FRM总复习计算题型汇总
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Answer: h = 0.925* 0.1 /0.125=0.74 number of contracts to hedge = 0.74*
6,250,000/62500=74 contracts
12
2.运用股指期货避险
公式：
注意：本公式隐含条件是将现货头寸的风险完全对冲
13
7
3. Tracking error, Information ratio, Sortino ratio
例二： Over a 10-year period, a manager uses a covered call strategy to enhance the return of the index fund she manages. The record of the fund’s return is:(0.095, 0.08, -0.022, 0.11, 0.09, -0.05, 0.035, 0.124, 0.072, 0.055).
Answer: 1.4 ×$20,000,000 /（1150×250）≈97contracts
14
二、Financial Markets and Products 3. Adjusting the Portfolio Beta
Let β be our portfolio beta, β* be our target beta after we implement the strategy with index futures, P be our portfolio value, and A be the value of the underlying asset.
To compute the appropriate number of futures:
number of contracts
ρ = CovS,F σSσF
and
CovS,F × σS σSσF σF
=
Cov S,F σF2
= βS,F
11
二、Financial Markets and Products 1. optimal hedge ratio
Suppose our currency trader from the previous example increased the sophistication of his trade and computed the correlation between the spot and futures to be 0.925, the annual standard deviation of the euro exchange rate to be $0.1, and the annual standard deviation of the British pound futures to be $0.125. Compute the number of futures to be sold to hedge the position.
measure 和Jensen measure 来看是优于市场的。但如果用 Sharpe measure 来看，结果则相反。 【基金经理人最在意Sharpe measure】这代表此投资组合相对于其 系统风险而言，有较佳的绩效，但这投资组合比起市场而言，是 较不风险分散的。这可由其标准差(25%)大于市场标准差(20.2%) 看出。
9
4. APT模型之运用 计算Excess return
An analyst believes equity prices are completely explained by the market risk premium, earnings growth, earnings yield (E/P), and the dividend payout ratio (D/E). The analyst uses beta to represent exposure to the market risk premium and standardizes earnings growth, earnings yield, and the dividend payout ratio. The analyst's forecasted factor returns are as follows:
Calculate the Treynor, Sharpe, and Jensen measures.
5
解析： Portfolio Treynor measure = (0.14-0.026) / 1.1 = 0.136 Sharpe measure = (0.14-0.026) / 0.25 = 0.456 Jensen measure = 0.14 – 0.026 – [0.125-0.026](1.1) = 0.0051 Market Treynor measure of the market = (0.125-0.026) / 1 = 0.099 Sharpe measure of the market = (0.125-0.026) / 0.202 = 0.4901 Jensen measure of the market = 0.125-0.026-(0.125-0.026)(1) = 0.0 由上面的例子可以看出，这10 支股票的投资组合绩效以Treynor
σp
RF
⎤ ⎥ ⎦
E (Rp ) − RF −
[E
(RM
)
−
RF
]β p
不同点
相同点
1，反映超额收益对系统性 风险的补偿 2，展望未来 3，更加适用于有效分散的 组合比较
1，反映超额收益对总风险 的补偿 2，回顾过去 3，标准差可用于所有组 合，应用更加广泛
1.值越大
越好 2.T和S均 代表单位 风险的超 额收益
FRM考试计算题型总复习
20100424
Financial Risk Management
2009 CopyRight by DMC
王刚
1
计算要求分布
风险管理基础 金融市场与产品 信用风险管理 操作风险管理 投资组合管理
2
一、风险管理基础 1. CAPM模型
z CML线
E(Rp)=
1，反映超额收益取得情况 2，绝对值指标，更加适用 于相同β的组合比较
4
例一：For a portfolio of 10 stocks, the estimates of the individual stocks, that the portfolio’s expected return is 14% with a standard deviation of 25%. The beta of the portfolio is 1.1.The expected return of the market is 12.5% with a standard deviation of 20.2%. The riskfree rate is 2.6%.
R F
+
⎡E(R )−
⎢⎣
M
σ M
R F
⎥⎦⎤σ P
z SML线（CAPM）
E(Ri)=RF+[E(RM)RF]βi
3
2. Performance Measurement
指标 Treynor
Sharpe
Jensen’s alpha
公式
⎡ ⎢ ⎣
E(Rp ) − βp
RF
⎤ ⎥ ⎦
⎡ ⎢ ⎣
E(Rp ) −
二、Financial Markets and Products 2. Hedging with stock index futures
You are a portfolio manager with a $20 million growth portfolio that has beta of 1.4, relative to the S&P 500. The S&P 500 futures are trading at 1,150, and the multiplier is 250. You would like to hedge your exposure to market risk over the next few months. Identify whether a long or short hedge is appropriate, and determine the number of S&P500 contracts you need to implement the hedge.
10
二、Financial Markets and Products 1. optimal hedge ratio
h
=
ρS,F
σS σF
,
which is also the beta of spot prices with respect to
futures contract prices since
Market risk premium = 3% Earnings growth = -2% Earnings yield = 2% What is the expected excess return for a stock with the following
factor exposures? Beta = 1 Earnings growth = -0.5 Earnings yield = 0.5 E(R) = (0.03 × 1) + (-0.02 × -0.5) + (0.02 × 0.5) = 0.05%
tracking error,the information ratio, and the Sortino ratio.
8
Tracking error 首先要先计算投资组合和benchmark 间的差异： (0.008, 0.002, 0.012, -0.014, -0.01, 0.014, 0.007, -0.007, 0.01, -0.004)。 差异数值的平均数 = 0.0018, Tracking error 就是差异数值的标准差 = tracking error = 0.00992 (样本标准差，分母是除以N – 1 = 9) Information ratio = 差异数值平均数 ÷ 差异数值的标准差 = 0.0018 / 0.00992 = 0.1815 Sortino ratio 10 个投资组合报酬的平均数是0.0519 Sortino ratio 分子 = 投资组合报酬平均数－minimum acceptable return = 0.0519 – 4% = 0.0119 Sortino ratio 分母 = 0.0017569 0.0419 min MSD = = 其中， min MSD 是(0, 0, -0.062, 0, 0, -0.09, -0.075, 0, 0, 0)的平方 和除以10，也就是0.0017569。 ∴ Sortino ratio = 0.0119 ÷ 0.0419 = 0.2840
The corresponding benchmark returns record is: (0.087, 0.078, -0.034, 0.124, 0.10, -0.064, -0.042,
0.131, 0.062, 0.059). The minimum acceptable return is 4%. Calculate the
6
3. Tracking error, Information ratio, Sortino ratio
Tracking error（跟踪误差）：组合偏离基准收 益的标准差
Information ratio(信息比率）：相对于基准的超 额收益除以跟踪误差
Sortino ratio:相对于基准（最低要求）的超额收 益除以下方标准差
6,250,000/62500=74 contracts
12
2.运用股指期货避险
公式：
注意：本公式隐含条件是将现货头寸的风险完全对冲
13
7
3. Tracking error, Information ratio, Sortino ratio
例二： Over a 10-year period, a manager uses a covered call strategy to enhance the return of the index fund she manages. The record of the fund’s return is:(0.095, 0.08, -0.022, 0.11, 0.09, -0.05, 0.035, 0.124, 0.072, 0.055).
Answer: 1.4 ×$20,000,000 /（1150×250）≈97contracts
14
二、Financial Markets and Products 3. Adjusting the Portfolio Beta
Let β be our portfolio beta, β* be our target beta after we implement the strategy with index futures, P be our portfolio value, and A be the value of the underlying asset.
To compute the appropriate number of futures:
number of contracts
ρ = CovS,F σSσF
and
CovS,F × σS σSσF σF
=
Cov S,F σF2
= βS,F
11
二、Financial Markets and Products 1. optimal hedge ratio
Suppose our currency trader from the previous example increased the sophistication of his trade and computed the correlation between the spot and futures to be 0.925, the annual standard deviation of the euro exchange rate to be $0.1, and the annual standard deviation of the British pound futures to be $0.125. Compute the number of futures to be sold to hedge the position.
measure 和Jensen measure 来看是优于市场的。但如果用 Sharpe measure 来看，结果则相反。 【基金经理人最在意Sharpe measure】这代表此投资组合相对于其 系统风险而言，有较佳的绩效，但这投资组合比起市场而言，是 较不风险分散的。这可由其标准差(25%)大于市场标准差(20.2%) 看出。
9
4. APT模型之运用 计算Excess return
An analyst believes equity prices are completely explained by the market risk premium, earnings growth, earnings yield (E/P), and the dividend payout ratio (D/E). The analyst uses beta to represent exposure to the market risk premium and standardizes earnings growth, earnings yield, and the dividend payout ratio. The analyst's forecasted factor returns are as follows:
Calculate the Treynor, Sharpe, and Jensen measures.
5
解析： Portfolio Treynor measure = (0.14-0.026) / 1.1 = 0.136 Sharpe measure = (0.14-0.026) / 0.25 = 0.456 Jensen measure = 0.14 – 0.026 – [0.125-0.026](1.1) = 0.0051 Market Treynor measure of the market = (0.125-0.026) / 1 = 0.099 Sharpe measure of the market = (0.125-0.026) / 0.202 = 0.4901 Jensen measure of the market = 0.125-0.026-(0.125-0.026)(1) = 0.0 由上面的例子可以看出，这10 支股票的投资组合绩效以Treynor
σp
RF
⎤ ⎥ ⎦
E (Rp ) − RF −
[E
(RM
)
−
RF
]β p
不同点
相同点
1，反映超额收益对系统性 风险的补偿 2，展望未来 3，更加适用于有效分散的 组合比较
1，反映超额收益对总风险 的补偿 2，回顾过去 3，标准差可用于所有组 合，应用更加广泛
1.值越大
越好 2.T和S均 代表单位 风险的超 额收益
FRM考试计算题型总复习
20100424
Financial Risk Management
2009 CopyRight by DMC
王刚
1
计算要求分布
风险管理基础 金融市场与产品 信用风险管理 操作风险管理 投资组合管理
2
一、风险管理基础 1. CAPM模型
z CML线
E(Rp)=
1，反映超额收益取得情况 2，绝对值指标，更加适用 于相同β的组合比较
4
例一：For a portfolio of 10 stocks, the estimates of the individual stocks, that the portfolio’s expected return is 14% with a standard deviation of 25%. The beta of the portfolio is 1.1.The expected return of the market is 12.5% with a standard deviation of 20.2%. The riskfree rate is 2.6%.
R F
+
⎡E(R )−
⎢⎣
M
σ M
R F
⎥⎦⎤σ P
z SML线（CAPM）
E(Ri)=RF+[E(RM)RF]βi
3
2. Performance Measurement
指标 Treynor
Sharpe
Jensen’s alpha
公式
⎡ ⎢ ⎣
E(Rp ) − βp
RF
⎤ ⎥ ⎦
⎡ ⎢ ⎣
E(Rp ) −
二、Financial Markets and Products 2. Hedging with stock index futures
You are a portfolio manager with a $20 million growth portfolio that has beta of 1.4, relative to the S&P 500. The S&P 500 futures are trading at 1,150, and the multiplier is 250. You would like to hedge your exposure to market risk over the next few months. Identify whether a long or short hedge is appropriate, and determine the number of S&P500 contracts you need to implement the hedge.
10
二、Financial Markets and Products 1. optimal hedge ratio
h
=
ρS,F
σS σF
,
which is also the beta of spot prices with respect to
futures contract prices since
Market risk premium = 3% Earnings growth = -2% Earnings yield = 2% What is the expected excess return for a stock with the following
factor exposures? Beta = 1 Earnings growth = -0.5 Earnings yield = 0.5 E(R) = (0.03 × 1) + (-0.02 × -0.5) + (0.02 × 0.5) = 0.05%
tracking error,the information ratio, and the Sortino ratio.
8
Tracking error 首先要先计算投资组合和benchmark 间的差异： (0.008, 0.002, 0.012, -0.014, -0.01, 0.014, 0.007, -0.007, 0.01, -0.004)。 差异数值的平均数 = 0.0018, Tracking error 就是差异数值的标准差 = tracking error = 0.00992 (样本标准差，分母是除以N – 1 = 9) Information ratio = 差异数值平均数 ÷ 差异数值的标准差 = 0.0018 / 0.00992 = 0.1815 Sortino ratio 10 个投资组合报酬的平均数是0.0519 Sortino ratio 分子 = 投资组合报酬平均数－minimum acceptable return = 0.0519 – 4% = 0.0119 Sortino ratio 分母 = 0.0017569 0.0419 min MSD = = 其中， min MSD 是(0, 0, -0.062, 0, 0, -0.09, -0.075, 0, 0, 0)的平方 和除以10，也就是0.0017569。 ∴ Sortino ratio = 0.0119 ÷ 0.0419 = 0.2840
The corresponding benchmark returns record is: (0.087, 0.078, -0.034, 0.124, 0.10, -0.064, -0.042,
0.131, 0.062, 0.059). The minimum acceptable return is 4%. Calculate the
6
3. Tracking error, Information ratio, Sortino ratio
Tracking error（跟踪误差）：组合偏离基准收 益的标准差
Information ratio(信息比率）：相对于基准的超 额收益除以跟踪误差
Sortino ratio:相对于基准（最低要求）的超额收 益除以下方标准差