50)(6,500)]/[555,000 + (.50)(6,500)] – 1

= 576,750/558,250 – 1 = .0331

5 (625,000 – (1 -.50)(10,000)]/[580,000 + (.50)(10,000)] – 1

= 620,000/585,00 – 1 = .0598

Manager M:

Periods HPY

1 [(692,000 – (1 -.50)(-35,000)]/[700,000 + (.50)(-35,000)] – 1

= (692,000 + 17,500/(700,000 – 17,500) – 1 = 709,500/682,500 – 1 = .0396

2 (663,000 – (1 -.50)(-35,000)]/[692,000 + (.50)(-35,000)] – 1

= 680,500/674,500 – 1 = .0089

3 (621,000 – (1 -.50)(-35,000)]/[663,000 + (.50)(-35,000)] – 1

= 638,500/645,500 – 1 = -.0108

4 (612,000 – (1 -.50)(-35,000)]/[621,000 + (.50)(-35,000)] – 1

= 629,500/603,500 – 1 = .0431

5 (625,000 – (1 -.50)(-35,000)]/[612,000 + (.50)(-35,000)] – 1

= 642,500/594,500 – 1 = .0807

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