Which of the following are true:
I. A interest rate cap is effectively a call option on an underlying interest rate
II. The premium on a cap is determined by the volatility of the underlying rate
III. A collar is more expensive than a cap or a floor
IV. A floor is effectively a put option on an underlying interest rate
- A . I, II, III and IV
- B . I, II and III
- C . III and IV
- D . I, II and IV
D
Explanation:
Interest rate caps are effectively call options on an underlying interest rate that protect the buyer of the cap against a rise in interest rates over the agreed exercise rate. As with options, the premium on the cap depends upon the volatility of the underlying rates as one of its variables. A floor is the exact opposite of a cap, ie it is effectively a put option on an underlying interest rate that protects the buyer of the floor against a fall in interest rates below the agreed exercise rate.
A cap protects a borrower against a rise in interest rates beyond a point, and a floor protects a lender against a fall in interest rates below a point.
A collar is a combination of a long cap and a short floor, the idea being that the premium due on the cap is offset partly by the premium earned on the short floor position. Therefore a collar is less expensive than a cap or a floor.
Which of the following are true:
I. A interest rate cap is effectively a call option on an underlying interest rate
II. The premium on a cap is determined by the volatility of the underlying rate
III. A collar is more expensive than a cap or a floor
IV. A floor is effectively a put option on an underlying interest rate
- A . I, II, III and IV
- B . I, II and III
- C . III and IV
- D . I, II and IV
D
Explanation:
Interest rate caps are effectively call options on an underlying interest rate that protect the buyer of the cap against a rise in interest rates over the agreed exercise rate. As with options, the premium on the cap depends upon the volatility of the underlying rates as one of its variables. A floor is the exact opposite of a cap, ie it is effectively a put option on an underlying interest rate that protects the buyer of the floor against a fall in interest rates below the agreed exercise rate.
A cap protects a borrower against a rise in interest rates beyond a point, and a floor protects a lender against a fall in interest rates below a point.
A collar is a combination of a long cap and a short floor, the idea being that the premium due on the cap is offset partly by the premium earned on the short floor position. Therefore a collar is less expensive than a cap or a floor.
The class intervals should be large enough so that they not obscure interesting variation within the group
- A . Statements 2 and 3 are correct
- B . Statements 1 and 2 are correct
- C . All three statements are correct
- D . Statements 1 and 3 are correct
An underlying asset price is at 100, its annual volatility is 25% and the risk free interest rate is 5%. A European call option has a strike of 85 and a maturity of 40 days. Its Black-Scholes price is 15.52. The options sensitivities are: delta = 0.98; gamma = 0.006 and vega = 1.55 .
What is the delta-gamma-vega approximation to the new option price when the underlying asset price changes to 105 and the volatility changes to 28%?
- A . 17.33
- B . 18.75
- C . 19.23
- D . 20.54
What is the maximum value for f(x)= 8-(x+3)(x-3)?
- A . 8
- B . -1
- C . 17
- D . None of these
What is a Hessian?
- A . Correlation matrix of market indices
- B . The vector of partial derivatives of a contingent claim
- C . A matrix of second derivatives of a function
- D . The point at which a minimum of a multidimensional function is achieved
You are investigating the relationship between weather and stock market performance. To do this, you pick 100 stock market locations all over the world. For each location, you collect yesterday’s mean temperature and humidity and yesterday’s local index return. Performing a regression analysis on this data is an example of…
- A . Simple time-series regression
- B . Multiple time-series regression
- C . Simple cross-section regression
- D . Multiple cross-section regression
The fundamental theorem of analysis establishes a relation between
- A . First and second derivative of a function
- B . The derivative of a function and the slope of its graph
- C . Integration and differentiation of functions
- D . The derivative of a function and the derivative of its inverse function
If the annual volatility of returns is 25% what is the variance of the quarterly returns?
- A . 0.1250
- B . 0.0156
- C . 0.0625
- D . None of the above
A 2-step binomial tree is used to value an American put option with strike 104, given that the underlying price is currently 100. At each step the underlying price can move up by 20% or down by 20% and the risk-neutral probability of an up move is 0.55. There are no dividends paid on the underlying and the discretely compounded risk free interest rate over each time step is 2% .
What is the value of the option in this model?
- A . 11.82
- B . 12.33
- C . 12.49
- D . 12.78
A typical leptokurtotic distribution can be described as a distribution that is relative to a normal distribution
- A . peaked and thin at the center and with heavy (fat) tails
- B . peaked and thin at the center and with thin tails
- C . flat and thick at the center and with heavy (fat) tails
- D . flat and thick at the center and with thin tails
In a portfolio there are 7 bonds: 2 AAA Corporate bonds, 2 AAA Agency bonds, 1 AA Corporate and 2 AA Agency bonds. By an unexplained characteristic the probability of any specific AAA bond outperforming the others is twice the probability of any specific AA bond outperforming the others .
What is the probability that an AA bond or a Corporate bond outperforms all of the others?
- A . 5/7
- B . 8/11
- C . 6/11
- D . None of these
Let X be a random variable distributed normally with mean 0 and standard deviation 1 .
What is the expected value of exp(X)?
- A . E(exp(X)) = 1.6487
- B . E(exp(X)) = 1
- C . E(exp(X)) = 2.7183
- D . E(exp(X)) = 0.6065
What is the total derivative of the function f(x,y) = ln(x+y), where ln() denotes the natural logarithmic function?
- A . 1 / (x+y)
- B . (x + y) / (x+y)
- C . -x/(x+y) – y/(x+y)
- D . ln(x+y) x + ln(x+y) y
Evaluate the derivative of ln(1+ x2) at the point x = 1
- A . 0.5
- B . 0
- C . 1
- D . 2
A linear regression gives the following output:
Figures in square brackets are estimated standard errors of the coefficient estimates.
What is the value of the test statistic for the hypothesis that the coefficient of is less than 1?
- A . 0.32
- B . 0.64
- C . 0.96
- D . 1.92
You invest $100 000 for 3 years at a continuously compounded rate of 3%. At the end of 3 years, you redeem the investment. Taxes of 22% are applied at the time of redemption .
What is your approximate after-tax profit from the investment, rounded to $10?
- A . $9420
- B . $7350
- C . $7230
- D . $7100
You are given the following values of a quadratic function f(x): f(0)=0, f(1)=-2, f(2)=-5.
On the basis of these data, the derivative f'(0) is …
- A . in the interval ]-2.5,-2[
- B . equal to -2
- C . in the interval ]-2,+[
- D . in the interval ]-,-2.5]
I have $5m to invest in two stocks: 75% of my capital is invested in stock 1 which has price 100 and the rest is invested in stock 2, which has price 125.
If the price of stock 1 falls to 90 and the price of stock 2 rises to 150, what is the return on my portfolio?
- A . -2.50%
- B . -5%
- C . 2.50%
- D . 5%
In a quadratic Taylor approximation, a function is approximated by:
- A . a constant
- B . a straight line
- C . a parabola
- D . a cubic polynomial
The determinant of a matrix X is equal 2 .
Which of the following statements is true?
- A . det(2X) =
- B . det(2X) = 2 det(X)
- C . det(2X) = det(X)2
- D . det(2X) = 4 det(X)
For the function f(x) =3x-x3 which of the following is true?
- A . x = 0 is a minimum
- B . x = -3 is a maximum
- C . x = 2 is a maximum
- D . None of these
You intend to invest $100 000 for five years. Four different interest payment options are available. Choose the interest option that yields the highest return over the five year period.
- A . a lump-sum payment of $22 500 on maturity (in five years)
- B . an annually compounded rate of 4.15%
- C . a quarterly-compounded rate of 4.1%
- D . a continuously-compounded rate of 4%
Suppose we perform a principle component analysis of the correlation matrix of the returns of 13 yields along the yield curve. The largest eigenvalue of the correlation matrix is 9.8 .
What percentage of return volatility is explained by the first component? (You may use the fact that the sum of the diagonal elements of a square matrix is always equal to the sum of its eigenvalues.)
- A . 64%
- B . 75%
- C . 98%
- D . Cannot be determined without estimates of the volatilities of the individual returns
Let N(.) denote the cumulative distribution function of the standard normal probability distribution, and N’ its derivative .
Which of the following is false?
- A . N(0) = 0.5
- B . N'(0) 0
- C . N(x) 0 as x
- D . N'(x) 0 as x
A 2-year bond has a yield of 5% and an annual coupon of 5% .
What is the Modified Duration of the bond?
- A . 2
- B . 1.95
- C . 1.86
- D . 1.75
Simple linear regression involves one dependent variable, one independent variable and one error variable. In contrast, multiple linear regression uses…
- A . One dependent variable, many independent variables, one error variable
- B . Many dependent variables, one independent variable, one error variable
- C . One dependent variable, one independent variable, many error variables
- D . Many dependent variables, many independent variables, many error variables
Which of the following is a false statement concerning the probability density function and the cumulative distribution function of a random variable?
- A . the PDF is non-negative.
- B . the definite integral of the CDF from minus infinity to plus infinity is undefined.
- C . the CDF approaches 1 as its argument approaches infinity.
- D . the definite integral of the PDF from minus infinity to plus infinity is zero.
Which of the provided answers solves this system of equations?
2y C 3x = 3y +x
y2 + x2 = 68
- A . x = 1; y = square root of 67
- B . x = 2; y = 8
- C . x = 2; y = -8
- D . x = -2; y = -8
In a binomial tree lattice, at each step the underlying price can move up by a factor of u = 1.1 or down by a factor of . The continuously compounded risk free interest rate over each time step is 1% and there are no dividends paid on the underlying.
The risk neutral probability for an up move is:
- A . 0.5290
- B . 0.5292
- C . 0.5286
- D . 0.5288
A simple linear regression is based on 100 data points. The total sum of squares is 1.5 and the correlation between the dependent and explanatory variables is 0.5 .
What is the explained sum of squares?
- A . 0.75
- B . 1.125
- C . 0.3333
- D . 0.375
Consider two securities X and Y with the following 5 annual returns:
X: +10%, +3%, -2%, +3%, +5%
Y: +7%, -2%, +3%, -5%, +10%
In this case the sample covariance between the two time series can be calculated as:
- A . 0.40729
- B . 0.00109
- C . 0.00087
- D . 0.32583
What is the angle between the following two three dimensional vectors: a=(1,2,3), b=(-4,2,0)?
- A . 90 degrees
- B . 180 degrees
- C . 57 degrees
- D . 45 degrees
Which of the following statements is true for symmetric positive definite matrices?
- A . Its eigenvalues are all positive
- B . One of its eigenvalues equals 0
- C . If a is its eigenvalue, then -a is also its eigenvalue
- D . If a is its eigenvalue, then is also its eigenvalue
Solve the simultaneous linear equations: x + 2y – 2 = 0 and y – 3x = 8
- A . x = 1, y = 0.5
- B . x = -2, y = 2
- C . x = 2, y = 0
- D . None of the above
Calculate the determinant of the following matrix:
- A . 4.25
- B . -4.25
- C . 4
- D . 2
At what point x does the function f(x) = x3 – 4×2 + 1 have a local minimum?
- A . -0.666666667
- B . 0
- C . 2.66667
- D . 2
A 95% confidence interval for a parameter estimate can be interpreted as follows:
- A . The probability that the real value of the parameter is within this interval is 95%.
- B . The probability that the real value of the parameter is outside this interval is 95%.
- C . The probability that the estimated value of the parameter is within this interval is 95%.
- D . The probability that the estimated value of the parameter is outside this interval is 95%.
Suppose I trade an option and I wish to hedge that option for delta and vega. Another option is available to trade.
To complete the hedge I would
- A . trade the underlying in such a way as to make the portfolio delta and vega neutral.
- B . trade the other option in such a way as to make the portfolio delta and vega neutral.
- C . trade the other option in such a way as to make the portfolio vega neutral, and then trade the underlying in such a way as to make the portfolio delta neutral.
- D . trade the underlying in such a way as to make the portfolio delta neutral, and then trade the other option in such a way as to make the portfolio vega neutral.
For each of the following functions, indicate whether its graph is concave or convex:
Y = 7×2 + 3x + 9
Y = 6 ln(3x)
Y = exp(-4x)
- A . concave, concave, concave
- B . concave, convex, convex
- C . convex, concave, concave
- D . convex, convex, concave
You are given the following regressions of the first difference of the log of a commodity price on the lagged price and of the first difference of the log return on the lagged log return. Each regression is based on 100 data points and figures in square brackets denote the estimated standard errors of the coefficient estimates:
Which of the following hypotheses can be accepted based on these regressions at the 5% confidence level (corresponding to a critical value of the Dickey Fuller test statistic of C 2.89)?
- A . The commodity prices are stationary
- B . The commodity returns are stationary
- C . The commodity returns are integrated of order 1
- D . None of the above
In a multiple linear regression, the significance of R2 can be tested using which distribution?
- A . Normal distribution
- B . Student’s t distribution
- C . F-distribution
- D . Binomial distribution
If A and B are two events with P(A) = 1/4, P(B) = 1/3 and P(A intersection B) =1/5, what is P(Bc | Ac) i.e. the probability of the complement of B when the complement of A is given?
- A . 12/29
- B . 37/45
- C . 3/4
- D . None of these
What is the maximum value of the function F(x, y)=x2+y2 in the domain defined by inequalities x 1, y -2, y-x 3 ?
- A . 29
- B . -25
- C . 1
- D . 17
Which of the following statements are true about Maximum Likelihood Estimation?
(i) MLE can be applied even if the error terms are not i.i.d. normal.
(ii) MLE involves integrating a likelihood function or a log-likelihood function.
(iii) MLE yields parameter estimates that are consistent.
- A . (i) and (ii)
- B . (i) only
- C . (i) and (iii)
- D . (i), (ii), and (iii)
An option has value 10 when the underlying price is 99 and value 9.5 when the underlying price is 101. Approximate the value of the option delta using a first order central finite difference.
- A . -4
- B . 0.25
- C . -0.5
- D . -0.25
Every covariance matrix must be positive semi-definite. If it were not then:
- A . Some portfolios could have a negative variance
- B . One or more of its eigenvalues would be negative
- C . There would be no Cholesky decomposition matrix
- D . All the above statements are true
Which of the following can be used to evaluate a regression model?
(i) Magnitude of R2
(ii) Magnitude of TSS (total sum of squares)
(iii) Tests for statistical significance
(iv) Sign and magnitude of each regression parameter
- A . (i) and (iv)
- B . (i), (ii), and (iii)
- C . (i), (iii), and (iv)
- D . (i), (ii), (iii), and (iv)
Which of the following properties is exhibited by multiplication, but not by addition?
- A . associativity
- B . commutativity
- C . distributivity
- D . invertibility