Assignment 1

Author

Jesse Perla, UBC

Student Name/Number: (doubleclick to edit)

Instructions

  • Edit the above cell to include your name and student number.
  • Submit just this ipynb to Canvas. Do not rename, it associates your student number with the submission automatically.
using Distributions, Plots, LaTeXStrings, LinearAlgebra, Statistics, Random

Question 1

Create the following variables:

  • D: A floating point number with the value 10,000
  • r: A floating point number with the value 0.025 (i.e., 2.5% net interest rate)

Compute the present discounted value of a payment (D) made in t years, assuming an interest rate of r = 2.5%. Save this value to a new variable called PDV and print your output.

Hint: The formula is

\[ \text{PDV}(D, t) = \frac{D}{(1 + r)^t} \]

For \(t = 10\), calculate this PDV

# add code here

Question 2

Now assume that you have an asset the pays \(D\) every year from \(t = 0,\ldots T\). Write code which will price this as the PDV of all payoffs,

\[ P_T(D) = \sum_{t=0}^{T}\left(\frac{1}{1+r} \right)^t D \]

Part (a)

Derive the analytic solution for the limit of \(P_{\infty}(D) \equiv \lim_{T\to \infty} P_T(D)\)

(doubleclick here to modify. Add other cells as required. No need to show all of your steps)

\[ P_{\infty}(D) = ? \]

Part (b)

Plot the price as the horizon increases

  • On the x-axis plot \(T = 1, \ldots 30\)
  • On the y-axis plot \(P_T(D)\) at that horizon
  • Plot a horizontal line at the asymptotic \(P_{\infty}\) you calculated
T_max = 30
T = 1:T_max
r = 0.025
D = 10_000

# add code here
10000

Question 3

Now instead of having constant dividends, assume that dividends follow the process

\[\log D_{t+1} = \log D_t + \sigma w_{t+1} \]

Where

  • \(w_{t+1} \sim N(0,1)\) are IID unit random normals
  • \(D_0 = 1.0\)
  • \(\sigma = 0.001\)

Part (a)

Write code to simulate a sequence of dividends with the process and initial condition for \(t = 0, \ldots T = 30\).

T = 30
D_0 = 1.0
sigma = 0.001
# reminder, can draw from N(0,1) with randn()

# add code here for simulation
0.001

Part (b)

Plot three simulated sequences of dividends (i.e, the \(D_{t}\) for \(t = 0, \ldots 30\)) on the same graph with the shared x-axis.

# add code here for plotting using your functions above

Question 4

Using the simulated sequences of dividends from Question 3, calculate the \(P_T\) assuming perfect foresight (i.e., they were able to know the sequence of \(w_{t+1}\) even for \(t \geq 0\)). The formula remains the same, except where \(\{D_0, \ldots D_T\}\) is an argument which allows for time-dependent dividends

\[ P_T(\{D_t\}_{t=0}^T) = \sum_{t=0}^{T}\left(\frac{1}{1+r} \right)^t D_t \]

All from the same \(D_0 = 1.0\) initial condition calculate the \(P_T(\{D^n_t\}_{t=0}^T)\) for \(n = 1, \ldots N\) simulated sequences of dividends (i.e. see Question 3)

Part (a)

Calculate the \(P_T\) above given a dividend sequence

T = 30
D_0 = 1.0
sigma = 0.001
# add code here for calculating P_T and check results
0.001

Part (b)

Plot a histogram of the prices for \(N = 100\) simulations and compare to the deterministic case, which is nested if \(\sigma = 0\). (Hint: see our lectures or Julia By Example for more on histograms)

N = 100
# add code here for plotting using your functions above
# hint: use the `histogram` function
100