A random variable X with an exponential distribution has the following probability density function: FX(x) = λe- xλYou are required to determine the probability density function of the random variable X, given that Y = X2 A 1 – e-λy1/2 B – e-λy1/2 C 1/2 λy- ½ e- λy1/2 D 1 The correct answer is: C From first principles, it can be shown that:FY(y) = P(Y ≤ y) = P(X2 ≤ y) = P(X ≤ y1/2) \(P[X\le { y }^{ \frac { 1 }{ 2 } }]=\int _{ 0 }^{ \frac { 1 }{ 2 } }{ \lambda { e }^{ -\lambda x }dx } \) From calculus, \(\int { { e }^{ -cx }dx=-\frac { 1 }{ c } } { e }^{ -cx }\quad\)Thus, \(P[X\le { y }^{ 1/2 }]=\int _{ 0 }^{ { y }^{ 1/2 } }{ \lambda { e }^{ -\lambda x }dx } ={ \left| \frac { -\lambda }{ \lambda } { e }^{ -\lambda x } \right| }_{ 0 }^{ { y }^{ 1/2 } }\\ =-{ e }^{ -\lambda { y }^{ 1/2 } }--{ e }^{ 0 }\\ =1-{ e }^{ -\lambda { y }^{ 1/2 } }\\ P[X\le { y }^{ 1/2 }]={ F }_{ Y }(y)=1-{ e }^{ -\lambda { y }^{ 1/2 } }\\ { f }_{ Y }(y)={ F }'_{ Y }(y)=0-\left( -\frac { 1 }{ 2 } \lambda { y }^{ -1/2 }{ e }^{ -\lambda { y }^{ 1/2 } } \right) \\ =\frac { 1 }{ 2 } \lambda { y }^{ -1/2 }{ e }^{ -\lambda { y }^{ 1/2 } }\) *User Question: I could not understand the problem and its solution at all. Can you please explain it from scratch?

FRM Part 1An analyst is comparing the STDEV or GARCH methodology with that of the RiskMetric® approach for estimating VaR using historical data. He wrote down the following similarities between both methods. Which of the following similarities is incorrect? A Both methods belong to the parametric class of risk assessing models B Both methods attempt to estimate conditional volatility C Both methods apply equal weights to all the periods D Both methods use recent historic data for assessing risk The correct answer is: C The standard deviation models STDEV or GARCH apply equal weights to all the windows of past data, while the RiskMetric® approach applies higher weights on more recent data. The weights decline exponentially to zero as returns become older. *User Question: Does GARCH apply equal weights to all past period? I think not

FRM Part 1Jasmine Forst is a risk manager at Lifelong Insurance Company. The company has a number of outstanding exposures in various foreign currencies. Today, she is analyzing the company’s current outstanding exposures in foreign currencies to derive the possible effects of exchange rates on these exposures. Which of the following is true regarding Lifelong Insurance Company? A If the company has a net short position in a specific foreign currency, then the company’s risk increases if the value of the foreign currency depreciates against the dollar B If the company has a net short position in a specific foreign currency, then the company’s risk increases if the value of the foreign currency appreciates against the dollar C If the company has a net long position in a specific foreign currency, then the company’s risk increases if the value of the foreign currency appreciates against the dollar D If the company has a net long position in a specific foreign currency, then the company’s risk increases if the value of the domestic currency depreciates against the dollar The correct answer is: B If the company has a net short position, then the risk related to the foreign exchange increases as the foreign currency rises in value. For instance, suppose the company has a net short position of -€100, and the exchange rate is $1.2/€. The firm can purchase 100 euros for 120 dollars to balance the equation. Now suppose the euro appreciates to $1.5/€, and it becomes more expensive to buy euros in terms of dollars. Then, the company has to spend $150 to balance the equation. Therefore, the company’s risk increases. *User Question: Why option D is not right?

FRM Part 1Which of the following statements is (are) correct?I. If the autoregressive (p) in an ARIMA model is 1, there is no autocorrelation in the series. II. If (d), the integrated component in an ARIMA model is 0, the series is not stationaryIII. There is autocorrelation in a series with lag 1 if the moving average component (q) in an ARIMA model is 1 A I and II B II only C III only D All the above The correct answer is: C P denotes the autoregressive (AR) parameter. If p = 0, then there is no auto-correlation in the series. Otherwise if p = 1, it means that there is autocorrelation with lag 1. If q = 1 in ARIMA, it means there is autocorrelation with 1 lag. *User Question: Why is II incorrect?

FRM Part 1Roderick Jaynes, FRM, analyzed historical sales \((S)\) for over 20 years and found that sales are increasing but its growth rate over the period is relatively constant. Which model is most suitable to forecast out-of-sample sales? A \(S_t=\beta_0+\beta_1 \times S_{t-1}\) B \(S_t=\beta_0+\beta_1 \times t\) C \(lnS_t=ln(\beta_0+\beta_1 \times t)\) D \(S_t=\beta_0+\beta_1 \times t+\beta_2 \times t^2\) The correct answer is: C The log linear model exhibits best fit for data having constant growth rates. *User Question: Why is answer B incorrect?

FRM Part 1An investment bank offers its customers the option to carry out leveraged trades. The investors are required to maintain a margin of 30% and pay a commission of 0.25% of the trade value. An investor acquires 2,000 shares each at a price of $30. If the shares are currently trading at $40 and the borrowing cost is 8%, then the return generated by the leveraged trade is closest to: A 89.75% B 9.08% C 90.75% The correct answer is: A) Total fund required to acquire 2,000 shares = 2,000 * $30 = $60,000 Margin required for the trade = $60,000 * 30% = $18,000 Commission = $60,000 * 0.25% = $150 Total out-of-pocket investment required for the trade = $18,000 + $150 = $18,150 Total funds required = $60,000 + $150 = $60,150 Funds borrowed = $60,150 - $18,150 = $42,000 Interest cost = $42,000 * 8% = $3,360 Profit earned in the leveraged trade = 2,000*($40-$30) - Commissions - Interest paid = 2,000*$10 - $60,000*0.25% - $80,000*0.25% - $3360 = $16,290 ROE = $16,290 / ($18,150) = 89.75% *User Question: Isnt the profit equation double counting the commissions? ?

How much money will you have if you invest $100,000 today in a project paying 8% interest rate compounded continuously for 3 years? A $127,124.90 B $108,328.70 C $125,971.20 The correct answer is: A) PV=100,000; r=8%=0.08; N=3;FV = PV*erN = 127,124.90Note: The question asks about continuous compounding. You don't have to use your financial calculator to solve this problem. You also have to use the the constant ''e'' which is 2.7182. *User Question: How do we calculate this on the Calculator because I put in all of those values and keep coming to $125,971.20. please explain? ?

An investment bank offers its customers the option to carry out leveraged trades. The investors are required to maintain a margin of 30% and pay a commission of 0.25% of the trade value. An investor acquires 2,000 shares each at a price of $30. If the shares are currently trading at $40 and the borrowing cost is 8%, then the return generated by the leveraged trade is closest to: A 89.75% B 9.08% C 90.75% The correct answer is: A) Total fund required to acquire 2,000 shares = 2,000 * $30 = $60,000 Margin required for the trade = $60,000 * 30% = $18,000 Commission = $60,000 * 0.25% = $150 Total out-of-pocket investment required for the trade = $18,000 + $150 = $18,150 Total funds required = $60,000 + $150 = $60,150 Funds borrowed = $60,150 - $18,150 = $42,000 Interest cost = $42,000 * 8% = $3,360 Profit earned in the leveraged trade = 2,000*($40-$30) - Commissions - Interest paid = 2,000*$10 - $60,000*0.25% - $80,000*0.25% - $3360 = $16,290 ROE = $16,290 / ($18,150) = 89.75% *User Question: There are two commissions one when you buy and the other when you sell. (you have to subtract them both) ?

Two random variables X and Y are such that V[X] = 4V[Y] and Cov[X,Y] = V[Y] Let E = X + Y and F = X – YFind Cov[E, F] A V[Y] – V[X] B Cov[X,Y] C V[Y] D 3V[Y] The correct answer is: D Cov[E,F] = Cov[X + Y,X – Y] = Cov[X,X] – Cov[X,Y] + Cov[Y,X] – Cov[Y,Y] = V[X] – V[Y] = 4V[Y] – V[Y] = 3V[Y] Logic applied: I. Given a random variable X, the covariance between X and itself is simply its varianceII. Cov[X,Y] = Cov[Y,X] *User Question: Why is Cov[X + Y,X – Y] = Cov[X,X] – Cov[X,Y] + Cov[Y,X] – Cov[Y,Y] ?

Which of the following is not a characteristic describing the dynamic nature of a white noise process? A The unconditional mean and variance must be constant for any covariance stationary process B The absence of any correlation means that all autocovariances and autocorrelations are not zero beyond displacement zero C Events in a white noise process do not exhibit any correlation between the past and the present D Both conditional and unconditional means and variances are the same for an independent white noise process The correct answer is: B The lack of any correlation means that all autocovariances and autocorrelations are zero beyond displacement zero. Displacement is the distance covered by a moving body from a central point. *User Question: Can anyone explain what beyond zero' imply? ?