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　　FRM一級(jí)數(shù)量分析

　　精選問(wèn)答1

　　題干

　　Monte Carlo simulation is a widely used technique in solving economic and financial problems. Which of the following statements is not a limitation of the Monte Carlo technique when solving problems of this nature?

　　選項(xiàng)A

　　High computational costs arise with complex problems.

　　選項(xiàng)B

　　Simulation results are experiment-specific because financial problems are analyzed based on a specific data generating process and set of equations.

　　選項(xiàng)C

　　Results of most Monte Carlo experiments are difficult to replicate.

　　選項(xiàng)D

　　If the input variables have fat tails, Monte Carlo simulations are not relevant because it always draws random variables from a normally distributed population.

　　答案解析

　　A disadvantage of Monte Carlo simulation is that imprecise results may be present when he assumptions of model inputs or data generating process are unrealistic.

　　The distribution of input variables does not need to be the normal distribution. The problem arises when a variable in the real world is fat-tailed, but a model could erroneously draw option prices from a normal distribution.

　　解題思路

　　選擇D是因?yàn)椋商乜迥M中的隨機(jī)變量，并不一定要服從正態(tài)分布，是可以根據(jù)實(shí)際情況，來(lái)設(shè)定它服從其他分布的。也因此，歷史上從未發(fā)生的數(shù)據(jù)，都可以被獲取到，解決wha t if問(wèn)題。D所說(shuō)的并不對(duì)，因此就不是一個(gè)limitation。

　　還有同學(xué)對(duì)B選項(xiàng)有疑問(wèn)：蒙特卡洛模擬的結(jié)果(輸出)，要依賴(lài)模擬之前對(duì)模型的假設(shè)，所以它每一次試驗(yàn)(模擬)都是specific的。所以說(shuō)對(duì)模型的假設(shè)、通過(guò)這個(gè)假設(shè)所輸入的數(shù)據(jù)，就會(huì)非常重要，影響模擬的結(jié)果。所以就比較容易出誤差。

　　易錯(cuò)點(diǎn)分析：

　　蒙特卡洛模擬中，并不要求變量服從正態(tài)分布。

　　精選問(wèn)答2

　　題干

　　Consider the following linear regression model: Y=a+bX+e. Suppose a=0.05, b=1.2, SD(Y) = 0.26, and SD(e) = 0.1. What is the correlation between X and Y?

　　選項(xiàng)A

　　0.923

　　選項(xiàng)B

　　0.852

　　選項(xiàng)C

　　0.701

　　選項(xiàng)D

　　0.462

　　答案解析

首先通過(guò)線性回歸式，在兩邊同取方差，得到解析中第二行的式子

　　解題思路

　　首先通過(guò)線性回歸式，在兩邊同取方差，得到解析中第二行的式子，即：

　　V(y)=V(a+bX+e)=b^2 * V(x)+V(e)，所以V(x)=0.04，然后把X的標(biāo)準(zhǔn)差求出：SDx=0.2

　　最后以下這個(gè)公式，即可計(jì)算得出相關(guān)系數(shù)rho。

最后以下這個(gè)公式，即可計(jì)算得出相關(guān)系數(shù)rho

　　易錯(cuò)點(diǎn)分析：

　　考察帶有殘差項(xiàng)的線性回歸算相關(guān)系數(shù)，本題將線性回歸、方差的性質(zhì)和協(xié)方差與相關(guān)系數(shù)的考點(diǎn)綜合在一起考察，稍微有點(diǎn)綜合。

　　精選問(wèn)答3

　　題干

　　A portfolio manager is interested in the systematic risk of a stock portfolio, so he estimates the linear regression: R_p–R_f = α + β * [ R_M –R_f ]+ ε ,where R_p is the return of the portfolio at time t, R_M is the return of the market portfolio at time t, and R_f is the risk-free rate, which is constant over time. Suppose that α = 0.008,β= 0.977, σ( R_p ) = 0.167, and σ( R_M ) = 0.156.

　　What is the approximate coefficient of determination in this regression?

　　選項(xiàng)A

　　0.913

　　選項(xiàng)B

　　0.834

　　選項(xiàng)C

　　0.977

　　選項(xiàng)D

　　0.955

　　答案解析

　　the R-squared is given by β^2 * σ_M^2 / σ_P^2 = 0.977^2 × 0.156^2 / 0.167^2 =0.83

　　解題思路

　　在一元線性回歸中，R^2=XY相關(guān)系數(shù)的平方，而相關(guān)系數(shù)=b*X的標(biāo)準(zhǔn)差/Y的標(biāo)準(zhǔn)差。把相關(guān)系數(shù)平方一下，就得到R方了。

　　即：β=ρ*σp/σm，變形后ρ=β*σm/σp，再等號(hào)兩邊同取平方。

　　精選問(wèn)答4

　　題干

　　Assume you are using a GARCH model to forecast volatility that you use to calculate the one-day VAR. If volatility is mean reverting, what can you say about the T-day VAR?

　　選項(xiàng)A

　　It is less than the T^0.5 × one-day VAR.

　　選項(xiàng)B

　　It is equal to T ^0.5× one-day VAR.

　　選項(xiàng)C

　　It is greater than the T^0.5 × one-day VAR.

　　選項(xiàng)D

　　It could be greater or less than the T^0.5 × one-day VAR

　　答案解析

　　If the initial volatility were equal to the long-run volatility, then the T-day VAR could be computed using the square root of time rule, assuming normal distributions.

　　If the starting volatility were higher, then the T-day VAR should be less than the T ^0.5× one-day VAR.

　　Conversely, if the starting volatility were lower, then the T-day VAR should be more than the long-run value. However, the question does not indicate the starting point. Hence, answer d. is correct.

　　解題思路：

　　波動(dòng)率是均值回歸的。如果初始的波動(dòng)率=長(zhǎng)期波動(dòng)率，那么因?yàn)樗芯祷貧w的特點(diǎn)，之后每天的波動(dòng)率都等于長(zhǎng)期的波動(dòng)率也就等于現(xiàn)在的波動(dòng)率，VaR可以使用平方根法則。

　　但是，如果現(xiàn)在的波動(dòng)率小于長(zhǎng)期波動(dòng)率，那么根據(jù)均值回歸的特點(diǎn)，它之后的波動(dòng)率會(huì)慢慢回歸，也就是要變大。波動(dòng)率要慢慢變大，那么VaR也就要慢慢變大。就會(huì)大于最開(kāi)始的VaR乘上根號(hào)n。

　　反之，也是一樣的道理。

　　易錯(cuò)點(diǎn)分析

　　要分清波動(dòng)率均值回歸還是return均值回歸。

　　如果是價(jià)格的均值回歸，也就是說(shuō)隨著時(shí)間的推移，它價(jià)格越來(lái)越回歸到長(zhǎng)期均值水平，那么它的波動(dòng)率就會(huì)越來(lái)越小。

　　精選問(wèn)答5

　　題干

　　Consider two stocks, A and B. Assume their annual returns are jointly normally distributed, the marginal distribution of each stock has mean 2% and standard deviation 10%, and the correlation is 0.9. What is the expected annual return of stock A if the annual return of stock B is 3%?

　　選項(xiàng)A

　　選項(xiàng)B

　　2.9%

　　選項(xiàng)C

　　4.7%

　　選項(xiàng)D

　　1.1%

　　答案解析

　　The information in this question can be used to construct a regression model of A on B. We have R A =2% + 0.9*( 10%/10% )*( R_B ?2% ) + ε . Next, replacing R_B by 3% gives R_A = 2% + 0.9*(3% - 2%) = 2.9%.

　　解題思路：

　　何老師在課上有講過(guò)這種題型，是應(yīng)該用線性回歸來(lái)做。當(dāng)時(shí)用的是Y=a+bX+隨機(jī)項(xiàng)ε這一形式。這個(gè)形式和本題解析只是形式上的區(qū)別。

　　本題把2%移到等式左邊，就相當(dāng)于是用Ra-2%=β(Rb-2%)+隨機(jī)項(xiàng) 來(lái)做回歸(這形式有點(diǎn)像單因素模型那的回歸)。

　　如果用Ra=a+β*Rb+隨機(jī)項(xiàng)ε 來(lái)做回歸，截距項(xiàng)就是0.2%。

　　因?yàn)镽a-2%=β(Rb-2%)+隨機(jī)項(xiàng)，也就是本題的回歸Ra=2%+β(Rb-2%)+隨機(jī)項(xiàng)中，β(Rb-2%)里，-2%*β=-1.8%，把它開(kāi)出來(lái)和2%相加，就變成0.2%，也就是何老師上課講的形式。

　　易錯(cuò)點(diǎn)分析：

　　求這種條件期望時(shí)，可以轉(zhuǎn)化成線性回歸的形式來(lái)做。

　　精選問(wèn)答6

　　題干

　　An analyst calculates the sum of squared residuals and total sum of squares from a multiple regression with four independent variables to be 4,320 and 9,105, respectively. There are 65 observations in the sample.

　　The critical F- value for testing H0 = B1 = B2 = B3 = B4 = 0 vs. HA: at least one Bj ≠ 0 at the 5% significance level is closest to:

　　選項(xiàng)A

　　2.37.

　　選項(xiàng)B

　　2.53.

　　選項(xiàng)C

　　2.76.

　　選項(xiàng)D

　　3.24.

　　答案解析

　　This is a one-tailed test, so the critical F-value at the 5% significance level with 4 and 60 degrees of freedom is approximately 2.53.

　　解題思路：

　　1、因?yàn)槭锹?lián)合檢驗(yàn)H0=B1=B2=...=0，所以用F-test。

　　2、F-test的自由度是一對(duì)數(shù)字：前一個(gè)數(shù)字，因?yàn)閒our independent variables (k)，所以是4，后一個(gè)數(shù)字=n-k-1=65-4-1=60 (65 observations, 4 independent variable)。

　　3、最后去查F(4,60)，得到2.53 (只需要你判斷出要求的是F(4,60)，之后就可以查表了。FRM考試是是給你幾個(gè)F(a,b)的值，讓你選)。并且注意，F(xiàn)檢驗(yàn)是單尾檢驗(yàn)，而不是雙尾。

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