# Q.2700 Statistical Concepts and Market Returns

Muller Investments has been investing in Curex Pharma for the last 6 years. The returns of Curex Pharma's shares are provided in the following table. As an analyst, determine the variance of the returns. $$\begin{array}{c|c} \textbf{Year} & \textbf{Returns} \\ \hline \text{2001} & \text{45%} \\ \text{2002} & \text{29%} \\ \text{2003} & \text{-16%} \\ \text{2004} & \text{-9%} \\ \text{2005} & \text{13%} \\ \text{2006} & \text{16%} \\ \end{array}$$ A 17% B 4.36% C 2.08% The correct answer is: B) The calculation of the Mean Absolute Deviation of the share's returns are provided in the following table: $$\begin{array}{c|c|c|c} \textbf{Year} & \textbf{Returns} & \textbf{Return - Arithmetic Mean} & \textbf{Variance} \\ \hline \text{2001} & \text{45%} & \text{|45% - 13%|} & \text{(45% - 13%)}^2 \\ \text{2002} & \text{29%} & \text{|29% - 13%|} & \text{(29% - 13%)}^2 \\ \text{2003} & \text{-16%} & \text{|-16% - 13%|} & \text{(-16% - 13%)}^2 \\ \text{2004} & \text{-9%} & \text{|-9% - 13%|} & \text{(-9% - 13%)}^2 \\ \text{2005} & \text{13%} & \text{|13% - 13%|} & \text{(13% - 13%)}^2 \\ \text{2006} & \text{16%} & \text{|16% - 13%|} & \text{(16% - 13%)}^2 \\ \hline \textbf{Arithmetic Mean} & \textbf{78%/6 = 13%} & \text{} \\ \textbf{MAD} & \textbf{102%/6 = 17%} & \text{} \\ \textbf{Variance} & \textbf{0.2573/6 = 4.356%} \end{array}$$ *User Question: 4.356% * 6 = 0.2614?

CFA Level 1

# Q.259 Bonds

One of the insurance company’s assets is an $$n$$-year annual coupon paying bond that is bought at a discount. This bond has the following features: The amount of  acccumulation of discount in the $$(n-12)^{th }$$ and $$(n-9 )^{th }$$ coupon payments were  $850 and$ 984 respectively The amount of discount is \$ 18,117 Calculate $$n$$. A 16 B 17 C 18 D 15 E 18 Denote the accumulation of discount at any time $$k$$ discount by $$P_k$$..So,we that the accumulation of discount form a geometric sequence so that \begin{align} & \cfrac { { p }_{ n-12 } }{ { p }_{ n-9 } } =\cfrac { 984 }{ 850 } ={ \left( 1+i \right) }^{ 3 } \\ & \Rightarrow { \left( 1+i \right) }^{ 3 }=1.157647\quad \therefore \quad i=0.05=5\% \\ \end{align} So, the equation of value is given by: \begin{align} 18117 & ={ P }_{ 1 }+{ P }_{ 1 }\left( 1.05 \right) +{ P }_{ 1 }{ \left( 1.05 \right) }^{ 2 }+\cdot \cdot \cdot +{ P }_{ 1 }{ \left( 1.05 \right) }^{ n-1 } \\ & ={ P }_{ 1 }\left[ 1+\left( 1.05 \right) +{ \left( 1.05 \right) }^{ 2 }+\cdot \cdot \cdot +{ \left( 1.05 \right) }^{ n-1 } \right] \\ & =P_1 s_{\overline{n|}} @5\% \end{align} Since the principal adjustments form a geometric sequence, then: $$P_k=P_1\left(1+i\right)^{k-1}$$ $$\Rightarrow P_{n-12}=P_1\left(1+i\right)^{n-13} \therefore P_1 =\frac{P_{n-12}}{\left(1+i\right)^{n-13}}$$ Therefore, the equation of value changes to: \begin{align*} 18117 & =\frac { 850 }{ { \left( 1.05 \right) }^{ n-13 } } \left[ { s }_{ \overline { n } | }&#64;5\% \right] =\frac { 850{ v }^{ n } }{ { \left( { 1.05 } \right) }^{ -13 } } \left[ { s }_{ \overline { n } | }&#64;5\% \right] \\ & \Rightarrow 850{ \left( { 1.05 } \right) }^{ 13 }\left[ { s }_{ \overline { n } | }&#64;5\% \right] =18117 \\ & { s}_{ \overline { n } | }&#64;5\%=11.3033\Rightarrow n=17.07\approx 17 \\ \end{align*} You can use the financial calculator in the last step. *User Question: I cannot figure out where the v to the n disappeared to in the last equation of value

Actuarial - FM(Financial Mathematics)

# Q.929 Financial Markets and Products

Karen Jacobs, a final year undergraduate student made the following points regarding mortgage pass-through securities. Which of Jacobs' statements is incorrect? A The mortgage pass-through security represents an investor’s claim against a pool of mortgages B The mortgage pass-through securities in the pool can have different maturities and mortgage rates C The cash flows of the pass-through security exactly matches the cash flows generated by the mortgages in the pool of pass-through securities D A single pool of mortgages can have different classes of pass-through mortgage securities The correct answer is: C It is not necessary that cash flows to the investors of the pass-through security exactly coincides with the cash flows generated by the mortgages in the pool or the pass-through securities. This is due to the difference between the timing when the mortgage providers receive the mortgage payments, and the payment is passed through to the investor of the mortgage pool. *User Question: Why is choice D correct? My understanding is that CDOs, as opposed to pass through s, have differentiating tranches.

FRM Part 1

# Q.276 Foundations of Risk Management

Green Belt Market Fund directs its two subsidiaries to simultaneously buy and sell emerging market stocks. In its monthly investment outlook literature, the company points to the overall emerging market volume as indicative of the market's liquidity. The move prompts more investors to increasingly participate in the emerging markets fund. Green Belt Market Fund most likely: A Did not violate the GARP Code of Conduct B Violated the GARP Code of Conduct regarding conflict of interest C Did not violate the GARP Code of Conduct but may have breached stock brokerage rules D Violated the GARP Code of Conduct regarding professional integrity and ethical conduct The correct answer is: D The Green Belt Market fund appears to be attracting investments in its own funds by manipulating the market's liquidity. The increased participation in the emerging markets fund does not emanate from market forces (supply and demand), nor does it indicate a genuine trading strategy meant to benefit investors. It's actually a veiled attempt to increase the assets under the management of the company. The action violates the Code of Conduct with regard to professional integrity and ethical conduct. *User Question: That´s called wash trading actually

FRM Part 1

# Q.29 Mean-Variance Portfolio Theory

You are given the following information about three security (A, B, and C) in a portfolio. The covariance matrix for each stock with each other security is given in the following table $$\begin{array}{lccc} & \textbf{A} & \textbf{B} & \textbf{C}\\ \textbf{A} & 0.040 & 0.018 & 0.016 \\ \textbf{B} & 0.018 & 0.090 & 0.021 \\ \textbf{C} & 0.016 & 0.021 & 0.010 \end{array}$$ (ii) The weighting of each stock in the portfolio is as follows: $$\begin{array}{lcc} \textbf{Security} & \textbf{Weight} & \textbf{Expected Return}\\ \textbf{A} & 30\% & 4\%\\ \textbf{B} &20\% & 1\% \\ \textbf{C} & 50\% & 6\% \end{array}$$ Calculate the variance of this portfolio. A 0.00814 B 0.00459 C 0.00567 D 0.00543 E 0.00675 We know that: $$\sigma^2=Var(R_p )=\sum _{ j }^{ }{ \sum _{ k }^{ }{ { w }_{ j }{ w }_{ k }{ C }_{ jk } } }$$ Where $$C_{jk}$$ is the covariance between the return of security $$i$$ and $$k$$. Since we have three stocks (A, B, and C): \begin{align*} Var(R_p ) =&w_{A}^{2}\sigma_{A}^{2} +w_{B}^{2}\sigma_{B}^{2}+w_{C}^{2}\sigma_{C}^{2}+2w_{A}w_{B}Cov(A,B)+2w_{A}w_{C}Cov(A,C)+2w_{B}w_{C}Cov(B,C)\\ & = 0.30^{2} (0.04)^{2}+0.20^2 (0.01)^{2}+0.50^2 (0.06)^{2}+2(0.30)(0.20)(0.018) \\ & +2(0.30)(0.50)(0.016) + 2(0.20)(0.50)(0.021) \\ & =0.012208 \\ &\Rightarrow \sigma_{P}=\sqrt{0.012208}=0.11049\approx 11.05\%\\ \end{align*} *User Question: if its a Cov matrix then AA would be the var so .04 would not be squared as well as the others and the expected return is also being used for this

Actuarial - IFM(Investment and Financial Markets)

## Q.4620 Valuation and Risk Models

The futures price of an asset is USD 40, and the annual volatility of the futures price is 20%. If the risk-free rate is 5%, what is the value of a put option to sell futures in 6 months for USD 45? A USD 0.028 B USD 4.498 C USD 0.026 D USD 5.520 The correct answer is: D In this case, $${\text{F}}_{0}$$=40, K=45, r=0.05, s=0.20, T=0.5 The following formula gives the value of the put option: $${ \text{P} }_{ 0 }=\text{K}{ \text{e} }^{ -{ \text{r} }\text{T} }\times \text{N}\left( -{ { \text{d} } }_{ 2 } \right) -{ \text{S} }_{ 0 }{ \text{e} }^{ -{ \text{r} }\text{T} }\times \text{N}\left( -{ { \text{d} } }_{ 1 } \right)$$ Where: $${\text{P}}_{0}$$= value of the put option $${\text{F}}_{0}$$= current futures price K= strike price s= volatility of the futures price r= risk-free rate T= time $${ \text{d} }_{ 1 }=\cfrac { \text{ln}\left( \cfrac { { \text{F} }_{ 0 } }{ \text{K} } \right) +\cfrac { { \sigma }^{ 2 }\text{T} }{ 2 } }{ \sigma \sqrt { \text{T} } } =\cfrac {\text{ln}\cfrac { 40 }{ 45 } + \cfrac{{ 0.20 }^{ 2 }}{2}\times 0.5 }{ 0.20\sqrt { 0.5 } } =-0.76214 \\ { \text{d} }_{ 2 }={ \text{d} }_{ 1 }-{ \sigma \sqrt { \text{T} } }=-0.9036 \\ \text{N}\left( -{ \text{d} }_{ 1 } \right) =\text{N}\left( 0.762 \right) =0.7764 \\ \text{N}\left( -{ \text{d} }_{ 2 } \right) =\text{N}\left( 0.9036 \right) =0.8159$$ The value of the put option is given by: $${ \text{P} }_{ 0 }=\text{K}{ \text{e} }^{ -{ \text{r} }\text{T} }\times \text{N}\left( -{ { \text{d} } }_{ 2 } \right) -{ \text{S} }_{ 0 }{ \text{e} }^{ -{ \text{r} }\text{T} }\times \text{N}\left( -{ { \text{d} } }_{ 1 } \right) \\ { P }_{ 0 }=45{ \text{e} }^{ -0.05\times 0.5 }\times 0.8159 -40{ \text{e} }^{ -0.05\times 0.5 }\times 0.7764=\text{ USD } 5.520$$ *User Question: I do not understand why the calculation of d1 does not take into account the risk free rate. Any thoughts as to why that is the case? My d1 = -0.5856 .

## Q.3190 Risk Management and Investment Management

Jack Marconi is an equity strategist at Gandhara Investment and is evaluating the performance of four large-cap equity portfolios: Azgard, Lambda, Tricky, and Jackpot. As part of his analysis, Jack computes the Sharpe ratio and the Treynor measure for all four funds. Based on his finding, the ranks assigned to the four funds are as follows: $$\begin{array}{|c|c|c|} \hline Fund & Treynor\quad Measure\quad Rank & Sharpe\quad Ratio\quad Rank \\ \hline Azgard & 1 & 4 \\ \hline Lambda & 2 & 3 \\ \hline Tricky & 3 & 2 \\ \hline Jackpot & 4 & 1 \\ \hline \end{array}$$ The difference in rankings for Funds Azgard and Jackpot is most likely due to: A Different benchmarks used to evaluate each fund's performance B A difference in risk premiums C A lack of diversification in Azgard Fund as compared to Jackpot Fund D None of the above The correct answer is: C The most likely reason for a difference in ranking is due to the absence of diversification in Azgard Fund. The Sharpe ratio measures excess return per unit of total risk, while the Treynor ratio measures excess return per unit of systematic risk. Since Azgard Fund performed well on the Treynor measure and so poorly on the Sharpe measure, it seems that the fund carries a greater amount of unsystematic risk, meaning it is not well diversified and unsystematic risk is not the relevant risk measure. *User Question: Hello, I believe the explanation is not correct (the answer is correct though). Azgard is the fund that performed poorly on the TREYNOR ratio and performed well on the Sharpe (not vice versa as you state it) Therefore, Azgard carries a greater amount of systematic risk (Beta is greater ,therefore Traenor is low) or it was not well diversified. Please comment because the current explanation is confusing - e.g. if a porftolio performes well on the Treynor it means it is WELL diversified.

## Q.3316 Valuation and Risk Models

A firm originally maintains a 1% VaR for a 250-day horizon. Recently, the firm shifted to the worst-case scenario (WCS) measure. What will be the capital requirement change if the firm targets a 1% WCS instead of the previous 1% VaR capital requirement? Refer to the following table for further information. $$E[Z_i &lt; -2.33]$$ $$E[Z_i &lt; -1.65]$$ Expected WCS 1% WCS 5% WCS H=250 2.5 12.5 -2.82 -3.92 -3.54 A 168% decrease B 68% decrease C 168% increase D 68% increase The correct answer is: D The 5% VaR is -2.33 and the 1% WCS is -3.92. Capital requirement = $$\frac{3.92}{2.33} = 1.68$$ Thus, it is a 68% increase from the original value of 1 $$(1.68 – 1 = 68\%).$$ *User Question: The question said the original target was 1% VAR and not 5% VAR. Why was the answer talking about 5% VAR originally?

## Q.987 Valuation and Risk Models

Donald Morisette, a risk analyst at TNZ Associates, collects 20 days data on the stock price of ANH. The analysis of the collected data gives an estimate of the yearly volatility of the stock as 15.45%. What is the standard error of the estimate (approx.)? A 0.0244 B 0.0039 C 0.0345 D 0.0691 The correct answer is: C Standard error of the estimate = Volatility/&radic;(n) Standard error of the estimate = 0.1545/&radic;(20) = 0.03454 *User Question: Shouldn't it be sqrt(2n) as per the GARP books?

## Q.528 Quantitative Analysis

An autoregressive process is considered stationary if: A The roots of the characteristic equation lie on the unit circle B The roots of the characteristic equation lie outside the unit circle C The roots of the characteristic equation lie inside the unit circle D The characteristic equation are of order 1 The correct answer is: B In any autoregressive process, the roots of the characteristic equation must lie outside the unit circle, which means the absolute value of the roots must be larger than one. *User Question: which roots and unit circle are these? what do they imply. ?