Mathematics And Statistics For Financial Risk Management 1st Edition Michael B. Miller - Solutions

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Assume that a random variable follows a normal distribution with a mean of 80 and a standard deviation of 24. What percentage of this distribution is not between 32 and 116?a. 4.56%b. 8.96%c. 13.36%d. 18.15%
The random variable Xwith density function f(x) = 1/(b −a) for a x b, and 0 otherwise, is said to have a uniform distribution over (a, b). Calculate its mean.a. (a + b)/2b. a − b/2c. a + b/4d. a − b/4
Given that x and y are random variables anda, b,c, and d are constants, which one of the following definitions is wrong?a. E(ax + by +c) = aE(x) + bE(y) +c, if x and y are correlated.b. V(ax + by +c) = V(ax + by) +c, if x and y are correlated.c. Cov(ax + by, cx + dy) = acV(x) + bdV(y) + (ad +
Suppose that A and B are random variables, each follows a standard normal distribution, and the covariance between A and B is 0.35. What is the variance of (3A+ 2B)?a. 14.47b. 17.20c. 9.20d. 15.10
The joint probability distribution of random variables X and Y is given by f(x, y) = k × x × y for x = 1, 2, 3, y = 1, 2, 3, and k is a positive constant.What is the probability that X + Y will exceed 5?a. 1/9b. 1/4c. 1/36d. Cannot be determined
Which one of the following statements about the correlation coefficient is false?a. It always ranges from −1 to +1.b. A correlation coefficient of zero means that two random variables are independent.c. It is a measure of linear relationship between two random variables.d. It can be calculated by
An analyst gathered the following information about the return distributions for two portfolios during the same time period:Portfolio Skewness Kurtosis A −1.6 1.9 B 0.8 3.2 The analyst states that the distribution for Portfolio A is more peaked than a normal distribution and that the distribution
By reducing the risk of financial distress and bankruptcy, a firm’s use of derivatives contracts to hedge its cash flow uncertainty willa. Lower its value due to the transaction costs of derivatives tradingb. Enhance its value since investors cannot hedge such risks by themselvesc. Have no impact
In perfect markets, risk management expenditures aimed at reducing a firm’s diversifiable risk serve toa. Make the firm more attractive to shareholders as long as costs of risk management are reasonableb. Increase the firm’s value by lowering its cost of equityc. Decrease the firm’s value
A portfolio manager returns 10% with a volatility of 20%. The benchmark returns 8% with risk of 14%. The correlation between the two is 0.98. The risk-free rate is 3%. Which of the following statements is correct?a. The portfolio has higher SR than the benchmark.b. The portfolio has negative IR.c.
Which of the following statements about the Sharpe ratio is false?a. The Sharpe ratio considers both the systematic and unsystematic risks of a portfolio.b. The Sharpe ratio is equal to the excess return of a portfolio over the risk-free rate divided by the total risk of the portfolio.c. The Sharpe
Suppose Portfolio A has an expected return of 8%, volatility of 20%, and beta of 0.5. Suppose the market has an expected return of 10% and volatility of 25%. Finally, suppose the risk-free rate is 5%. What is Jensen’s alpha for Portfolio A?a. 10.0%b. 1.0%c. 0.5%d. 15%
An analyst at CAPM Research Inc. is projecting a return of 21% on Portfolio A. The market risk premium is 11%, the volatility of the market portfolio is 14%, and the risk-free rate is 4.5%. Portfolio A has a beta of 1.5. According to the capital asset pricing model, which of the following
Based on the risk assessment of the CRO, Bank United’s CEO decided to make a large investment in a levered portfolio of CDOs. The CRO had estimated that the portfolio had a 1% chance of losing $1 billion or more over one year, a loss that would make the bank insolvent. At the end of the first
An investment manager is given the task of beating a benchmark. Hence the risk should be measured in terms ofa. Loss relative to the initial investmentb. Loss relative to the expected portfolio valuec. Loss relative to the benchmarkd. Loss attributed to the benchmark
Assume that the mean of a data generating process is known and equal to zero. Your initial estimate of the standard deviation is 10%. You observe the following returns (t = 6 is the most recent period). Assume that the initial estimator was generated from an infinitely long series, and use a decay
Assume that the mean of a data generating process is known and equal to 10%. The initial estimate of the standard deviation is 20%, when you observe a return of 15%. What is your updated estimate of the mean?Assume the data series is of infinite length, and use a decay factor of 0.97.
You are estimating the expected value of the annual return of a stock market index using an EWMA estimator with a decay factor of 0.98.The current estimate of the mean is 10%. Over the next three years, the index returns 15%, −4%, and finally 8%. Recalculate the estimate of the mean in each of
Calculate the sample standard deviation for the data set in problem #3, also using no decay factor, a decay factor of 0.99, and a decay factor of 0.90.
Using a decay factor of 0.95, calculate the mean, sample variance, and sample standard deviation of the following series. Assume t = 7 is the most recent data point, and use all eight points:
For an estimator based on n data points, with a decay factor of , prove that the half-life, h, is given by:h = ln(0.5 + 0.5n)ln()
You are provided with 10 years of monthly log returns for a mutual fund. The mean monthly log return is 2.0%, the standard deviation of the returns is 1.5%, the skewness is –1.0, and the kurtosis is 2.4.Assuming the log returns are i.i.d., what is the expected annualized mean, standard deviation,
Given our AR(1) process, Equation 9.11:rt = + rt−1 + εt Prove that the serial correlation is equal to .
For the time series equation in the previous problem, what is the correlation between rt and a previous return, rt–n?
Assume the spread between two equity indexes can be described by the following time series equation:rt = rt−1 + εt ε ∼ N(0, 2), | | ≤ 1 Cov[εt−1, εt−j] = 0 ∀i = j Derive equations for the unconditional mean and variance of this process?
Assume that interest rates evolve according to the following Vasicek model:rt = rt−1 + (1 − ) + εt ε ∼ N(0, 1) = 0.50 = 0.04= 0.02 What is the unconditional expected value of rt? Assuming interest rates are currently 6%, what is the conditional expected value of rt over the next three
Assume that daily stock returns for Reversion Corporation can be described by the following time series equation:rt = + rt−1 + εt where rt is a log return; the error term, εt, is i.i.d., with a mean of zero and a standard deviation of 1.5%; alpha is a constant equal to 0.10%;and is a
Assume that daily stock returns for Drift Corporation can be described by the following equation:rt = pt = + εt where rt is a log return; the error term, εt, is i.i.d., with a mean of zero and a standard deviation of 1.5%; and alpha is a constant equal to 0.10%. What is the expected log return
Assume that daily stock returns for RW Corporation can be described by the following equation:rt = pt = εt where rt is a log return, and the error term, εt, is i.i.d., with a mean of zero and a standard deviation of 1.5%. What is the expected log return over one year? What is the standard
Assume that a credit spread evolves according to the following time series equation:rt = 0.01 + 0.30rt−1 − 0.20rt−2 + εt Further, assume rt was 2% during the most recent period, and 4% the period before that. What is the expected value of rt in the next period?The period after that?
You are given the following time series model. What is the long-term expected value of rt?rt = 0.02 + 0.8rt−1 + εt
Classify each of the following time series models in terms of the random variable rt:a. rt = + 1rt−1 + 2rt−2 + εtb. rt = + (1 − )rt−1 + εtc. rt = 1rt−1 + 2rt−2 + εt + 1εt−1d. rt = + εt
The equation for the residual sum of squares (RSS) of the univariate regression model is:RSS = n i=1ε2 i = n i=1(yi − − xi)2 Take the derivatives of this equation, first with respect to , then with respect to . Set the two resulting equations to zero, and solve to get the standard equations
Prove Equation 8.9. That is, given the standard univariate regression:Y = + X+ εProve that:= Cov[X,Y]2 X= XYYX where X and Y are the standard deviations of X and Y, respectively, and XY is the correlation between the two. Hint: start by calculating the covariance between X and Y.
You perform the following multivariate regression:r = 1 + 2X2 + 3X3 + ε1 Upon closer inspection you notice that X2 and X3 are, in fact, correlated.Their relationship can be described in the following regression:X3 = 4 + 5X2 + ε2 Suggest a new model that avoids the problem of multicollinearity.
Based on your analysis of Company ABC’s stock returns, rABC, you develop the following OLS regression model:rABC = 0.01 + 1.25rA + 0.34rB + εwhere rA and rB are two uncorrelated indexes, and ε is a mean zero disturbance term. If rA = 10% and rB = 50%, what is the expected value of rABC?
An initial univariate regression produces an R2 of 60%. There are 20 data points. In an effort to improve the model, two additional regressors are added, which boost the R2 to 64%. Determine which model is better based on adjusted R2.
The following regression equation describes the daily returns of stock XYZ, rXYZ, in terms of an index return, rIndex, and a mean zero disturbance term, ε:rXYZ = + rIndex + εwhere and are constants; ε is mean zero with a standard deviation of 1.0%, is 0.01%, and is 1.20. If the index return on a
Given the following matrix B, whose columns form a vector space in R3, find the coordinate vector for the vector x: 4 2 -46 -170 B= 1 -18 -1 x= -19 5 2 37 165
Given the following matrix B, whose columns are orthonormal and form a vector space in R2, find the coordinate vector for the vector x:B =⎡⎣1√2 −1√2 1√2 1√2⎤⎦ x =6 4
Find x and y such that B is an orthonormal basis: B = x y +15 26 5
Find x such that A is an orthonormal basis: A X * Im 1 22 13
Given the following vectors,a, b, andc, are a and b orthogonal? b and c? a 10 -5 b = 2 c= 5 -0-0-0 4 -4 10
Calculate the Cholesky decomposition for the following covariance matrix:
Using the one-year ratings transition matrix from the previous question, calculate the corresponding two-year transition matrix.
You are the risk manager for a large corporate bond portfolio. At the start of the year, 60% of the bonds in the portfolio are rated A, and 40% are rated B. Given the following one-year rating transition matrix, what is the expected distribution of ratings after one year?1-Year To A B C D A 95% 4%
Given the matrix M, what is M5?M =2 0 0 2
Given the following matrices, prove that J is the inverse of K.J =4 1 9 2 K =−2 1 9 −4
Given the following matrices, what is UI? I 2? U2? AU?A =−10 9 8 7 U =1 1 1 1 I =1 0 0 1
Given the following matrices, what is F + G? FG? F G?F =⎡⎣−6 1−8 2−6 −3⎤⎦ G =⎡⎣5 0 0 −1−8 −7⎤⎦
Using the same matrices from Question 1, what is the transpose of A?of C?
Using the same matrices from Question 1, what is B + (A + C)? What is B(A – C)?
Given the following matrices, what is A + B? BC? CB?A =−10 9 8 7 B =2 9 1 1 C = −5 7−10 7
The probability density function for daily profits at Pyramid Asset Management can be described by the following functions:p = 3 80 +1 400 − 15 ≤ ≤ 5 p = 5 80 − 1 400 5 ≤ 25 0.00 0.05–25 –20 –15 –10 –5 0 5 10 15 20 25πp Probability Density Function for Pyramid Asset
Continuing with our example of Box Asset Management, find the expected shortfall, using the same PDF and the calculated VaR from the previous answer.
The probability density function for daily profits at Box Asset Management can be described by the following function:p = 1 200 − 100 ≤ ≤ 100 p = 0 − 100 100 0.00 0.01–120 –100 –80 –60 –40 –20 0 20 40 60 80 100 120πp Probability Density Function for Box Asset Management
A fund of funds has investments in 36 hedge funds. At the end of the year, the mean return of the constituent hedge funds was 18%. The standard deviation of the funds’ returns was 12%. The benchmark return for the fund of funds was 14%. Is the difference between the average return and the
A hedge fund targets a mean annual return of 15% with a 10% standard deviation. Last year, the fund returned –5%. What is the probability of a result this bad or worse happening, given the target mean and standard deviation? Assume the distribution is symmetrical.
In the previous question, you were told that the actual standard deviation was 40%. If, instead of 40%, the measured standard deviation turns out to be 50%, how confident can you be in the initial assumption?State a null hypothesis and calculate the corresponding probability
You are told that the log annual returns of a commodities index are normally distributed with a standard deviation of 40%. You have 33 years of data, from which you calculate the sample variance. What is the standard deviation of this estimate of the sample variance?
Suppose you invest in a product whose returns follow a uniform distribution between –40% and +60%. What is the expected return? What is the 95% VaR? The expected shortfall?
You are presented with an investment strategy with a mean return of 20% and a standard deviation of 10%. What is the probability of a negative return if the returns are normally distributed? What if the distribution is symmetrical, but otherwise unknown?
You are given the following sample of annual returns for a portfolio manager. The mean return is positive, but how confident should you be about this?–7% 7% 19% 23% –18% –12% 49% 34% –6% –20%
Given the following data set, how confident can we be that the mean is greater than 40?64 70 20 3 58 13 74 84 47 17
Imagine we have two independent uniform distributions, A and B. A ranges between –2 and –1, and is zero everywhere else. B ranges between+1 and +2, and is zero everywhere else. What are the mean and standard deviation of a portfolio that consists of 50% A and 50% B? What are the mean and
Prove that the correlation between XA and XB is , where:XA = √ X1 +1 − X2 XB = √ X1 +1 − X3 and X1, X2, and X3 are uncorrelated standard normal variables.
Prove that the variance of a normal distribution, as specified in Equation 4.12, is 2. You may find the following result useful:∞−∞x2e−x2 dx = 1 2√
Prove that the mean of the normal distribution, as specified in Equation 4.12, is . That is, show that:∞−∞x 1√22 e(x−)2 22 dx =
Prove that the normal distribution is a proper probability distribution.That is, show that:∞−∞1√22 e(x−)2 22 dx = 1 You may find it necessary to use the Gaussian integral:∞−∞e−x2 dx = √
For a uniform distribution with a lower bound x1 and an upper bound x2, prove that the formulas for calculating the mean and variance are: (1x+7x)=1 2 1 12(x-x1) -
The annual returns of an emerging markets exchange-traded fund (ETF)have an expected return of 20.60% and a standard deviation of 30.85%.You are asked to estimate the likelihood of extreme return scenarios.Assume the returns are normally distributed. What is the probability that returns are worse
The annual returns of an emerging markets bond fund have a mean return of 10% and a standard deviation of 15%. Your firm invests$200 million into the fund. What is the probability of losing more than $18.4 million? Assume the returns are normally distributed, and ignore the limited liability
The number of defaults per month in a large bond portfolio follows a Poisson distribution. On average, there are two defaults per month.The number of defaults is independent from one month to the next.What is the probability that there are five defaults over five months?Ten defaults? Fifteen
Assume the annual returns of Fund A are normally distributed with a mean and standard deviation of 30%. The annual returns of Fund B are also normally distributed, but with a mean and standard deviation of 40%. The returns of both funds are uncorrelated to each other. What is the mean and standard
The market risk group at your firm has developed a Value at Risk(VaR) model. In Chapter 5 we examine VaR models more closely. In the meantime, assume the probability of an exceedance event on any given day is 5%, and the probability of an exceedance event occurring on any given day is independent
XYZ Corporation announces its earnings four times per year. Based on historical data, you estimate that in any given quarter the probability that XYZ Corporation’s earnings will exceed consensus estimates is 30%. Also, the probability of exceeding the consensus in any one quarter is independent
A $100 notional, zero-coupon bond has one year to expiry. The probability of default is 10%. In the event of default, assume that the recovery rate is 40%. The continuously compounded discount rate is 5%. What is the present value of this bond?
Given two random variables, XA and XB, with corresponding means A and B and standard deviations A and B, prove that the variance of XA plus XB is:Var[XA + XB] = 2 A + 2 B + 2ABAB where AB is the correlation between XA and XB.
Prove that Equation 3.19 is an unbiased estimator of variance.E 2 r= ˆ 2 r = 1 n − 1 ni=1(ri − ˆ r)2
Given the probability density function for a random variable X:f(x) = x 50 s.t. 0 ≤ x ≤ 10 Find the variance of X.
Calculate the population mean, standard deviation, and kurtosis for each of the following two series:Series #1 −23 −7 7 23 Series #2 −17 −17 17 17
Calculate the population covariance and correlation of the following series:Series #1 21% 53% 83% 19%Series #2 20% 32% 80% 40%
What is the standard deviation of the estimator in Equation 3.2? Assume the various data points are i.i.d.
Prove that Equation 3.2 is an unbiased estimator of the mean. That is, show that ˆ = E[r].
Compute the sample mean and the standard deviation following returns:7% 2% 6% −4% −4% 3% 0% 18% −1%
Compute the mean and the median of the following series of returns:12% 5% −8% 20% 4% 10% 2%
The following table is a one-year ratings transition matrix. Given a bond’s rating now, the matrix gives the probability associated with the bond having a given rating in a year’s time. For example, a bond that starts the year with an A rating has a 90% chance of maintaining that rating and an
As a risk analyst, you are asked to look at EB Corporation, which has issued both equity and bonds. The bonds can either be downgraded, be upgraded, or have no change in rating. The stock can either outperform the market or underperform the market. You are given the following probability matrix
Given the probability density function, f(x):f(x) = c xwhere 1 ≤ x ≤e. Calculate the cumulative distribution function, F(x), and solve for the constant c.
Given the following cumulative distribution function, F(x), for 0 ≤x ≤ 10:F (x) = x 100(20 − x)Check that this is a valid CDF (i.e., show that F(0) = 0 and F(10) = 1). Calculate the probability density function, f(x).
Given the following density function:f(x) =c(100 − x2) for − 10 ≤ x ≤ 10 0 otherwise Calculate the value of c.
An analyst develops a model for forecasting bond defaults. The model is 90% accurate. In other words, of the bonds that actually default, the model identifies 90% of them; likewise, of the bonds that don’t default, the model correctly predicts that 90% will not default. You have a portfolio of
The probability that gross domestic product (GDP) decreases is 20%.The probability that unemployment increases is 10%. The probability that unemployment increases given that GDP has decreased is 40%.What is the probability that GDP decreases given that unemployment has increased?
There is a 30% chance that oil prices will increase over the next six months. If oil prices increase, there is a 60% chance that the stock market will be down. What is the probability that oil prices increase and the stock market is down over the next six months?
What is the present value of a newly issued 10-year bond, with a notional value of $100, and a 2% annual coupon? Assume a constant 5% annual discount rate, and no risk of default.
The risk department of your firm has 10 analysts. You need to select four analysts to serve on a special audit committee. How many possible groupings of four analysts can be put together?
Calculate the following summation: 9 i=0(−0.5)i
Solve for x, where eex= 10.
What is the present value of a 10-year bond with a $100 face value, which pays a 6% coupon annually? Use an 8% annual discount rate.
ABC stock will pay a $1 dividend in one year. Assume the dividend will continue to be paid annually forever and the dividend payments will increase in size at a rate of 5%. Value this stream of dividends using a 6% annual discount rate.

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