AIM 1: Understand the natural logarithm, differentiation and Taylor series, and distinguish between expectation and variance.

1、Consider a portfolio of derivatives on fixed income securities and interest rates. If a Taylor Series approximation is used to estimate the delta normal value at risk for the individual derivatives in the portfolio, which of the following positions will have a substantially improved estimate of value at risk?

1. Interest rate cap on 3-month LIBOR
2. Forward rate agreement on 6-month LIBOR
3. 6-month call option on Treasury bonds

A) I and III.

B) III only.

C) II only.

D) I and II.

1、The correct Answer is A

The Taylor Series approximation adds a second order term (i.e., the second derivative of the value function) to the slope (i.e., the first derivative of the value function) to estimate the rate of change in the value of the non-linear derivative. Doing so improves the estimated value change for large changes in the underlying asset value. The slope by itself only provides a reasonable estimate of price sensitivity for small changes but when combined with the rate of change, the convexity of the value function for the non-linear derivative is accounted for. Applying this methodology in the context of a VAR calculation improves the estimate of potential value loss. For linear derivatives, such as forwards and futures, the linear approximation and Taylor Series approximation should be equivalent.

[此贴子已经被作者于2009-6-24 14:20:40编辑过]

2、Which of the following statements are correct for the natural log?

1. The natural log is defined for all numbers, positive and negative.
2. It is the inverse function of the exponential function: e^ln(x) = ln(e^x) = x.
3. The slope of the log function equals x¹.
4. The natural logarithm of 1 is 0, since e⁰ = 1.

A) II and IV.

B) II only.

C) I and II.

D) II and III.

2、The correct Answer is A

Both statements II and IV are correct. Statement I is incorrect because the natural log is defined for positive numbers only. Statement III is incorrect because the slope of the log function is equal to x^?1, not x¹.

[此贴子已经被作者于2009-6-24 14:20:59编辑过]

3、Finding the rate of change is called finding the derivative or differentiating. Which of the following statements about differentiation is incorrect?

A) The Taylor series is useful in that it provides an accurate approximation to a function as a finite sum of derivative values.

B) At every point along a curve, the derivative is the slope of the line that is tangent to the curve.

C) The rate of change of the slope is determined by taking the second derivative:

D) The derivative of a function y = f(x) is:

3、The correct Answer is A

The Taylor series is useful in that it provides an accurate approximation to a function as an infinite, not finite, sum of derivative values.

4、Expected values have properties that can be useful to know. Which of the following properties are correct concerning expected values?

1. Expected value of a constant times a random variable is the constant times the expected value of the random variable.
2. Expected value of the sum of random variables is the sum of the expected value of the random variables.
3. If the random variables are independent of each other then the expected value of the product of random variables is the product of the expected values.

A) I and III.

B) I and II.

C) I, II, and III.

D) None are correct.

4、The correct Answer is C

All of the statements are correct.

谢谢！[em78]

好帖子