Problem Set 1

Stochastic Foundations for Finance

Solutions

Instructions: This problem set is due on 10/30 at 11:59 pm CST and is an individual assignment. All problems must be handwritten. Scan your work and submit a PDF file.

Problem 1 Suppose the evolution of y_{t} given by: y_{t} = 0.1 + 0.90 y_{t-1}.

  1. Compute y_{10} if you know that y_{1} = 3.
  2. Sketch a graph of y_{t} as a function of t, specifying the value of \bar{y} = \lim_{t \rightarrow \infty} y_{t}.

Problem 2 (Exponential Growth) Let y_{t} = e^{x_{t}}, and consider the process x_{t+1} = a + x_{t}, \tag{1} where a > 1.

  1. Compute y_{t} as function of y_{0}, a and t.
  2. Sketch a graph of y_{t} as function of t.

Problem 3 (Exponential Growth with Capacity) Let y_{t} = e^{x_{t}}, and consider the process x_{t+1} = x_{t} \left(1 + r \left(1 - \frac{x_{t}}{K}\right)\right), \tag{2} given 0 < x_{0} < K and 0 < r < 1.

  1. Explain what should happen to the growth rate of x_{t} as t \rightarrow \infty.
  2. Assuming that \bar{x} = \lim_{t \rightarrow \infty} x_{t} exists, compute \bar{x}.
  3. Sketch a graph of y_{t} as a function of t, specifying the value of \lim_{t \rightarrow \infty} y_{t}. How this plot differs from the one in Exercise 2?

Problem 4 (Applying the DDM) ACME last year paid a dividend of $3.40 per share. This dividend is expected to grow at 20% per year for the next five years, after which it is expected to grow at 3% in perpetuity.

  1. What is the stock’s value if your required rate of return is 10%?
  2. Would the price change if you expected to hold the share for only three years?

Problem 5 (Fibonacci Numbers) The Fibonacci numbers are defined by the recurrence relation F_{0} = 0, \quad F_{1} = 1, \tag{3} and F_{n} = F_{n-1} + F_{n-2} \tag{4} for n > 1.

  1. Compute the first 10 Fibonacci numbers using (3) and (4).
  2. Does the Fibonacci sequence F_{n} converges to a finite number as n \rightarrow \infty?
  3. Write (4) as a system of two difference equations and indicate the initial conditions of each variable.

Problem 6 Consider the system x_{t} = 0.95 x_{t-1} + w_{t}, where x_{-1} = 0, and w_{t} = \begin{cases} 1 \quad \text{if $t = 0$}, \\ 0 \quad \text{if $t > 0$}. \end{cases}

  1. Sketch a graph of x_{t} as a function of t, specifying the value of \bar{x} = \lim_{t \rightarrow \infty} x_{t}.
  2. Determine t \in \mathbb{N} such that x_{t} is the closest to 0.5.

Problem 7 A bank account gives you an interest rate of r per period. Denote by x_{t} the balance of your bank account at time t, and assume that at the end of each period you withdraw the interest paid by the bank account, leaving the principal intact.

  1. Write down the difference equation relating x_{t} to x_{t-1}.
  2. Using the previous result, derive the present value of a perpetuity paying a constant cash flow c at the end of each period indefinitely.