EVALUATION OF PORTFOLIO PERFORMANCE
1. The two major factors would be: (1) attempt to derive risk-adjusted returns that exceed a naive buy-and-hold policy and (2) completely diversify – i.e., eliminated all unsystematic risk from the portfolio. A portfolio manager can do one or both of two things to derive superior risk-adjusted returns. The first is to have superior timing regarding market cycles and adjust your portfolio accordingly. Alternatively, one can consistently select undervalued stocks. As long as you do not make major mistakes with the rest of the portfolio, these actions should result in superior risk-adjusted returns.
2. Treynor (1965) divided a fund’s excess return (return less risk-free rate) by its beta. For a fund not completely diversified, Treynor’s “T” value will understate risk and overstate performance. Sharpe (1966) divided a fund’s excess return by its standard deviation. Sharpe’s “S” value will produce evaluations very similar to Treynor’s for funds that are well diversified. Jensen (1968) measures performance as the difference between a fund’s actual and required returns. Since the latter return is based on the CAPM and a fund’s beta, Jensen makes the same implicit assumptions as Treynor – namely, that funds are completely diversified. The information ratio (IR) measures a portfolio’s average return in excess of that of a benchmark, divided by the standard deviation of this excess return.
3. For portfolios with R2 values noticeably less than 1.0, it would make sense to compute both measures. Differences in the rankings generated by the two measures would suggest less-than-complete diversification by some funds – specifically, those that were ranked higher by Treynor than by Sharpe.
4. Jensen’s alpha () is found from the equation Rjt – RFRt == j + j[Rmt – RFRt] +ejt. The aj indicates whether a manager has superior (j > 0) or inferior (j < 0) ability in market timing or stock selection, or both. As suggested above, Jensen defines superior (inferior) performance as a positive (negative) difference between a manager’s actual return and his CAPM-based required return. For poorly diversified funds, Jensen’s rankings would more closely resemble Treynor’s. For well-diversified funds, Jensen’s rankings would follow those of both Treynor and Sharpe. By replacing the CAPM with the APT, differences between funds’ actual and required returns (or “alphas”) could provide fresh evaluations of funds.
5. The Information Ratio (IR) is calculated by dividing the average return on the portfolio less a benchmark return by the standard deviation of the excess return. The IR can be viewed as a benefit-cost ratio in that the standard deviation of return can be viewed as a cost associated in the sense that it measures the unsystematic risk taken on by active management. Thus IR is a cost-benefit ratio that assesses the quality of the investor’s information deflated by unsystematic risk generated by the investment process.
6. Since the return for selectivity is the difference between overall performance and the required return for risk, if the overall performance exceeds the required return for risk, the portfolio experiences a positive return for selectivity. In the example, the required return would have to be less than -0.50 in order to experience a positive selectivity value. Common sense tells us that a negative required return for assuming risk is not realistic.
7. A high R2 value of .95 implies that the portfolio is highly diversified and, thus, the diversification term will be minimal. By definition, if we have a selectivity value of a positive 2.5 percent and a minimal diversification term, net selectivity will be a positive value.
8(a). CFA Examination I (1991)
The returns of a well-diversified portfolio (within an asset class) are highly correlated with the returns of the asset class itself. Over time, diversified portfolios of securities within an asset class tend to produce similar returns. In contrast, returns between different asset classes are often much less correlated, and over time, different asset classes are very likely to produce quite different returns. This expected difference in returns arising from differences in asset class exposures (i.e., from differences in asset allocation) is, thus, the key performance variable.
8(b). Three reasons why successful implementation of asset allocation decisions is more difficult in practice than in theory are:
A. Transaction Costs – investing or rebalancing a portfolio to reflect a chosen asset allocation is not cost-free; expected benefits are reduced by the costs of implementation.
B. Changes in Economic and Market Factors – changing economic backgrounds, changing market price levels and changing relationships within and across asset classes all act to reduce the optimality of a given allocation decision and to create requirements for eventual rebalancing. Changes in economic and market factors change the expected risk/reward relationships of the allocation on a continuing basis.
C. Changes in Investor Factors – the passage of time often gives rise to changes in investor needs, circumstances or preferences which, in turn, give rise to the need to reallocate, with the attendant costs of doing so.
In summary, even the “perfect” asset allocation is altered by the very act of implementation, due to transaction costs and/or changes in the original economic/market conditions and, as time passes, changes in the investor’s situation. These impediments to successful implementation are inherent in the process, mandating ongoing monitoring of the relevant input factors. In practice, the fact of change in one or more of these factors is a “given;” constant attention of the degree and the importance of the effects required.
9. The difference by which a manager’s overall actual return beats his/her overall benchmark return is termed the total value-added return and decomposes into an allocation effect and a selection effect. The former effect measures differences in weights assigned by the actual and benchmark portfolios to stocks, bonds and cash times the respective differences between market-specific benchmark returns and the overall benchmark return. The latter effect focuses on the market-specific actual returns less the corresponding market-specific benchmark returns times the weights assigned to each market by the actual portfolio. Of course, the foregoing analysis implicitly assumes that the actual and benchmark market-specific portfolios (e.g., stocks) are risk-equivalent. If this is not true the analysis would not be valid.
10. CFA Examination III (2001)
Benchmark Explain two different weaknesses of using each of the benchmarks to measure the performance of the portfolio.
Index • A market index may exhibit survivorship bias; firms that have gone out of business are removed from the index resulting in a performance measure that overstates the actual performance had the failed firms been included.
• A market index may exhibit double counting that arises because of companies owning other companies and both being represented in the index.
• It is often difficult to exactly and continually replicate the holdings in the market index without incurring substantial trading costs.
• The chosen index may not be an appropriate proxy for the management style of the managers.
• The chosen index may not represent the entire universe of securities (e.g., S&P 500Index represents 65-70 percent of U.S. equity market capitalization).
• The chosen index may have a large capitalization bias (e.g., S&P 500 has a large capitalization bias).
• The chosen index may not be investable. There may be securities in the index that cannot be held in the portfolio.
Benchmark Normal Portfolio • This is the most difficult performance measurement method to develop and calculate.
• The normal portfolio must be continually updated, requiring substantial resources.
• Consultants and clients are concerned that managers who are involved in developing and calculating their benchmark portfolio may produce an easily-beaten normal portfolio making their performance appear better than it actually is.
Median of the Manager Universe • It can be difficult to identify a universe of managers appropriate for the investment style of the plan’s managers.
• Selection of a manager universe for comparison involves some, perhaps much, subjective judgment.
• Comparison with a manager universe does not take into account the risk taken in the portfolio.
• The median of a manager universe does not represent an “investable” portfolio, meaning a portfolio manager may not be able to invest in the median manager portfolio.
• Such a benchmark may be ambiguous. The names and weights of the securities constituting the benchmark are not clearly delineated.
• The benchmark is not constructed prior to the start of an evaluation period; it is not specified in advance.
• A manager universe may exhibit survivorship bias; managers that have gone out of business are removed from the universe resulting in a performance measure that overstates the actual performance had those managers been included.
The Sharpe ratio is calculated by dividing the portfolio risk premium, (i.e., actual portfolio return minus risk-free return), by the portfolio standard deviation of return.
Sharpe Ratio = (Rp – Rf)/p
Where: Rp = Actual portfolio return
Rf = Risk-free return
p = Standard deviation of portfolio return
The Treynor measure is calculated by dividing the portfolio risk premium (i.e., actual portfolio return minus risk-free return), by the portfolio beta.
Treynor measure = (Rp – Rf)/p
Where: p = Portfolio beta
Jensen’s alpha is calculated by subtracting the market premium, adjusted for risk by the portfolio’s beta, from the actual portfolio’s excess return (risk premium). It can be described as the difference in return earned by the portfolio compared to the return implied by the Capital Asset Pricing Model or Security Market Line.
p = Rp – Rf – p(Rm- Rf)
or p = Rp – [Rf +p(Rm – Rf)]
The Sharpe ratio assumes that the relevant risk is total risk and measures excess return per unit of total risk.
The Treynor measure assumes that the relevant risk is systematic risk and measures excess return per unit of systematic risk.
Jensen’s alpha assumes that the relevant risk is systematic risk and measures excess return at a given level of systematic risk.
11. CFA Examination III (1981)
11(a). The basic procedure in portfolio evaluation is to compare the return on a managed portfolio to the return expected on an unmanaged portfolio having the same risk, via use of the CAPM. That is, expected return (Ep) is calculated from:
Ep = Ef + p(Em – Ef)
Where Ef is the risk free rate, Em is the unmanaged portfolio or the market return and p is the beta coefficient or systematic risk of the managed portfolio. The benchmark of performance then is the unmanaged portfolio. The typical proxy for this unmanaged portfolio is some aggregate stock market index such as the S&P 500.
11(b). The benchmark error often occurs because the unmanaged portfolio used in the evaluation process is not “optimized.” That is, market indices, such as the S&P 500, chosen as benchmarks are not on the evaluator’s ex ante mean/variance efficient frontier. Benchmark error may also occur because of an error in the estimation of the risk free return. Together, these two sources of error will cause the implied Security Market Line (SML) to be mispositioned.
11(c). The main ingredients are that the true risk free rate is lower than the measured risk free rate and the true market is above the measured market. The result is under performance relative to the true SML rather than superior performance relative to the measured SML.
11(d). The fact that the portfolio manager has been judged superior based on several different benchmarks should not make me feel any more comfortable because all the benchmarks could have errors, which means that you are simply computing different errors. It is shown by Roll that if the various indexes are perfectly correlated, a proportionate difference will exist in the error. Notably, all of these indexes are very highly correlated.
11(e). All of the discussion by Roll is not directed against the CAPM theory, but is concerned with a measurement problem involved in finding a valid benchmark, i.e., an unmanaged portfolio that is mean/variance efficient. The theory is correct and valid. The problem is implementing the theory in the real world where it is difficult to construct a true “market portfolio.”
12. When measuring the performance of an equity portfolio manager, overall returns can be related to a common total risk or systematic risk. Factors influencing the returns achieved by the bond portfolio manager are more complex. In order to evaluate performance based on a common risk measure (i.e., market index), four components must be considered that differentiate the individual portfolio from the market index. These components include: (1) a policy effect, (2) a rate anticipation effect, (3) an analysis effect, and (4) a trading effect. Decision variables involved include the impact of duration decisions, anticipation of sector/quality factors, and the impact of individual bond selection.
13. CFA Examination III (1982)
13(a). Yield-to-maturity. This is the expected return on the bond based upon the beginning price. Assuming no changes in the market, it is made up of the accrued coupon payments; an expected price change to amortize the difference between par and the beginning market price; and the “roll effect,” which is due to changes in yield-to-maturity due to the slope of the yield curve and the fact that the bond’s maturity declined during the holding period.
13(b). The interest rate effect is an analysis of what happened to the bond’s price due to a change in market interest rates during the period. Specifically, the analysis involves relating what should have happened to the price of the portfolio bond taking into account the change in yields for Treasury securities of a comparable maturity and assuming the same spread as at the beginning of the period.
13(c). The sector/quality effect examines what should have happened to returns based upon changes in sector/quality differentials during the period. You begin with a matrix of differential returns for the bonds in different sectors (corporates, utilities, financial, telephone) and quality (Aaa, Aa, Baa) relative to the returns for Treasury bonds of the same maturity. As an example, the matrix will indicate that the return difference for an Aa corporate bond during the period was one percent more or less than a similar maturity Treasury bond. Put another way, it indicates what happened to bonds of this quality and sector during this period relative to Treasury bonds.
13(d). The residual return is what is left of total return after taking account of yield to maturity, the interest rate effect, and the sector/quality effect. It is as follows:
Total Return = Yield to Maturity+Interest Rate Effect+Sector / Quality Effect + Residual Return
Answers to Problems
1. CFA Examination III (1985)
1(a). The risk adjusted returns of the two equity portfolios are computed as follows:
(Realized returns risk free rate)
Risk Adj Returns = + risk free rate
Good Samaritan Equity Portfolio:
(11.8% 7.8%)/1.20 + 7.8% = 3.3% + 7.8% = 11.1%
Mrs. Atkins’ Equity Portfolio:
(10.7% 7.8%)/1.05 + 7.8% = 2.8% + 7.8% = 10.6%
Both portfolios outperformed the S&P 500 both on an absolute basis and on a risk adjusted basis. The Good Samaritan portfolio outperformed Mrs. Atkins’ portfolio by more than a full percentage point before risk adjustment, but by only one half percentage point after risk adjustment. These differences are small enough to be within the range of normal statistical variation and are therefore not meaningful in judging performance.
1(b). Factors which could account for the differences in total account performance would include the following:
Different asset mixes between stocks, bonds, and short-term reserves. Clearly, a higher proportion of equity investments would have improved the total portfolio return for Mrs. Atkins.
The use of taxable bonds versus tax exempt bonds. Since Mrs. Atkins’ bonds were tax exempt whereas the Good Samaritan bonds were undoubtedly taxable, Mrs. Atkins’ portfolio return would be adversely affected unless an adjustment were made for after tax returns.
Mrs. Atkins’ portfolio is not diversified since it contains only eight equity issues (other than Merit Enterprises). This creates a higher potential for specific risk to affect the portfolio return in any given year.
The objectives and constraints under which the two portfolios are operating are probably quite different. The higher beta of the Good Samaritan portfolio suggests that it may have been managed with less restrictive constraints than Mrs. Atkins’ portfolio.
The relatively short time period (i.e., twelve months) is too short to make a truly meaningful evaluation of relative performance of the two portfolios. A complete market cycle would be more appropriate.
P 2 3
Q 4 2
R 5 5
S 1 1
Market 3 4
2(c). It is apparent from the rankings above that Portfolio Q was poorly diversified since Treynor ranked it #2 and Sharpe ranked it #4. Otherwise, the rankings are similar.
3. CFA Examination I (1994)
3(a). The Treynor measure (T) relates the rate of return earned above the risk free rate to the portfolio beta during the period under consideration. Therefore, the Treynor measure shows the risk premium (excess return) earned per unit of systematic risk:
where: Ri = average rate of return for portfolio i during the specified period
Rf = average rate of return on a risk free investment during the specified period
i = beta of portfolio i during the specified period.
Treynor Measure Performance Relative to the Market (S&P 500)
10% – 6%
T = = 6.7% Outperformed
Market (S&P 500)
12% – 6%
TM = = 6.0%
The Treynor measure examines portfolio performance in relation to the security market line (SML). Because the portfolio would plot above the SML, it outperformed the S&P 500 Index. Because T was greater than TM, 6.7 percent versus 6.0 percent, respectively, the portfolio clearly outperformed the market index.
The Sharpe measure (S) relates the rate of return earned above the risk free rate to the total risk of a portfolio by including the standard deviation of returns. Therefore, the Sharpe measure indicates the risk premium (excess return) per unit of total risk:
where: Ri = average rate of return for portfolio i during the specified period
Rf = average rate of return on a risk free investment during the specified period
i = standard deviation of the rate of return for portfolio i during the specified
Sharpe Measure Performance Relative to the Market (S&P 500)
10% – 6%
S = = 0.222% Underperformed
Market (S&P 500)
12% – 6%
SM= = 0.462%
The Sharpe measure uses total risk to compare portfolios with the capital market line (CML). The portfolio would plot below the CML, indicating that it underperformed the market. Because S was less than SM, 0.222 versus 0.462, respectively, the portfolio underperformed the market. I
3(b). The Treynor measure assumes that the appropriate risk measure for a portfolio is its systematic risk, or beta. Hence, the Treynor measure implicitly assumes that the portfolio being measured is fully diversified. The Sharpe measure is similar to the Treynor measure except that the excess return on a portfolio is divided by the standard deviation of the portfolio.
For perfectly diversified portfolios (that is, those without any unsystematic or specific risk), the Treynor and Sharpe measures would give consistent results relative to the market index because the total variance of the portfolio would be the same as its systematic variance (beta). A poorly diversified portfolio could show better performance relative to the market if the Treynor measure is used but lower performance relative to the market if the Sharpe measure is used. Any difference between the two measures relative to the markets would come directly from a difference in diversification.
In particular, Portfolio X outperformed the market if measured by the Treynor measure but did not perform as well as the market using the Sharpe measure. The reason is that Portfolio X has a large amount of unsystematic risk. Such risk is not a factor in determining the value of the Treynor measure for the portfolio, because the Treynor measure considers only systematic risk. The Sharpe measure, however, considers total risk (that is, both systematic and unsystematic risk). Portfolio X, which has a low amount of systematic risk, could have a high amount of total risk, because of its lack of diversification. Hence, Portfolio X would have a high Treynor measure (because of low systematic risk) and a low Sharpe measure (because of high total risk).
4(a). Portfolio MNO enjoyed the highest degree of diversification since it had the highest R2 (94.8%). The statistical logic behind this conclusion comes from the CAPM which says that all fully diversified portfolios should be priced along the security market line. R2 is a measure of how well assets conform to the security market line, so R2 is also a measure of diversification.
Fund Treynor Sharpe Jensen
ABC 0.975(4) 0.857(4) 0.192(4)
DEF 0.715(5) 0.619(5) -0.053(5)
GHI 1.574(1) 1.179(1) 0.463(1)
JKL 1.262(2) 0.915(3) 0.355(2)
MNO 1.134(3) 1.000(2) 0.296(3)
Only GHI and MNO have significantly positive alphas at a 95% level of confidence.
5(a). (Information ratio) IRj = j/u where u = standard error of the regression
IRA = .058/.533 = 0.1088
IRB = .115/5.884 = 0.0195
IRC = .250/2.165 = 0.1155
5(b). Annualized IR = (T)1/2(IR)
Annualized IRA = (52)1/2(0.1088) = 0.7846
Annualized IRB = (26)1/2(0.0195) = 0.0994
Annualized IRC = (12)1/2(0.1155) = 0.4001
5(c). The higher the ratio, the better. Based upon the answers to part a, Manager C would be rated the highest followed by Managers A and B, respectively. However, once the values are annualized, the ranking change. Specifically, based upon the annualized IR, Manger A is rated the highest, followed by C and B. (In both cases, Manager C is rated last). Based upon the Grinold-Kahn standard for “good” performance (0.500 or greater), only Manager A meets that test.
6(b). Overall Performance =Ra RFR = .15 – .05 = .10
6(c). Selectivity = Ra Rx (a) = .15 – .11 = .04
6(d). Risk = [Rx(a) – RFR] = .11 – .05 = .06
where Rx(a) = .05 + 1.2 (.10 – .05) = .11
7(a). Overall performance (Fund 1) = 26.40% – 6.20% = 20.20%
Overall performance (Fund 2) = 13.22% – 6.20% = 7.02%
7(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
7(c)(i). Selectivity1 = 20.2% – 12.85% = 7.35%
Selectivity2 = 7.02% – 8.61% = -1.59%
7(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%
7(c)(iii). Net Selectivity = Selectivity – Diversification
Net Selectivity1 = 7.35% – 1.99% = 5.36%
Net Selectivity2 = -1.59% – 1.57% = -3.16%
7(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.
8. CFA Examination III (1995)
8(a). The following briefly describes one strength and one weakness of each manager.
1. Manager A
Strength. Although Manager A’s one-year total return was slightly below the EAFE Index return (-6.0 percent versus -5.0 percent, respectively), this manager apparently has some country/security return expertise. This large local market return advantage of 2.0 percent exceeds the 0.2 percent return for the EAFE Index.
Weakness. Manager A has an obvious weakness in the currency management area. This manager experienced a marked currency return shortfall compared with the EAFE Index of 8.0 percent versus -5.2 percent, respectively.
2. Manager B
Strength. Manager B’s total return slightly exceeded that of the index, with a marked positive increment apparent in the currency return. Manager B had a -l.0 percent currency return versus a -5.2 percent currency return on the EAFE index. Based on this outcome, Manager B’s strength appears to be some expertise in the currency selection area.
Weakness. Manager B had a marked shortfall in local market return. Manager B’s country/security return was -l.0 percent versus 0.2 percent on the EAFE Index. Therefore, Manager B appears to be weak in security/market selection ability.
8(b). The following strategies would enable the Fund to take advantage of the strengths of the two managers and simultaneously minimize their weaknesses.
1. Recommendation: One strategy would be to direct Manager A to make no currency bets relative to the EAFE Index and to direct Manager B to make only currency decisions, and no active country or security selection bets.
Justification: This strategy would mitigate Manager A’s weakness by hedging all currency exposures into index like weights. This would allow capture of Manager A’s country and stock selection skills while avoiding losses from poor currency management. This strategy would also mitigate Manager B’s weakness, leaving an index-like portfolio construct and capitalizing on the apparent skill in currency management.
2. Recommendation: Another strategy would be to combine the portfolios of Manager A and Manager B. with Manager A making country exposure and security selection decisions and Manager B managing the currency exposures created by Manager A’s decisions (providing a “currency overlay”).
Justification: This recommendation would capture the strengths of both Manager A and Manager B and would minimize their collective weaknesses.
9(a)(i). .6(-5) + .3(-3.5) + .1(0.3) = -4.02%
9(a)(ii). .5(-4) + .2(-2.5) + .3(0.3) = -2.41%
9(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.
9(b)(i). [.5(-4 + 5) + .2(-2.5 + 3.5) + .3(.3 -.3)] = 0.70%
9(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. CFA Examination III (June 1985)
10(a). Overall, both managers added value by mitigating the currency effects present in the Index. Both exhibited an ability to “pick stocks” in the markets they chose to be in (Manager B in particular). Manager B used his opportunities not to be in stocks quite effectively (via the cash/bond contribution to return), but neither of them matched the passive index in picking the country markets in which to be invested (Manager B in particular).
Manager A Manager B
Strengths Currency Management Currency Management
Stock Selection Stock Selection
Use of Cash/Bond Flexibility
Weaknesses Country Selection Country Selection
(to a limited degree)
10(b). The column reveals the effect on performance in local currency terms after adjustment for movements in the U.S. dollar and, therefore, the effect on the portfolio. Currency gains/losses arise from translating changes in currency exchange rates versus the U.S. dollar over the measuring period (3 years in this case) into U.S. dollars for the U.S. pension plan. The Index mix lost 12.9% to the dollar reducing what would otherwise have been a very favorable return from the various country markets of 19.9% to a net return of only 7.0%.
11. I = E + U
11% = 10 + U
1% = U
10-Year 5-Year 25 Year
AA Bonds A Bonds B Bonds
M 1.50 1.00 3.00
S -0.40 -0.60 -1.20
B .25 .50 .75
C 1.35 .90 2.55
I 11.00 11.00 11.00
R 12.35 11.90 13.65
M = .2 x 5 + .1 x 5 = 1.50 (10 Yr.)
M = .2 x 5 + .1 x 0 = 1.00 ( 5 Yr.)
M = .2 x 5 + .1 x 20 = 3.00 (25 Yr.)
12. CFA Examination III (1994)
12(a). Evaluation begins with selection of the appropriate benchmark against which to measure the firms’ results:
Firm A. The Aggregate Index and “Managers using the Aggregate Index” benchmark are appropriate here, because Firm A maintains marketlike sector exposures. Performance has been strong; Firm A outperformed the Aggregate Index by 50 basis points and placed in the first quartile of managers’ results.
Firm B. Firm B does not use mortgages; therefore, the Government/Corporate for both index and universe comparisons would be the appropriate benchmark. Although Firm B produced the highest absolute performance (9.3 percent), it did not perform up to either the Index (9.5 percent) or other managers investing only in the Government/Corporate sectors (third quartile).
Firm C. Like Firm A, this firm maintains broad market exposures and should be compared with the Aggregate Index for both index and universe comparisons. Performance has been good (30 basis points ahead of the index and in the second quartile of manager results) but not as good as Firm A’s showing during this relatively short measurement period.
12(b). Firm A does not show an observable degree of security selection skill ( 10 basis points); nor does it appear to be managing in line with its stated marketlike approach. Some large nonmarketlike bets are driving return production (e.g., duration bets, +100 basis points; yield curve bets, +30 basis points; and sector-weighting bets, -70 basis points) and account for its +50 basis-point better-than-benchmark total return.
Firm C’s ability to anticipate shifts in the yield curve correctly is confirmed by the analysis (its +30 basis points accounts for all of its observed better than benchmark outcome). In addition, its claim n to maintain marketlike exposures is also confirmed (e.g., the nominal differences from benchmark in the other three attribution areas).
12(c). Firm C produced the best results because its style and its expertise were confirmed by the analysis; Firm A’s were not.
13(i). Dollar-Weighted Return
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+
Solving for r, the internal rate of return or DWRR is 2.74%
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 +
Solving for r, the internal rate of return or DWRR is 2.98%.
13(ii). Time-weighted return
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%
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
= (1.1658) 1/5 – 1= 1.03094 – 1 = .03094 = 3.094%
EV – (1 – DW)(Contribution)
13(iii). Dietz approximation method = – 1
BV + (DW)(Contribution)
In this case, DW = (91 – 45.5)/91 = 0.50
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
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