CFA Lv2 组合

CFA 2级 组合 思维答题 例题，通过清晰的逻辑结构和丰富的例题展示，帮助考生构建全面而深入的理解体系。

CFA Lv2 组合

• M1 Exchange-Traded Funds: Mechanics and Applications
  • ETF mechanics
    • Primary market
      • creation
      • redemption
        • 例题
          • Assuming arbitrage costs are minimal, which of the following is most likely to occur when the share price of an ETF is trading at a premium to its intraday NAV? A. New ETF shares will be created by the ETF sponsor. B. Redemption baskets will be received by APs from the ETF sponsor. C. Retail investors will exchange baskets of securities that the ETF tracks for creation units
            • A is correct. When the share price of an ETF is trading at a premium to its intraday NAV and assuming arbitrage costs are minimal, APs will step in and take advantage of the arbitrage. Specifically, APs will step in and buy the basket of securities that the ETF tracks (the creation basket) and exchange it with the ETF provider for new ETF shares (a creation unit). These new shares received by APs can then be sold on the open market to realize arbitrage profits.
              • 超经典题，必须理解和掌握 低买高卖决定是redemption还是creation；只有AP才能入场
    • Secondary market
      • 例题
        • Which of the following statements regarding applications of ETFs in portfolio management is correct? A. Equity ETFs tend to be more active than fixed-income ETFs. B. The range of risk exposures available in the futures market is more diverse than that available in the ETF space. C. ETFs that have the highest trading volumes in their asset class category are generally preferred for tactical trading applications.
          • C is correct. ETFs that have the highest trading volumes in their asset class category are generally preferred for tactical trading applications.
            • 一般股票市场比较活跃，所以大家不买ETF，直接买股票就好了。但是债券市场流动性不好，不容易买到，所以大家通过买ETF来投资债券市场，所以债券ETF市场更活跃。 Future市场参与者不多，但是ETF市场参与者更多，更多元化，所以风险更分散。 TAA (tactical)的投资者发现短期机会，不一定直接购买底层资产，可能通过投资EFT获得收益，收益交易的频繁，volume也大。
  • Issues in ETF investing
    • Costs
      • Management Fees
        • Expense ratios
      • Trading costs
        • Commission
        • Bid-ask spread
          • Amount of orders
          • Amount of competition
          • Actual cost
          • less liquid
            • cannot matched quickly
              • creation/redemption fees
              • bid-ask spreads of underlying
              • compensation to market maker
              • 例题
                • The bid–ask spread for very liquid, high-volume ETFs will be least influenced by the: A. market maker's desired profit spread. B. creation/redemption fees and other direct costs. C. likelihood of receiving an offsetting ETF order in a short time frame.
                  • B is correct. ETF bid–ask spreads are generally less than or equal to the combination of the following: ■ ± Creation/redemption fees and other direct costs, such as brokerage and exchange fees ■ + Bid–ask spread of the underlying securities held by the ETF ■ + Compensation for the risk of hedging or carrying positions by liquidity providers (market makers) for the remainder of the trading day ■ + Market maker’s desired profit spread ■ − Discount related to the likelihood of receiving an offsetting ETF order in a short time frame For very liquid and high-volume ETFs, buyers and sellers are active throughout the trading day. Therefore, because most of these ETF trades are matched extremely quickly and never involve the creation/redemption process, the first three factors listed do not contribute heavily to their bid–ask spreads. So, creation/redemption fees and other direct costs are not likely to have much influence on these ETFs' bid–ask spreads.
                    • 这里要注意，很快就能匹配到，收益大概率不需要creation/redemption过程，所以less liquid的前三个基本不影响。
            • Market maker's desired profit spread
            • Discount related to receiving offsetting order
        • Premiums and discounts of NAV
        • 例题
          • A portfolio manager at DKIF wants to take an active view on the energy sector of the market and he considers the other three index-based ETFs in Exhibit 2
            • Base on information in Exhibit 2, which is the best ETF for the portfolio manager's tactical adjustment? A. ETF I B. ETF II C. ETF III
              • A is correct ETFs that have the highest trading volumes in their asset class category are preferred for tactical trading applications. Thematic ETFs hold stock passively but allow investors to take an active view on a market segment they believe will deliver strong returns. Therefore, as a thematic ETF with the highest trading volume, ETF I is most suited for the portfolio manager's tactical adjustment.
                • tactical adjustment只是短期偏离，所以要找trading cost低的。因为可能频繁交易。本题没直接给出trading cost低的。但是给出trading volume大的，变相可以得到ETF I的trading cost最低。
            • Which of the ETFs in Exhibit 2 most likely has the lowest tracking error? A. ETF I B. ETF II C. ETF III
              • C is correct ETF III applies a full replication approach, which will generate less potential tracking error than representative sampling. Engaging in security lending and the lowest management fee also support the argument that ETF III has the tracking error.
                • Trading error要找里面股票最多的，ETF III有180个，所以最低。
        • 例题
          • Investors buying ETFs: A. incur management fees that decrease with the length of the holding period. B. are assured of paying a price equal to the NAV if they purchase shares at the market close. C. incur trading costs in the form of commissions and bid–ask spreads at the time of purchase.
            • 答案：C 解析：ETF trading costs in the form of commissions and bid–ask spreads are paid by investors buying or selling ETF shares on an exchange. These trading costs are influenced by the bid–ask spread of the ETF, the size of the trade relative to the normal trading activity of the ETF, and the ease of hedging the ETF by the marketmaking community. Even the closing price of the ETF on the exchange includes a premium or discount to the NAV, driven by supply and demand factors on the exchange and the market impact costs of executing an exchange transaction. The purchase and sale trading costs of an ETF are paid regardless of holding period, whereas other costs, such as management fees, increase as the holding period lengthens.
              • 管理费是时间越长越多
    • Risks
      • Tracking error
        • Daily difference tracking error
        • Rolling return tracking difference
          • 例题
            • To best assess an ETF's performance, which reflects the impact of portfolio rebalancing expenses and other fees, an investor should: A. review daily return differences between the ETF and its benchmark. B. perform a rolling return assessment between the ETF and its benchmark. C. compare the ETF's annual expense ratio with that of other ETFs in its asset class category
              • B is correct. A rolling return assessment, referred to in the ETF industry as the "tracking difference," provides a more informative picture of the investment outcome for an investor in an ETF. Such an analysis allows investors to see the cumulative effect of portfolio management and expenses over an extended period. It also allows for comparison with other annual metrics such as a fund's expense ratio. Tracking error, as a statistic, reveals only ETF tracking variability; it does not reveal to investors whether the fund is over- or underperforming its index or whether that tracking error is concentrated over a few days or is more consistently experienced. An ETFs expense ratio does not fully reflect the investor experience. That is, the expense ratio does not reflect the cost of portfolio rebalancing or other fees, making it an inferior assessment measure relative to a rolling return assessment.
                • 这里也可以当一个结论记，我们用rolling return来评估
        • Source of tracking error
          • Fees and expenses
          • Representative sampling
          • Index changes
            • 例题
              • For a typical ETF, which of the following sources of tracking error is most likely to be the smallest contributor to tracking error? A. Representative sampling B. Fees and expenses incurred by the ETF C. Changes to the underlying index securities
                • C is correct. Although additions and deletions of securities from the underlying benchmark index may occur and result in tracking error, such index changes generally occur infrequently (often quarterly). In addition, ETF portfolio managers may work with APs for index rebalance trades to ensure market-on-close pricing to minimize this source of tracking error. Therefore, theresulting tracking error caused by index changes will not likely be as large as the tracking error caused by representative sampling or by fees and expenses incurred by the ETF.
                  • 要注意，ETF的index充分不会天天变，所以这个虽然会导致误差，但是和其他的比，相对影响最小。
          • Fund accounting practices
          • Regulatoryu and tax requirements
        • 例题
          • An ETF's reported tracking error is typically measured as the: A. standard deviation of the difference in daily returns between an ETF and its benchmark. B. difference in annual return between an ETF and its benchmark over the past 12 months. C. annualized standard deviation of the difference in daily returns between an ETF and its benchmark.
            • C is correct. An ETF's tracking error is typically reported as the annualized standard deviation of the daily differential returns of the ETF and its benchmark
              • 这个可以直接当成结论记
      • Other unique risks
        • Counterparty risk
        • Fund closures
          • 例题
            • Such factors as regulations, competition, and corporate actions relate to: A. fund-closure risk. B. counterparty risk. C. expectation-related risk.
              • A is correct. Fund-closure risk is the risk that an ETF may shut down. The reasons that lead to an ETF closing down often have to do with changes in regulations, increased competition, and corporate activity (merger and acquisition activity within the ETF industry).
                • 送分题
        • Investor-related risk
    • Tax consideration
      • Funds must distribute any capital gains realized during the year
      • ETFs distribute less in capital gains than competing mutual funds
      • 例题
        • Which of the following statements relating to capital gains in ETFs and mutual funds is correct? A. ETFs tend to distribute less in capital gains than mutual funds do. B. Mutual funds may elect not to distribute all realized capital gains in a given year. C. The selling of ETF shares by some investors may create capital gains that affect the remaining ETF investors in terms of taxes.
          • A is correct. ETFs tend to distribute far less in capital gains relative to mutual funds. This is mostly due to the fact that ETFs have historically had significantly lower turnover than mutual funds have had.
            • Funds一般回报都要分配，但是ETF分配的少
  • ETFs in portfolio management
    • ETF strategies
      • Efficient portfolio management
      • Asset class exposure management
      • Active and factor investing
• M2 Using Multifactor Models
  • Arbitrage pricing theory (APT)
    • Review of CAPM
    • APT
      • Basic form
      • Assumptions
        • factor model
          • factor sensitivity
            • 例题
              • Martinez next estimates Fund D's sensitivity to time horizon risk and inflation risk. She wants to further understand the drivers of Fund D's risk relative to its benchmark. For this analysis, she calculates the active risk for each factor and assumes that the correlation between the model's factors is negligible. She also assumes a zero value for the error term when working with the selected two-factor model, given in Exhibit 2. The expected return for Fund D is 6.0% and it has a tracking error of 8.0%. Martinez considers whether Fund D would be a suitable investment for Lakemont University's endowment. The endowment has an investment horizon of 30 years. The activities it supports have historically been subject to cost increases running above the average rate of inflation. Based solely on the factor model, is Fund D an appropriate investment for Lakemont University? A. Yes B. No, because of Fund D's sensitivity to inflation risk C. No, because of Fund D's sensitivity to time horizon risk
                • 正确答案：B A.Incorrect because the activities supported by Lakemont University are sensitive to inflation risk. Therefore, Fund D's negative factor sensitivity to inflation risk does not help Lakemont University in reducing its exposure to inflation risk. B.Correct because Fund D has a positive factor sensitivity to time horizon risk and a negative factor sensitivity to inflation risk. Investors "may tilt their strategic asset allocation or investments within an asset class to capture the associated risk premiums for risks that do not much affect them." Lakemont University has an investment time horizon of 30 years and is not much affected by time horizon risk. Therefore, it has a comparative advantage in bearing time horizon risk. However, the activities supported by Lakemont University are sensitive to inflation risk. Therefore, Fund D's negative factor sensitivity to inflation risk does not help Lakemont University in reducing its exposure to inflation risk. C.Incorrect because Lakemont University has an investment time horizon of 30 years and is not much affected by time horizon risk. Therefore, it has a comparative advantage in bearing time horizon risk.
                  • 这里主要考察对factor sensitivity的理解。 大学基金时间长，并且成本增长高于通知，所以需要对时间和通胀都有一定的补偿。但是对通胀的敏感因子是负数，所以不合适。
            • 例题
              • Then, White develops a multifactor model to analyze Fund M, a population fund of PAM. Fund M returned 7.30% for the last year, its benchmark returned 5.8% during the same period. White is performing an analysis of the sources of Fund M 's active return over the last year. He assumes that any return not explained by the model factors is considered security selection. Related information is presented in Exhibit 1. The return from security selection for Fund M was closest to: A. 1.67% B. 1.50% C. 1.33%
                • A is correct Active return =Return from factor tilts + Return from security selection 7.3% - 5.8% = (0.7-0.85)×5.93% + (0.57-0.2)×3.14% + (1-1.2)×2.22% + Return from security selection The return from security selection for Fund M = (0.7-0.85)×5.93% + (0.57-0.2)×3.14% + (1-1.2)×2.22% -7.3%+5.8% = 1.67%
        • well-diversified
        • no arbitrage
        • 例题
          • Which of the following is not a key assumption of APT, which is used by Altuve to evaluate strategies and manage risks? A. A factor model describes asset returns. B. Asset-specific risk can be eliminated through diversification. C. Arbitrage opportunities exist among well-diversified portfolios.
            • C is correct. Arbitrage pricing theory (APT) is a framework that explains the expected return of a portfolio in equilibrium as a linear function of the risk of the portfolio with respect to a set of factors capturing systematic risk. A key assumption of APT is that, in equilibrium, there are no arbitrage opportunities.
              • 送分题
      • 例题
        • The arbitrage opportunity identified by Zapata can be exploited with: A. Strategy 1: Buy $50,000 Fund A and $50,000 Fund B; sell short $100,000 Fund C. B. Strategy 2: Buy $60,000 Fund A and $40,000 Fund B; sell short $100,000 Fund C. C. Strategy 3: Sell short $60,000 of Fund A and $40,000 of Fund B; buy $100,000 Fund C.
          • C is correct. The expected return and factor sensitivities of a portfolio with a 60% weight in Fund A and a 40% weight in Fund B are calculated as weighted averages of the expected returns and factor sensitivities of Funds A and B: Expected return of Portfolio 60/40 = (0.60)(0.02) + (0.40)(0.04) = 0.028, or 2.8% Factor sensitivity of Portfolio 60/40 = (0.60)(0.5) + (0.40)(1.5) = 0.9 The factor sensitivity of Portfolio 60/40 is identical to that of Fund C; therefore, this strategy results in no factor risk relative to Portfolio C. However, Fund C's expected return of 3.0% is higher than Portfolio 60/40’s expected return of 2.8%. This difference supports Strategy 3: buying Fund C and selling short Portfolio 60/40 to exploit the arbitrage opportunity.
            • 送分题
      • 例题
        • Hunt evaluates ETF EU1, a smart-beta European ETF, along with its benchmark index which is market-cap weighted. Based on a risk-free rate of 0.55% and the annualized factor risk premiums and factor sensitives in Exhibit 2, Hunt uses the Carhart model to estimate ETF EU1's expected annual return. Based on the Carhart model, the expected annual return for ETF EU1 is closest to: A. 2.17%. B. 2.30%. C. 2.85%.
          • C is correct Because the expected annual return for ETF EU1 using the Carhart model is computed as follows: E[RETF EU1] = RF + βETF EU1,1(RMRF) + βETF EU1,2(SMB)+ βETF EU1,3(HML)+ βETF EU1,4(WML) where RF is the annualized risk-free rate, β is the factor sensitivity for each risk factor, RMRF is the market risk premium, SMB (small minus big) is the size (market capitalization) risk premium, HML (high minus low) is the book-to-market (value) risk premium, and WML (winners minus losers) is the momentum risk premium. E[RETF EU1] = 0.55% + (1.12 × 4.48%) + (–1.08 × 1.65%) + (0.55 × –1.72%) + (–0.02 × –0.63%) E[RETF EU1] = 0.5500% + 5.0176% – 1.7820% – 0.9460% + 0.0126% E[RETF EU1] = 2.8522% ≈ 2.85%
            • 著名的Carhart model要了解 注意这里问的是expected return，而不是value added。
  • Multifactor models
    • Macroeconomic factor models
      • a
        • E(R)
      • b
        • sensitivity
        • 回归
      • F
        • Surprise = actual - predicted
          • 例题
            • Based on Exhibit 1, the active risk for Portfolio 2 is explained by surprises in: A. GDP. B. consumer spending. C. all four model factors.
              • C is correct. Active risk, also referred to as tracking risk or tracking error, is the sample standard deviation of the time series of active returns, where the active returns consist of the differences between the portfolio return and the benchmark return. Whereas GDP is the only portfolio non-zero sensitivity for Portfolio 2, the contribution to the portfolio's active return is the sum of the differences between the portfolio's and the benchmark's sensitivities multiplied by the factor return. Because all four of the factor sensitivities of Portfolio 2 are different from the factor sensitivities of the benchmark, all four factors contribute to the portfolio's active return and, therefore, to its active risk.
                • 要注意这里的F是actual - predicted，所以4个因子都会影响
      • e
        • company specific risk
      • Data
        • time-series
      • 解释某一公司股价
      • 例题
        • Francesca Martinez is a portfolio manager. She decides to review several fund managers who use macroeconomic multifactor models. The intercept term in the type of multifactor model used by Martinez represents the: A. risk-free rate B. portion of a portfolio's total return not explained by the model's factors C. expected return of a portfolio with zero sensitivity to all of the model's factors
          • C is correct A.Incorrect because the intercept represents the risk-free rate in the general APT equation. However, in macroeconomic factor models the intercept represents the expected return on a risky portfolio with zero sensitivity to all the factors. B.Incorrect because it describes the model's error term, not the intercept. In a factor model, εi is an error term with a zero mean that represents the portion of the return not explained by the factor model. C.Correct because in macroeconomic factor models, if no risk-free asset exists the intercept represents the expected return on a risky portfolio with zero sensitivity to all the factors.
            • 送分题，必须掌握
    • Fundamental factor models
      • a
        • 是的e=0的截距
      • b
        • 基本面因素（P/B，P/E，Leverage）
        • Standardized beta of attributes
      • F
        • sensitivity
        • 回归
      • e
        • 回归残差
      • Data
        • cross-sectional
      • 解释同行业公司股价差异
    • Macroeconomic vs fundamental
      • 例题
        • Lam and Cheung discuss similarities and differences between macroeconomic factor models and fundamental factor models, and Lam offers a comparison of those models to statistical factor models. Lam makes the following statements. Statement 1 The factors in fundamental factor models are based on attributes of stocks or companies, whereas the factors in macroeconomic factor models are based on surprises in economic variables. Statement 2 The factor sensitivities are generally determined first in fundamental factor models, whereas the factor sensitivities are estimated last in macroeconomic factor models. Which of Lam's statements regarding macroeconomic factor models and fundamental factor models is correct? A. Only Statement 1 B. Only Statement 2 C. Both Statements 1 and 2
          • C is correct. In a macroeconomic factor model, the factors are surprises in macroeconomic variables that significantly explain returns. Factor sensitivities are generally specified first in fundamental factor models, whereas factor sensitivities are estimated last in macroeconomic factor models.
            • 所有参数的对比必须熟练掌握
            • marcoeconomic宏观因子模型主要解释某一个公司的股价回报，fundamental基本面因子模型主要解释同行业公司股票价格的差异。所以statement 1是正确的，基本面模型主要是看公司，宏观经济主要是看经济。宏观经济模型是先有factor sensitivity，然后回归出来beta，而基本面模型是反过来。所以statement 2也是对的。
            • 【PS】2个模型本质都是回归模型，公式一样R=A+B1F1+B2F2+...+e
            • 【PS2】宏观模型的alpha是预期收益，基本面是使得残差为0的截距。
            • 【PS3】宏观模型的beta是回归的sensitivity，是先有F，然后回归beta。而基本面模型是基本面要素，比如P/B, P/E，所以beta直接可以先算出来。
            • 【PS4】宏观模型的factor sensitivity是宏观经济要素，最后得到suprise = actual - predicted。而基本面是回归的sensitivity，是先有beta，然后回归F。二者区别的本质原因是，宏观经济模型算factor sensitivity比较容易，有了F后再回归具体公司的beta。而基本面模型有公司具体的P/B比较容易，然后再回归出来F。2个模型都是哪个好算先算哪个。
            • 【PS4】宏观经济模型的残差是公司特有风险，而基本面模型是回归出来的残差。
            • 【PS4 Pro】宏观经济模式是时间序列，基本面模型是横截面序列。
            • 【PS5】除了这2个模型，还有一个statistical factor model统计因子模型，主打放飞自我。优点是假设条件少，贼容易算。缺点是没经济学含义，算出来干蛋。
    • Statistical factor models
      • pros
        • 假设少
      • cons
        • 经济意义不明
      • 例题
        • An advantage of statistical factor models is that they make minimal assumptions, and therefore, statistical factor model estimation lends itself to easier interpretation than macroeconomic and fundamental factor models. Is Lam's comment regarding statistical factor models correct? A. Yes B. No, because he is incorrect with respect to interpretation of the models' results C. No, because he is incorrect with respect to the models' assumptions
          • B is correct. An advantage of statistical factor models is that they make minimal assumptions. However, the interpretation of statistical factors is generally more difficult than the interpretation of macroeconomic and fundamental factor models.
            • 越简单的模型，越难被操控
  • Applications
    • Active attribution
      • Return attribution
        • Factor return/tilts
        • Security selection
      • Risk attribution
        • Active risk squared
          • Active factor risk (squared)
          • Active specific risk (squared)
        • Information ratio
    • Portfolio construction
    • Strategic portfolio decisions
    • 例题
      • Lam tells Cheung that multifactor models can be useful in active portfolio management, but not in passive management. Cheung disagrees; she tells Lam that multifactor models can be useful in both active and passive management. Whose statement regarding the use of multifactor models in active and passive portfolio management is correct? A. Lam only B. Cheung only C. Both Lam and Cheung
        • B is correct. Analysts can use multifactor models in passively managed portfolios to replicate an index fund's factor exposures.
          • 送分题
• M3 Measuring and Managing Market Risk
  • Value at Risk (VaR)
    • Definition
      • certain time period
      • position
      • min $
      • 例题
        • Which of the following statements regarding the VaR of the Index Plus Fund is correct? A. The expected maximum loss for the portfolio is $6.5 million. B. Five percent of the time, the portfolio can be expected to experience a loss of at least $6.5 million. C. Ninety-five percent of the time, the portfolio can be expected to experience a oneday loss of no more than $6.5 million.
          • B is correct. VaR measures the frequency of losses of a given minimum magnitude. Here the VaR indicates that on 5% of trading days, the portfolio will experience a loss of at least $6.5 million. (Although C may appear to say the same thing as B, it actually implies that the portfolio will experience a loss on 95% of trading days.) The correct interpretation is that returns will be equal to or greater than -$6.5 million on 95% of trading days; those returns include gains as well as losses.
            • VaR的3个要素，时间，概率，金额。A只有金额，不行。C在暗示95%都在损失，也是错的，只有B是对的。
            • 【PS】C想修改正确的话，需要这么说：return...at least -6.5M...on 95% of trading days。
            • 【PS2】严格来说B也是不完美的，但是CFA选择题，3个选个最合适的。B想要严谨的话，需要修改成5% of time for one day...
            • 【PS3】坊间传闻，CFA的VaR定义是从左向右而不是从右向左，所以如果B和C都改的非常严谨完美，也是选B，因为是书上对VaR的定义。
    • Estimate VaR
      • Parametric method
        • Assumption
          • normal distribution
        • pros
          • simple
        • cons
          • sensitive to covariance
          • cannot use when contain options
        • One-tailed test
          • 90%: 1.28
          • 95%: 1.65
          • 99%: 2.33
      • Historical simulation method
        • Assumption
          • history data can represent future
        • pros
          • can do with option
          • based on what actually happened
        • cons
          • not certain that historical event will reoccur
      • Monte Carlo Simulation method
        • Assumption
          • probability distribution
        • pros
          • flexibility to complex distribution
          • can do with option
        • cons
          • time and spending-consuming
      • Comparison betwqeen three methods
        • 例题
          • The markets' volatility during the last 12 months has been significantly higher than the historical norm, with increased frequency of large daily losses, and Hamilton expects the next 12 months to be equally volatile. Given Hamilton's expectations, which of the following models is most appropriate to use in estimating portfolio VaR? A. Parametric method B. Historical simulation method C. Monte Carlo simulation method
            • C is correct. The Monte Carlo simulation method can accommodate virtually any distribution, an important factor given the increased frequency of large daily losses. This method can also more easily accommodate the large number of portfolio holdings. The Monte Carlo method allows the user to develop her own forward-looking assumptions about the portfolio’s risk and return characteristics, unlike the historical simulation method, which uses the current portfolio and re-prices it using the actual historical changes in the key factors experienced during the lookback period. Given the limited return history for infrastructure investments and Hamilton's expectations for higher-than-normal volatility, the historical simulation method would be a suboptimal choice.
              • 注意历史不代表未来就不能用历史法，参数会发生改变也不能用参数法。没方法可以选的时候，又没特别说要考虑复杂性包括时间和成本，就用蒙特卡洛
    • Adv and limitation
      • pros
        • simple
        • easy to communicate
        • accpeted by regulators
        • can be verified
      • cons
        • subjectivity
        • underestimate extreme events (left tail)
        • cannot reflect return (right tail)
          • 例题
            • Statement 2 In last year's annual client performance report, IMA stated that a hypothetical $6 million Equity Opportunities Fund account had a daily 5% VaR of approximately 1.5% of portfolio value Based only on Statement 2, the risk measurement approach: A. ignores right-tail events in the return distribution. B. is similar to the Sharpe ratio because it is backward looking. C. provides a relatively accurate risk estimate in both trending and volatile regimes.
              • 解析：VaR一般都是关注左尾也就是损失，很少会用VaR来关注右尾的盈利部分，所以本题选A
              • 【PS1】VaR的优点：1. 容易沟通；2. 监管接受
              • 【PS2】VaR的缺点：1. 主观；2. 忽视右尾收益；3. 忽视左尾极端损失；4. 不考虑流动性
        • not consider liquidity
        • sensitivity to correlation risk
        • 例题
          • After reading McKee's report, Ming asks why the number of daily VaR breaches over the last year is zero even though the portfolio has accumulated a substantial loss. The number of Flusk's VaR breaches most likely resulted from: A. using a standard normal distribution in the VaR model. B. using a 95% confidence interval instead of a 99% confidence interval. C. lower market volatility during the last year compared with the lookback period.
            • C is correct. Flusk experienced zero daily VaR breaches over the last year yet incurred a substantial loss. A limitation of VaR is its vulnerability to different volatility regimes. A portfolio might remain under its VaR limit every day but lose an amount approaching this limit each day. If market volatility during the last year is lower than in the lookback period, the portfolio could accumulate a substantial loss without technically breaching the VaR constraint. A is incorrect because VaR was calculated using historical simulation, so the distribution used was based on actual historical changes in the key risk factors experienced during the lookback period. Thus, the distribution is not characterized using estimates of the mean return, the standard deviation, or the correlations among the risk factors in the portfolio. In contrast, the parametric method of estimating VaR generally assumes that the distribution of returns for the risk factors is normal.
              • VaR只是一个边界，即使没到VaR的边界，也是有可能一直有损失产生。
    • Extensions of VaR
      • Conditional
        • 例题
          • The committee is likely to have questions in a number of key areas—the limitations of the VaR report, potential losses in an extreme adverse event, and the reliability of the VaR numbers if the market continues to exhibit higher-than-normal volatility. Hamilton wants to be certain that she has thoroughly evaluated the risks inherent in the LICIA portfolio and compares them with the risks in Hi-ram's present portfolio. Which risk measure is Hamilton most likely to present when addressing the committee's concerns regarding potential losses in extreme stress events? A. Relative VaR B. Incremental VaR C. Conditional VaR
            • C is correct. Conditional VaR is a measure of tail risk that provides an estimate of the average loss that would be incurred if the VaR cutoff is exceeded.
              • 这题关键是读题，因为选项都是VaR，又问极端场景下的极端损失，所以问的是左边的极端损失，是超过了极限的conditional VaR
        • 例题
          • Upon reviewing the VaR report, the supervisor requested further details on the potential average loss if the VaR threshold were expanded. White replied that he would incorporate incremental VaR, conditional VaR, and marginal VaR in the report. Which extension of VaR will most likely meet supervisor’s needs? A. Incremental VaR. B. Conditional VaR. C. Marginal VaR.
            • B is correct Conditional VaR, also referred to as expected tail loss or expected shortfall, is used to determine the average loss that would be incurred if the VaR threshold were expanded. Incremental VaR is used to determine how VaR will change if a position size is changed relative to the remaining positions. Marginal VaR is conceptually similar to incremental VaR in that it reflects the effect of an anticipated change in the portfolio, but it uses formulas derived from calculus to reflect the effect of a very small change in the position.
              • 不同VaR的作用
      • Incremental
      • Marginal
      • Relative
        • Ex-ante tracking error
        • 关键词：benchmark
        • 例题
          • In response, ARC's management asks Newman what metric can best estimate the degree to which the firm's portfolios could underperform their respective benchmarks in the future. Which metric best addresses management's concern about estimating portfolio underperformance? A. Relative VaR B. Surplus at risk C. Maximum drawdown
            • A is correct A. Correct because ex-ante tracking error, also known as relative VaR, tracks the deviation of performance from the benchmark. As Newman's metric needs to track the future performance of the portfolios against benchmarks, the suitable metric needs to be forward looking and also based on the portfolios' current holdings. Ex-ante tracking error or relative VaR is a measure of that percentage of the portfolio that differs from the benchmark (i.e., a deviation from the benchmark). It is often monitored to help limit tracking error of the portfolio. Ex ante tracking error or relative VaR is a measure of the degree to which the performance of a given investment portfolio might deviate from its benchmark. Ex ante tracking error takes today’s benchmark-relative position and exposes it to the variability of past markets to estimate what kind of benchmark-relative performance could arise from the current portfolio. B. Incorrect because surplus at risk is a risk measure that is used by pensions and estimates how much their assets might underperform the liabilities, but not usually used for tracking returns versus benchmarks. This measure is an application of VaR. It is computed by entering the assets in the portfolio into a VaR model as long positions and the pension liabilities as short fixed-income positions. It estimates how much the assets might underperform the liabilities, usually over one year, and pension plan sponsors may vary with respect to how high a level of confidence they choose to use (e.g., 84%, 95%, 99%). The more volatile the investments in the pension fund and the less well correlated these assets are with the liabilities, the higher the surplus at risk. C. Incorrect because drawdown is a measure based on historical data used to identify the worst returning month or quarter and will not help tracking the underperformance relative to a benchmark for the future. Drawdown is often helpful in replacing standard deviation and historical beta measures where return distributions are non-normal. It can be helpful in assessing what the worst peak-to-trough decline in a portfolio’s historical returns is for hedge fund strategies like equity market neutral, long–short credit, or investments in illiquid asset classes with asymmetric return profiles. It is not useful for tracking the expected underperformance of a portfolio compared to its benchmark, which is Newman's requirement.
              • 要注意，relative VaR可以用来做对比。surplus at risk一般是看养老金。maximum drawdown一般是历史数据。
  • Sensitivity and scenario measures
    • Sensitivity risk measures
      • pros
        • can inform exposure to risk factors
        • can modify exposure to risk factor
      • cons
        • can only estimate small changes
    • Scenario risk measures
      • Historical
      • Hypothetical
        • stress test
        • 例题
          • Hamilton believes the possibility of a ratings downgrade on Indian sovereign debt is high and not yet fully reflected in securities prices. If the rating is lowered, many of the portfolio's holdings will no longer meet Hiram's minimum ratings requirement. A downgrade's effect is unlikely to be limited to the government bond portfolio. All asset classes can be expected to be affected to some degree. Hamilton plans to include a scenario analysis that reflects this possibility to ensure that management has the broadest possible view of the risk exposures in the India portfolio.
            • The scenario analysis that Hamilton prepares for the committee is most likely a: A. stress test. B. historical scenario. C. hypothetical scenario.
              • C is correct. A hypothetical scenario analysis allows the risk manager to estimate the likely effect of the scenario on a range of portfolio risk factors. A sovereign ratings downgrade would affect Hiram's India equity and corporate bond exposures as well as the government bond exposure. In addition, the assumptions used in constructing the scenario analysis can specifically address the effect of a need to sell large position sizes under decreased liquidity conditions resulting from a ratings downgrade. VaR alone does not accurately reflect the risk of large position sizes, which may be difficult to trade.
                • 假设未发生的是hypothetical scenario，没到极端情况，不用stress test
            • The scenario analysis that Hamilton prepares for the committee is a valuable tool to supplement VaR because it: A. incorporates historical data to evaluate the risk in the tail of the VaR distribution. B. enables Hamilton to isolate the risk stemming from a single risk factor—the ratings downgrade. C. allows the committee to assess the effect of low liquidity in the event of a ratings downgrade.
              • C is correct. A hypothetical scenario analysis allows Hamilton to estimate the direct effect of a ratings downgrade on the portfolio's government bond holdings and the resulting need to sell a number of the portfolio's holdings because they no longer meet the ratings guidelines. VaR alone does not accurately reflect the risk of large position sizes, which may be difficult to trade. The hypothetical scenario analysis will also highlight the effect of increased economic turmoil on all of the portfolio's exposures, not only the government bond exposures.
                • 情景假设不需要历史数据，所以A不对。 单因子模型是敏感性分析，所以B不对。 低流动性是C选项自己加进去的，3个选项选个最合适的，只能选C。
            • What additional risk measures would be most appropriate to add to Hamilton's risk assessment? A. Delta B. Duration C. Tracking error
              • B is correct. Given the large fixed-income exposure in the LICIA portfolio, examining the portfolio duration more closely would be prudent. Duration is the primary sensitivity exposure measure for fixed-income investments.
                • 这题的破题点是，现在的产品是gov bond。 A是衍生品，C是fund，只有B是用来衡量bond的。
      • pros
        • not bound by historical events or assumptions about probability distribution
      • cons
        • historical scenarios not happen exactly the same way again
        • may incorrectly specify how assets will co-move
          • 例题
            • The limitation of using a hypothetical scenario analysis is that: A. cannot simulate a scenario that didn't happen in history. B. may incorrectly estimate the correlation of asset returns. C. fail to incorporate portfolio manager actions.
              • 答案：B 假设情景法可能无法精确模拟资产是如何共同变化的，对相关系数的建模可能 不准确。因此，B选项正确。 假设情景法并不是只考虑历史中已经发生的情景。因此，A选项错误。 假设情景法并不是不会考虑投资组合经理的行为。因此，C选项错误。
        • difficult to create and maintain
    • Sensitivity vs scenario vs VaR
      • VaR
        • Probability of loss
      • Sensitivity risk measures
        • No estimate of probabilities
        • Not downside risk measures
      • Scenario risk measures
        • No estimate of probabilities
  • Applications of risk measures
    • Banks
    • Asset Managers
      • Traditional asset managers
      • Hedge funds
    • Pension funds
    • Issurers
  • Using constraints in market risk management
    • Risk budgeting
    • Positioin limits / scenario limits / stop-loss limits
      • 例题
        • Ben, risk analyst, believes that the economy will go into recession soon and the price of corporate-bonds will decrease, so he suggests limiting exposure to corporate-bonds. This constraint is best described as a: A.Position limit. B.Scenario limit. C.Stop-loss limit.
          • 答案：A 限制敞口也就是限制持仓，所以是position limit，故正确答案是A。
            • 送分题
    • Risk measures and capital allocation
    • 例题
      • Use of Options: The trust department of Eastern Regional Bank manages an equity fund called the Index Plus Fund, with $325 million in assets. This fund's objective is to track the S&P 500 Index price return while producing an income return 1.5 times that of the S&P 500. The bank's chief investment officer (CIO) uses put and call options on S&P 500 stock index futures to adjust the risk exposure of certain client accounts that have an investment in this fund. The portfolio of a 60-year-old widow with a below-average risk tolerance has an investment in this fund, and the CIO has asked his assistant, Janet Ferrell, to propose an options strategy to bring the portfolio's delta to 0.90. Which of the following options strategies is Ferrell most likely to recommend for the client's portfolio? A. Long calls B. Short calls C. Short puts
        • B is correct. An index-tracking portfolio without options has a delta of 1. To achieve a delta of 0.9, the delta of the options position must be negative. Of the three choices, only short calls have a negative delta. Long call options have deltas ranging from 0 to 1. Short calls, therefore, have deltas ranging from 0 to -1. The short call position lowers the portfolio’s overall delta as desired.
          • 因为这是一个index tracking profolio，所以现在的delta为1，要降低到0.9，需要让option的delta为负，只有long put或者short call，题目B符合，选B。
          • 【PS】如果不记得profolio的delta和put call的delta，这题也可以做。60岁孤寡老太太，肯定是要降低下行风险，所以找long put或者short call。
• M4 Backtesting and Simulation
  • Backtesting
    • Objectives
    • Steps
    • Problems
      • Survivorship bias
        • 例题
          • Yuen tasks Ruckey with specifying the investment universe and determining the availability of appropriate reporting data in vendor databases. Ruckey selects a vendor database that does not provide point-in-time data, so he adjusts the database to include point-in-time constituent stocks and a reporting lag of four months. Following Ruckey's adjustments to the initial vendor database, backtested returns will most likely be subject to: A. stale data. B. data snooping C. p-hacking
            • A is correct. A reporting lag of four months is likely to introduce stale data into the backtest because many large-capitalization companies report earnings within 30–50 days of quarter end. Although assuming four months (120 days) of reporting lag will eliminate a source of look-ahead bias, it introduces a new problem (i.e., stale data). B and C are incorrect. Data snooping and p-hacking refer to the same problem: a flawed approach to using data to make decisions. Data snooping and p-hacking are not characteristics of data, nor can they be added to a dataset by making an adjustment.
              • 送分题
      • Look-ahead bias
      • Data snooping
    • Historical scenario analysis
      • 例题
        • The portfolio manager at Fastlane told Galic that the company selects the top-performing strategies after performing thousands of backtests. Comparing the two strategies in Exhibit 2, the best risk-adjusted performance is demonstrated by: A. Strategy II in periods of low volatility and recession. B. Strategy I in periods of high volatility and non-recession. C. Strategy II in periods of high volatility and non-recession.
          • A is correct. Using the Sharpe ratio, the best risk-adjusted relative performance can be determined by comparing the sensitivity of the two strategies under differing macroeconomic regimes: recession versus non-recession and high volatility versus low volatility. The best risk-adjusted return will exhibit the highest Sharpe ratio. Strategy II demonstrates higher risk-adjusted returns compared with Strategy I under all four macroeconomic conditions, particularly in periods of low volatility, when the Sharpe ratio outperformance is 0.96, and recessions, when the Sharpe ratio outperformance is 1.56.
            • Sharpe ratio是越高越好，要找明显比另外一个高的点。
    • 例题
      • Kevin is a hedge fund manager at Prosperity Asset Management (PAM). White met with Kevin and asked him to describe how hedge funds develop relevant investment strategies and explain the backtesting process of hedge funds. Kevin notes the following: Statement 3: uses historical data and backtesting to approximate the real investment process, in order to demonstrate the risk return trade-off of a specific proposed investment strategy. Statement 4: backtesting is almost exclusively used for quantitative investment managers and rarely for fundamental investment managers, who are more concerned with forward-looking estimates of company returns, macroeconomic factors, and intrinsic value information. Which of Kevin’s statements concerning backtesting is correct? A. Only Statement 3 B. Only Statement 4 C. Both Statement 3 and Statement 4
        • A is correct Statement 3 is correct because the main objective of backtesting is to understand the risk–return tradeoff of an investment strategy by approximating the real-life investment process. B is incorrect because Statement 4 is inaccurate. Although backtesting fits quantitative and systematic investment styles more naturally, it has also been heavily used by fundamental managers. C is incorrect because Statement 4 is not accurate. Backtesting, used in quantitative and systematic investment styles, is also heavily used by fundamental managers.
          • 大家都用backtesting
  • Simulation
    • Historical simulation
    • Monte Carlo simulation
      • 例题
        • Stone reviews Brockway's fixed-income strategy, which uses a multifactor model to pick undervalued securities. He estimates downside risk measures for the strategy using Monte Carlo simulation. To perform the simulation, Stone specifies a functional form using ten factors and calibrates the model to a multivariate Student's t-distribution. The model is complex and several factors have limited historical data. The model used in Stone's Monte Carlo simulation most likely has a: A. large estimation error. B. high specification error. C. low probability of producing extreme return estimates.
          • A is correct A. Correct because several factors have limited empirical/historical data, which means that it may not be possible to specify these factors accurately. We can specify a highly complex model with many parameters (all of which need to be estimated/calibrated from historical data) that describe the empirical properties of the data well. Given limited historical data, however, we may be unable to estimate all the underlying parameters with sufficient precision. Such models tend to have low specification errors, but they suffer from large estimation errors. B. Incorrect because high specification error arises when the model is overly simplistic. Overly simplistic models require fewer parameters (therefore, they might have low estimation errors), but they may not fit the data well (because they are mis-specified). Stone's model is complex and has ten key decision variables. C. Incorrect because a multivariate Student's t-distribution is used, which has fatter tails than the normal distribution and accounts for a higher probability of extreme events. The Student's t-distribution is a natural extension of the multivariate normal distribution, because it has the ability to account for the skewness and the excess kurtosis often observed in factor and asset return data. Excess kurtosis indicates a higher probability of extreme events.
            • 模型需要许多参数，但是可用的数据有限，就会导致estimation error，所以A是对的。
            • 模型参数很多很复杂，所以一般specification error比较下。B说反了。
            • T分布一般肥尾，更大的极端值，所以C也说反了
    • Sensitivity analysis
    • 例题
      • Yuen and Ruckey discuss the differences between the two approaches and then design the simulations, making key decisions at various steps. During the process, Yuen expresses a number of concerns: Concern 1: Returns from six of the nine factors are correlated. Concern 2: The distribution of Factor 1 returns exhibits excess kurtosis and negative skewness. Concern 3: The number of simulations needed for Approach 1 is larger than the size of the historical dataset. For each approach, Yuen and Ruckey run 1,000 trials to obtain 1,000 returns for Portfolios A and B. To help understand the effect of the skewness and excess kurtosis observed in the Factor 1 returns on the performance of Portfolios A and B, Ruckey suggests simulating an additional 1,000 factor returns using a multivariate skewed Student's t-distribution, then repeating the Approach 2 simulation.
        • To address Concern 1 when designing Approach 2 (Monte Carlo simulation), Yuen should: A. model each factor or asset on a standalone basis. B. calculate the 15 covariance matrix elements needed to calibrate the model. C. specify a multivariate distribution rather than modeling each factor or asset on a standalone basis.
          • C is correct. Approach 2 is a Monte Carlo simulation. The returns of Portfolios A and B are driven by the returns of the nine underlying factor portfolios (based on nine common growth factors). In the case of asset or factor allocation strategies, the returns from six of the nine factors are correlated, and therefore it is necessary to specify a multivariate distribution rather than modeling each factor or asset on a standalone basis. A is incorrect because Approach 2 is a Monte Carlo simulation to generate investment performance data for the nine underlying factor portfolios. The returns of six of the nine factors are correlated, which means specifying a multivariate distribution rather than modeling each factor or asset on a standalone basis. B is incorrect because the analyst should calculate the elements of the covariance matrix for all factors, not only the correlated factors. Doing so entails calculating 36, not 15, elements of the covariance matrix. Approach 2 is a Monte Carlo simulation using the factor allocation strategies for Portfolios A and B for the nine factor portfolios, the returns of which are correlated, which means specifying a multivariate distribution. To calibrate the model, a few key parameters need to be calculated: the mean, the standard deviation, and the covariance matrix. For 9 assets, we need to estimate 9 mean returns, 9 standard deviations, and 9 × (9 - 1)/2 = 36 elements of the covariance matrix. Assuming just the 6 correlated assets, the calculation is 6 × (6 - 1)/2 = 15.
            • 送分题
        • Based on Concern 2, the Factor 1 strategy is most likely to: A. be favored by risk-averse investors. B. generate surprises in the form of negative returns. C. have return data that line up tightly around a trend line.
          • B is correct. The distribution of Factor 1 returns exhibits excess kurtosis and negative skewness (relative to the normal distribution). The excess kurtosis implies that these strategies are more likely to generate surprises, meaning extreme returns, whereas the negative skewness suggests those surprises are more likely to be negative (than positive). A is incorrect because risk-averse investors are more likely to prefer distribution properties such as positive skew (higher probability of positive returns) and lower to moderate kurtosis (lower probability of extreme negative surprises). The distribution of Factor 1 returns exhibits excess kurtosis and negative skewness. C is incorrect because the distribution of Factor 1 returns exhibits excess kurtosis and negative skewness. The joint distribution of such returns is rarely multivariate normal—so, typically the means and variances of these returns and the correlations between them are insufficient to describe the joint return distribution. In other words, the return data do not line up tightly around a trend line because of fat tails and outliers.
            • 送分题
            • 这题的关键是看懂题目，excess kurtosis，说明不是正太分布，存在肥尾。negative skewness说明左斜，损失端比较大。
            • A说风险厌恶者希望这种损失，明显扯淡。B说损失端有超额回报，也就是损失，这个是对的。C比较有趣，line up tightly around a trend line，这个说的其实是正态分布。正态分布的一个重要特性是数据点围绕均值（即趋势线）对称分布，且大多数数据点集中在均值附近，远离均值的数据点较少。如果数据“line up tightly around a trend line”，这意味着数据点紧密且对称地分布在均值周围，这与正态分布的特性相符。
            • 【PS】在投资中，gain和loss可以用return来统一代指，所以surprise也可以是超额损失。要能正确的把B翻译成“有极端损失"。
            • 【PS2】C说紧贴，说明波动不大。但是已经肥尾了，暗示波动大，所以不能选C。
        • To address Concern 3 when designing Approach 1 (Historical simulation), Yuen should: A. add monthly return observations to the dataset using interpolation. B. randomly sample from the historical returns with replacement. C. choose the multivariate normal distribution as the initial functional form.
          • B is correct. Random sampling with replacement, also known as bootstrapping, is often used in historical simulations because the number of simulations needed is often larger than the size of the historical dataset. Because Approach 1 is a historical simulation and Concern 3 notes that the number of simulations needed is larger than the size of the historical dataset, bootstrapping should be used. A is incorrect because this approach would result in creating observations that do not exist in the historical record. Doing so would violate the assumption and procedures of historical simulation. C is incorrect because choosing the multivariate normal distribution as the initial functional form is typically done in a Monte Carlo simulation (Approach 2), not in a historical simulation (Approach 1). Historical simulation randomly samples from the historical dataset by drawing a number from a uniform distribution so that there is equal probability of being selected. Choice of distribution would not address the concern about the size of the dataset.
            • 送分题
            • 样本不多，需要多次做，最简单的就是bootstrap，关键字replacement，选B。
            • 【PS】A的问题在于，我们是历史模拟，数据应该从历史来，而A用插值法，在创造历史不存在的数据。
            • 【PS2】如果B选项改成without replacement，说明用的是jackknife。这时候要看样本数够不够。如果实在没前提选项，也可以选。如果bootstrap vs jackknife，还是建议选bootstrap。
            • 【PS3】C潜台词是蒙特卡洛，但是题目明确说了historical，所以不能选。历史模拟法只能有啥用啥。而不是用function去预测。
• M5 Economics and Investment Markets
  • The discount rate on real default-free bonds
    • Inter-temporal rate of substitution
      • l=1/m(t) - 1
        • Inter-temporal rate of substitution = m(t)/m(0)
          • 例题
            • The covariance between a risk-averse investor's inter-temporal rate of substitution and the expected future price of a risky asset is typically: A. negative. B. zero. C. positive.
              • 答案：A 解析：For risk-averse investors, when the expected future price of the investment is high (low), the marginal utility of future consumption relative to that of current consumption is low (high). Hence, the covariance of the inter-temporal rate of substitution with asset price is expected to be negative for risk-averse investors.
                • 替代率=m(t)/m(0)，和m(t)同向变化。m(t)和p(t)反向变化，所以最后结论是负相关。
      • 例题
        • Julie Carlisle is a financial planner at a large wealth management firm. One of her clients, Esteban Blake, just received a sizable inheritance. He invests a portion of the inheritance in an annuity that will immediately increase his income by a substantial amount. He enlists Carlisle's help to invest the remaining amount of the inheritance. Holding all else constant, the change in Blake's income will most likely result in: A. an increase in his marginal utility of consumption. B. an increase in his inter-temporal rate of substitution. C. a decrease in his required risk premium for investing in risky assets.
          • C is correct. The additional annuity payment substantially increases Blake's income and wealth, which decreases his marginal utility of consumption. As a result, the average loss of marginal utility from any risk taking decreases as his wealth increases. Thus, he requires a lower risk premium and is willing to buy more risky assets.
            • 钱越多，utility越低
    • Default-free interest rates and economic growth
      • 例题
        • The prices of one-period, real default-free government bonds are likely to be most sensitive to changes in: A. investors' inflation expectations. B. the expected volatility of economic growth. C. the covariance between investors' inter-temporal rates of substitution and the expected future prices of the bonds.
          • B is correct. Only changes in default-free real interest rates will affect the price of real, default-free bonds. The average level of default-free real interest rates is positively related to the volatility of economic growth in the economy; thus, changes in the expected volatility of economic growth would likely lead to changes in default-free real interest rates, which in turn would affect the prices of real, default-free government bonds.
            • 真实无风险利率里面没通胀，A是错的 经济波动影响影响真实无风险利率 协方差是有风险利率里的一部分，是无风险加上协方差得到有风险利率。
            • 一般nominal rate = real rate + inflation，题目问的是real rate，所以没inflation，排除A。B选项期待的波动变高会要求风险补充，所以会影响实际利率。C选项提到的的确会有影响，但是本题是defaul-free government bonds，也就是是risk-free bond，这个covariance是0，所以不会有rf bond任何影响。
            • 【PS】即使C选项把rf gov bond改成普通的企业债券，也不建议选。因为跨期替代率和预期未来债券价格之间的协方差，这个更多的是影响长期利率，但是题目中说one-period，说明是短期，所以不能选C。
    • 例题
      • GERA forecasts that Country A and Country C will experience the same long-term real GDP growth rate, which is significantly higher than the growth rate estimate for Country B. GERA's expectations are that Country A and Country B will have similar levels of volatility of real GDP growth, with volatility expected to be significantly higher than that of Country C. Based on GERA's GDP growth forecasts, which country will most likely have the highest short-term interest rates, all else being equal? A. Country A B. Country B C. Country C
        • A is correct A.Correct because Country A is expected to have the highest level of real GDP growth rates (tied with Country C) but also high levels of volatility of real GDP growth, both of which are positively correlated with the level of real default-free interest rates. B.Incorrect because Country B is expected to have the lowest level of real GDP growth by a significant amount. Real default-free interest rates over time are positively correlated with both real GDP growth and the volatility of real GDP growth. C.Incorrect because Country C is expected to have the highest level of real GDP growth rates (tied with Country A) but with lower levels of volatility of real GDP growth. Since both of these factors are positively correlated with the level of real default-free interest rates, we would expect Country C to have lower interest rates than Country A, all else being equal.
          • 理解不了就记结论，real GDP和volatility和利率正相关
          • 想挣扎下，可以这么理解
            • 考点Taylor Rule，具体见原版书。判断影响从policy rate出发
            • 如果高增长，一般会加息降热。高波动，提高利率来应对潜在的不稳定性。
  • The yield curve on nominal default-free bonds
    • Short-term nominal interest rates
      • r(short-term) = l + θ
        • θ: expected inflation (θ, theta)
      • Expected inflation
    • Long-term nominal interest rates
      • r(long-term) = Ɩ + θ + π = Ɩ + BEI
        • π: uncertainty about actual inflation (π, pi)
        • 例题
          • An analyst, who measures yield as a combination of interest rates and premiums, observes an upward-sloping, default-free government bond nominal yield curve. Which of the following statements is correct? A. Interest rates must be expected to rise in the future. B. Bond risk premiums must be expected to rise in the future. C. Expectations relating to the future direction of interest rates are indeterminate
            • C is correct. An upward-sloping yield curve may be caused by a combination of expected rate increases and positive bond risk premiums. It may also be a combination of expectations that interest rates will be unchanged in the future coupled with positive bond risk premiums. Lastly, an upward-sloping yield curve may actually be a reflection of expected rate cuts that are more than offset by the existence of positive bond risk premiums. So, expectations relating to the future direction of interest rates are indeterminate.
              • A说的是Ɩ，B说的是θ + π。题目只说3个加起来是涨，不代表某一个或者2个一定涨。
      • Break-even inflation rate (BEI)
      • Policy rate
      • The slope of yield curve
    • 例题
      • • Based on the data in Exhibit 1, current real short-term interest rates would most likely be highest in: A. Country #1. B. Country #2. C. Country #3.
          • B is correct. Real short-term interest rates are positively related to both real GDP growth and the volatility of real GDP growth. Country 1 and Country 2 have the highest real GDP growth, as estimated by the difference between nominal GDP growth and average inflation (6.5% - 4.0% = 2.5% and 5.0% - 2.5% = 2.5%, respectively), while Country 3 has the lowest real GDP growth (3.5% - 2.0% = 1.5%). Looking at the volatility of real GDP growth, Country 2 has high real GDP growth volatility, whereas Country 1 and Country 3 have low real GDP growth volatility. Therefore, Country 2 would most likely have the highest real short-term interest rates.
            • Nominal - inflation是real的一部分，另外一部分是波动
        • The recent change in Country #3's break-even inflation rate suggests that the expected rate of inflation over the next three years is: A. less than 2.0%. B. equal to 2.0%. C. greater than 2.0%.
          • A is correct. The difference, or spread, between the yields on the country’s three-year default-free nominal bond and on the default-free real zero-coupon bond is 2.0%. This spread is known as the break-even rate of inflation (BEI), which is composed of the expected rate of inflation plus a risk premium for the uncertainty of future inflation. Because this risk premium component is most likely positive, because investors are unlikely to be very confident in their ability to predict inflation accurately, the expected rate of inflation component would be less than 2.0%.
            • Inflation有2部分，一部分是expect，一部分是uncertainty
        • Blake expresses concern that stocks may be currently overvalued in Country 3 given its 20-year historical equity index P/E of 16.0. Carlisle comments, I think the equilibrium P/E in Country #3 has increased because of changes in market conditions. Which of the following changes in market conditions best supports Carlisle's comment regarding the equilibrium P/E for Country #3? A. An increase in the equity risk premium B. A decrease in uncertainty about future inflation C. A decrease in expectation of future real earnings growth
          • B is correct. Stock prices are a function of expected cash flows discounted by inflation expectations, the uncertainty of future inflation, and the equity risk premium, among other factors. Holding all else equal, a decline in the uncertainty of future inflation would result in lower discount rates and higher valuations. This result would support a higher equilibrium P/E, thus justifying Country 3's current trailing P/E being higher than its historical average.
            • 价格上升说明是好事，只有B不确定性下降才是好事
    • 例题
      • Chan's previous experience was working as a junior analyst for an equity fund. Morgan is impressed with Chan's ability to model earnings but realizes he is still relatively inexperienced with macro forecasting models. Morgan mentions to Chan that the uncertainty of future cash flows is reflected by the discount rate, which is composed of three key components. He asks Chan if he knows what they are. Chan responds that the three distinct components can be thought of as 1. the additional return required from investing in a nominal defaultfree investment for investing in a real default-free investment; 2. the expected return on an inflation-linked bond issued by the government of a developed country; and 3. the increased premium for more actively traded securities relative to less actively traded securities. Which of Chan's comments regarding components of the discount rate is most likely correct? A. 3 B. 2 C. 1
        • B is correct. The expected return on an inflation-linked bond issued by the government of a developed country represents the return that an investor requires on a real default-free fixed-income security and is a key component of the discount rate. A is incorrect. The required risk premium is an input in determining present value. C is incorrect. The real risk-free rate is an input in determining present value.
          • 利率一般分为3部分，实际无风险+通胀+流动性 3的问题在于更流动的不应该是增加premium，而是减少。 1说的是名义-实际=通胀，2说的是实际无风险。 如果抠字眼的花，1不应该说default-free，用risk-free会更好。并且如果用government bond代替investment，会更加严谨。 所以1和2其实都是对的，不过如果抠字眼，1还是有些地方可以更好的表达，题目要求选个最合适的，所以选2。 这题建议当脏题处理，不用细究。
  • Credit risky bonds
    • Credit spread
      • Credit quality
        • Yield for credit risky bond = Ɩ + π + θ + γ
          • (gamma) γ: credit spread, or risk premium for credit risk
        • 例题
          • A corporate bond has a remaining maturity of 1 year, has a face value of EUR100, and is currently priced at EUR90.90. The real risk-free rate is 3.25%. Inflation is expected to be 2.0% next year, and the premium required by investors for inflation uncertainty is 0.25%. The implied credit risk premium embedded in the bond's price is best described as: A. equal to (100/90.90) − 1 = 10%. B. 10% reduced by the real risk-free rate and expected inflation. C. 10% reduced by the real risk-free rate, expected inflation, and the premium for inflation uncertainty.
            • C is correct. The implied credit risk premium embedded in the bond’s price is the yield (10%) less the default-risk-free nominal interest rate, which includes a premium for inflation uncertainty. See Example 15. The credit risk premium can be calculated as 4.51% in this case: γti,s = _90.90 100 - (1 + 0.0325 + 0.02 + 0.0025) . γti,s = 4.51 % .
              • 送分题
      • Company-specific factors
      • Industry sector
      • Sovereign credit risk
  • Equity risk premium
    • Yield for equity
      • Yield for equity = Ɩ + π + θ + γ + κ
        • (kappa) κ: equity risk premium relative to credit risky bond.
        • (lambda) λ= γ + κ: equity risk premium
      • Economic cylcle
        • 例题
          • Maria Silva is an equity strategist at GERA. Silva believes it is important to study the relationships between economic cycles and corporate earnings, and she has made the following observations: Observation 1: Corporate profitability tends to be a lagging indicator of the business cycle and improvements are not typically evident until a recovery is well underway. Observation 2: An understanding of the business cycle is more important for making earnings projections over the long term than the short term. Observation 3: Durable goods manufacturers have much more volatile earnings profiles than non-durable goods manufacturers because of the dramatic shifts in economic cycle-related demand. Which of Silva's observations regarding economic cycles and corporate earnings is most accurate? A. Observation 1 B. Observation 2 C. Observation 3
            • C is correct A.Incorrect because corporate profitability is considered a leading, not lagging, indicator of the business cycle. Improvements in profitability are often evident before a recovery is underway. B.Incorrect because an understanding of the business cycle is crucial for making earnings projections, particularly in the short term. C.Correct because durable goods manufacturers do have more volatile earnings profiles than non-durable goods manufacturers because of the dramatic shifts in economic cycle-related demand.
              • A说反了，盈利能力是leading指标
              • B也说反了，经济周期对短期的影响更大
              • C是对的，经济周期对耐用品影响大。因为非耐用品，不管经济怎么样，你都要买
        • Consumption hedge
          • 经济下行时，依然有稳定现金流。i.e. gold, T-bill
          • 例题
            • Which of the following financial assets is likely to offer the most effective hedge against bad consumption outcomes? A. Equities. B. Short-dated, default-free government bonds. C. Long-dated, default-free government bonds.
              • B is correct. The relative certainty about the real payoff from short-dated, default-free government bonds and, therefore, the relative certainty about the amount of consumption that the investor will be able to undertake with the payoff indicate that an investment in such bonds would be a good hedge against bad consumption outcomes.
                • 短期无风险国债才能对冲经济下行风险，长期的不行
      • Valuation multiples
      • Style strategy
  • Commercial real estate
    • Yield for commercial real estate = Ɩ + π + θ + γ + κ + φ
      • (phi) φ: risk premium for illiquidity
    • Characteristics
      • weak consumption-hedging
      • 例题
        • Which of the following statements relating to commercial real estate is correct? A. Rental income from commercial real estate is generally unstable across business cycles. B. Commercial real estate investments generally offer a good hedge against bad consumption outcomes. C. The key difference in the discount rates applied to the cash flows of equity investments and commercial real estate investments relates to liquidity.
          • C is correct. To arrive at an appropriate discount rate to be used to discount the cash flows from a commercial real estate investment, a liquidity premium is added to the discount rate applicable to equity investments. The added liquidity premium provides additional compensation for the risk that the real estate investment may be very illiquid in bad economic times.
            • 和股票比，商业房地产多了流动性的风险
• M6 Analysis of Active Portfolio Management
  • Active management and value added
    • Measuing value added
    • Decomposition of value added
      • Security selection
        • 例题
          • Bauer also evaluates Portfolio D. The portfolio has no investment constraints and is invested in an equity fund and a fixed-income fund. Exhibit 2 shows Portfolio D's strategic asset allocation and its current allocation to these funds as well as the return of the funds and their two benchmarks. The value added to Portfolio D that is attributable to security selection is closest to: A. –0.5% B. 1.4% C. 1.9%
            • C is correct Because the value added due to security selection is the product of the active return RA = RP – RB and the allocated weight to the fixed-income and equity funds, respectively; 70%(10.00% – 8.00%) + 30%(12.00% – 10.50%) = 1.40% + 0.45% = 1.85% ≈ 1.9%.
              • Security selection 和 asset allocation的计算必须熟练掌握
              • Security selection = Sum(W(p) * (R(p) - R(B)))
              • Asset allocation = Sum((W(p) - W(B)) * R(B))
        • 例题
          • • Based on Exhibit 1, the value added to the diversified asset portfolio attributable to the security selection decision in 2019 was closest to: A. 2.3%. B. 3.9%. C. 6.1%.
              • B is correct. Based on the differences in returns for the portfolio and benchmark in Exhibit 1, the value added by each asset class within the portfolio is shown in the following table: The value added from security selection is calculated as the sum of the actual portfolio weights multiplied by each sub-portfolio’s value added measure. Thus, the value added from security selection is calculated as: Value added from security selection = 0.63(5.3%) + 0.28(0.2%) + 0.09(5.1%) = 3.9%.
                • 画图，横坐标w，纵坐标R。上面的是SS，下面右边是AA。 这里要注意，strategic asset allocation的strategic代表benchmark
            • Based on Exhibit 1, the value added of the diversified asset portfolio attributable to the asset allocation decision in 2019 was closest to: A. 2.3%. B. 3.9%. C. 6.1%.
              • A is correct. The value added from asset allocation is calculated as the sum of the differences in the weights between the strategic (benchmark) allocation and the actual sub-portfolio allocation multiplied by each sub-portfolio’s benchmark return. Thus, the value added by the active asset allocation decision is calculated as Value added from asset allocation decision = 0.03(31.6%) - 0.07(-2.6%) + 0.04(28.3%) = 2.3%.
      • Asset allocation
      • 例题
        • • Based on Exhibit 1, the value added to the diversified asset portfolio attributable to the security selection decision in 2019 was closest to: A. 2.3%. B. 3.9%. C. 6.1%.
            • B is correct. Based on the differences in returns for the portfolio and benchmark in Exhibit 1, the value added by each asset class within the portfolio is shown in the following table: The value added from security selection is calculated as the sum of the actual portfolio weights multiplied by each sub-portfolio’s value added measure. Thus, the value added from security selection is calculated as: Value added from security selection = 0.63(5.3%) + 0.28(0.2%) + 0.09(5.1%) = 3.9%.
              • 画图，横坐标w，纵坐标R。上面的是SS，下面右边是AA。 这里要注意，strategic asset allocation的strategic代表benchmark
          • Based on Exhibit 1, the value added of the diversified asset portfolio attributable to the asset allocation decision in 2019 was closest to: A. 2.3%. B. 3.9%. C. 6.1%.
            • A is correct. The value added from asset allocation is calculated as the sum of the differences in the weights between the strategic (benchmark) allocation and the actual sub-portfolio allocation multiplied by each sub-portfolio’s benchmark return. Thus, the value added by the active asset allocation decision is calculated as Value added from asset allocation decision = 0.03(31.6%) - 0.07(-2.6%) + 0.04(28.3%) = 2.3%.
  • Comparing risk and return
    • Sharpe ratio
      • unaffected by leverage
      • affected by active weight
      • 例题
        • Will Smith, CFA, is a research analyst at Dark Knight Investment Firm (DKIF). During a regular meeting, Smith mentions the Frontier fund and collects the following information for S&P and Frontier fund shown in Exhibit 1.
        • Using the data in Exhibit 1, the maximum Sharpe ratio obtained by investing in both the S&P 500 benchmark portfolio and the Frontier fund is closest to: A. 0.379. B. 0.418. C. 0.565.
          • B is correct The highest squared Sharpe ratio of an actively managed portfolio is: SR2P = SR2B + IR2 SR2P = (0.379)2 + (0.176)2 SR2P = 0.174617 SRp = 0.418.
            • 核心公式，必须掌握
      • 例题
        • Bauer next examines the annual performance data in Exhibit 3 in order to determine the level of active risk that would maximize the Sharpe ratio of Portfolio D. The level of active risk that would maximize the Sharpe ratio of Portfolio D is closest to: A. 4.0% B. 4.3% C. 5.9%
          • A is correct Because "[f]or unconstrained portfolios, the level of active risk that leads to the optimal result in Equation 7 [the highest Sharpe ratio] is σA = IR/SRB × σB, where σB is the standard deviation of the benchmark's return," IR is the portfolio's information ratio and SRB is the benchmark's Sharpe ratio. Using the information presented in Exhibit 2, σA = 0.23/0.71 × 12.30% = 0.0398 ≈ 4.0%.
            • Highest Sharpe ratio的active risk的公式必须掌握
    • Information ratio
      • affected by leverage
      • unaffected by active weight
      • 例题
        • Which of Singh's statements regarding the information ratio is correct? A. Only Statement 1 B. Only Statement 2 C. Both Statement 1 and Statement 2
          • C is correct. The information ratio for a portfolio of risky assets will generally shrink if cash is added to the portfolio. Because the diversified asset portfolio is an unconstrained portfolio, its information ratio would be unaffected by an increase in the aggressiveness of active weights.
            • SR和IR的对比一定要记清楚
      • 例题
        • An analyst is given the following information about a portfolio and its benchmark. In particular, the analyst is concerned that the portfolio is a closet index fund. The T-bill return chosen to represent the risk-free rate is 0.50%. Which of the following three statements does not justify your belief that the portfolio is a closet index? i. The Sharpe ratio of the portfolio is close to the Sharpe ratio of the benchmark. ii. The information ratio of the portfolio is relatively small. iii. The active risk of the portfolio is very low. A. Statement I B. Statement II C. Statement III
          • B is correct. A closet index will have a very low active risk and will also have a Sharpe ratio very close to the benchmark. Therefore, Statements I and III are consistent with a closet index portfolio. A closet index's information ratio can be indeterminate (because the active risk is so low) and is often negative due to management fees.
            • 要注意，closet index fund的IR没特别要求
    • Construct optimal portfolios
      • SR(P)^2 = SR(B)^2 + IR^2
        • 例题
          • Martinez next estimates Fund D's sensitivity to time horizon risk and inflation risk. She wants to further understand the drivers of Fund D's risk relative to its benchmark. For this analysis, she calculates the active risk for each factor and assumes that the correlation between the model's factors is negligible. She also assumes a zero value for the error term when working with the selected two-factor model, given in Exhibit 2. The expected return for Fund D is 6.0% and it has a tracking error of 8.0%. The contribution of inflation risk to Fund D's active risk squared is closest to: A. 25% B. 31% C. 40%
            • 正确答案：A A.Correct because "[i]n analyzing risk, it is more convenient to use variances rather than standard deviations because the variances of uncorrelated variables are additive. We refer to the variance of active return as active risk squared: Active risk squared = s2(Rp – RB)", where Rp is the portfolio return and RB is the benchmark return. Active risk squared is the square of tracking error, i.e. TE = s(Rp – RB). Also, "[a]ctive factor risk is the contribution to active risk squared resulting from the portfolio’s different-from-benchmark exposures relative to factors specified in the risk model." The tracking risk is given as 8%, so the active risk squared is 82 or 64. Therefore, the contribution of inflation risk to Fund D's active risk is calculated as 16/82 = 16/64 = 0.25 = 25%.
              • 这题的难点在于公式的掌握：Active Risk Squared = Active Factor Risk Squared + Active Specific Risk Squared。所以如果用题目给的条件24+16，只能得到Active Factor Risk Squared。
              • 这题解题的关键在于理解Tracking Error是什么，tracking error本质是active risk SD，所以直接8^2=64可以得到真正的Active Risk Squared.
      • SD(A') = IR / SR(B) * SD(B)
        • SD(A'): 最优主动管理风险
      • SD(A)^2 = SD(P)^2 - SD(B)^2
        • SD(A): 实际主动管理风险
      • w(P) = SD(A') / SD(A)
        • w(B) = 1 - w(P)
      • 例题
        • • What is the maximum Sharpe ratio that a manager can achieve by combining the S&P 500 benchmark portfolio and the Indigo Fund? A. 0.333 B. 0.365 C. 0.448
            • B is correct. The highest squared Sharpe ratio of an actively managed portfolio is SRP^2 = SRB^2 + IR^2 = 0.3332 + 0.152 = 0.1334. The highest Sharpe ratio is SRP = √_0.1334 = 0.365.
              • 核心公式，必须掌握。
          • Which of the following pairs of weights would be used to achieve the highest Sharpe ratio and optimal amount of active risk through combining the Indigo Fund and benchmark portfolio, respectively? A. 1.014 on Indigo and −0.014 on the benchmark B. 1.450 on Indigo and −0.450 on the benchmark C. 1.500 on Indigo and −0.500 on the benchmark
            • A is correct. The optimal amount of active risk isσA =IR_SRB σB =0.15_0.33318.0 % = 8.11%. The weight on the active portfolio (Indigo) would be 8.11%/8.0% = 1.014, and the weight on the benchmark portfolio would be 1 - 1.014 = -0.014. We can demonstrate that these weights achieve the maximum Sharpe ratio (of 0.365). Note that 8.11% is the optimal level of active risk and that Indigo has an expected active return of 1.014(1.2%) = 1.217% over the benchmark and a total excess return of 6.0% + 1.217% = 7.217%. The portfolio total risk is σP2 = σB2 + σA2 = 18.02 + 8.1112 = 389.788. Taking the square root, σP = 19.743, and the optimal Sharpe ratio is indeed 7.217/19.743 = 0.365.
              • 核心公式，必须掌握。 要注意，这里实际主动风险已经给了，active risk = 8%，题目给的另外一个方差是return的方差，不少risk的方差，不要用错了参数。
  • Fundamental law of active management
    • Key parameters
      • The correlation triangle
        • TC: correlation between forecasted active return and active weight
          • 执行能力
        • IC: Correlation between forecast return and realized return
          • 预测能力
          • 例题
            • Based on the fundamental law of active management, which coefficient measures the correlation between forecasted active returns and realized active returns? A. Information coefficient. B. Transfer coefficient. C. Breadth.
              • 答案：A 解析：IC衡量的是基金经理预测的准确程度，可以用预测的主动收益和真实的 主动收益之间的相关性来衡量。因此，A选项正确。 TC衡量的是基金经理的执行力，可以用预测的主动收益和操作时的主动权重之 间的相关性来衡量。因此，B选项错误。 BR衡量的是基金经理的努力程度，可以用一年中做独立主动决策的次数来衡量。 因此，C选项错误。
                • IC是用来衡量预估和实际的差距
        • BR
          • 努力程度
        • 例题
          • Two analysts make the following statements about the transfer coefficient in the expanded fundamental law of active management: Analyst One says, "The transfer coefficient measures how well the realized returns correlate with the anticipated returns, adjusted for risk." Analyst Two says, "The transfer coefficient measures how well the realized returns correlate with the active weights, adjusted for risk." Which, if either, analyst is correct? A. Only Analyst One is correct. B. Only Analyst Two is correct. C. Neither analyst is correct.
            • C is correct. The transfer coefficient measures how well the anticipated (ex ante), risk-adjusted returns correlate with the risk-adjusted active weights. This is also expressed in the equation for the transfer coefficient: TC = ρ(μi/σi,Δwiσi).
              • IC是看实际和预估的转化率，重点的信息质量 TC是预估的回报和权重的关系，受到对权重的限制影响
      • Basic & full fundamental law
        • Transfer coefficient (TC)
          • 例题
            • Bauer meets with the manager of Portfolio D. She informs the manager that the family office will require ESG considerations to be incorporated into all of their investments going forward. The manager considers how the imposition of ESG constraints will affect their performance as measured by the fundamental law of active management. In terms of the fundamental law, the family office's investment requirements will most likely lower Portfolio D's: A. active risk only. B. transfer coefficient only. C. active risk and transfer coefficient.
              • B is correct A.Incorrect because Portfolio D will be under the constraint of having fewer eligible securities available; the risk level he prefers to take with the available securities is not constrained. Active risk is "the volatility of the active return." B.Correct because Portfolio D is currently unconstrained and therefore its transfer coefficient equals 1; the planned addition of ESG constraint would reduce this. "Like the information coefficient, the transfer coefficient, TC, is a simple multiplicative factor in the fundamental law. It measures the extent to which constraints reduce the expected value added of the investor’s forecasting ability. In the absence of constraints, the transfer coefficient is approximately 1.00, resulting in the basic form of the fundamental law. However, in practice, investors often work under constraints that result in TC values between 0.20 and 0.80." C.Incorrect because while Portfolio D's transfer coefficient will be lowered by the added constraints, its active risk will be unaffected.
                • 要理解active risk和transfer coefficient的定义和作用
                • 这里active risk不变的原因是，虽然可选的数量再减少，但是只是影响actual和optimal active weights的相关性，即transfer coefficient。但是我们可选数量少了，还是可以调整到active risk不变的。
        • Information coefficient (IC)
        • Breadth (BR)
          • 例题
            • Based on the fundamental law of active management, if a portfolio manager has an information ratio of 0.3, an information coefficient of 0.2, and a breadth of 9, the portfolio manager will most likely: A. face no constraints during execution steps. B. have an portfolio return which is higher than the benchmark return. C. make nine independent active decisions per month.
              • 答案：B 解析：信息比率等于主动收益除以主动风险，由于本题中信息比率大于0，而主 动风险一定大于等于0，可以得出，主动回报大于0，组合收益高于基准收益。 因此，B选项正确。 根据FLOAM公式，计算可得TC为0.5，意味着该基金经理在执行决策时受到一定 的限制。因此，A选项错误。 Breadth代表的是一年中独立主动决策的次数，并非一个月中的次数。因此，C 选项错误。
                • IR=Active Return / Active Risk。因为IR大于0，说明active return一定大于0. BR是一年中的独立次数
        • 例题
          • Manager 1 has an information coefficient of 0.15, a transfer coefficient of 1.0, and invests in 50 securities. Manager 2 has a different strategy, investing in more securities; however, he is subject to investment constraints that reduce his transfer coefficient. Manager 2 has an information coefficient of 0.10, a transfer coefficient of 0.8, and invests in 100 securities. The investment selections of each manager are independent decisions. If both managers target an active risk of 5.0%, which manager will have the greater expected active return? A. Manager 1 B. Manager 2 C. Both managers will have the same active return.
            • A is correct. Manager 1's IR = TC × IC × √_BR = 1.0 × 0.15 × √_50 = 1.06. Manager2's IR = 0.8 × 0.10 × √_100 = 0.80. Manager 1's active return is 1.06(5.0) = 5.3%, and Manager 2's expected active return is 0.80(5.0) = 4.0%. Manager 1 has the greater expected active return.
              • 这里要注意，得到IR后，还不是题目问的active return。因为IR = Active Return / Active risk。所以用IR * Active Risk才能得到Active Return。
    • Applications
      • 例题
        • Bauer identifies three unconstrained, diversified fixed-income portfolios with three different managers as potential investments. Each manager has the same benchmark. To evaluate the managers, Bauer reviews the annual performance data in Exhibit 1 and assumes a risk-free rate of 0.50%. Which portfolio in Exhibit 1 has the highest information ratio? A. Portfolio A B. Portfolio B C. Portfolio C
          • B is correct Because "[t]he information ratio tells the investor how much active return has been earned, or is expected to be earned, for incurring the level of active risk. Active return, RA, is the difference between the managed portfolio return, RP, and the benchmark portfolio return, RB. The information ratio of an actively managed portfolio, IR, is calculated by dividing the active return, RA, by active risk σA:" IR = (RP – RB)/σ(RP – RB) = RA/σA. Using the figures presented in Exhibit 1: For Portfolio A, IR = (8.80% – 8.75%)/3.06% = 0.05%/3.06% = 0.016 For Portfolio B, IR = (9.40% – 8.75%)/3.32% = 0.65%/3.32% = 0.196 For Portfolio C, IR = (10.50% – 8.75%)/9.46% = 1.75%/9.46% = 0.185 Therefore, Portfolio B has the highest information ratio.
            • 灵活运用IR的公式来计算
    • Limitations
• 特别提醒
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  • 的是重要知识点，颜色越深越重要
  • 是习题，颜色越深越重要
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