Examples from Applied Finance

• Long-run risk
• Retirement planning
• Retirement planning with simulation
• Monte-Carlo option valuation
• Monte-Carlo stock valuation

Long-Run Risk

• Ask Julius to simulate how much a $1 investment would grow to in 10 years if the investment return is normally distributed with a mean of 6% and a standard deviation of 20%.
• Ask Julius to calculate the fraction of times the investment outperforms a risk-free return of 1% over 10 years.
• Ask Julius to generate boxplots and histograms of the ending investment account balance.

Retirement Planning

• Tell Julius you want to check if a retirement savings plan is feasible.
  
• Ask Julius what information you need to provide and provide it.
• Ask Julius to calculate the ending balance as a function of the rate of return over some range and plot it.

Retirement Planning with Simulation

• Ask Julius to simulate the retirement plan assuming the annual returns are normally distributed with some mean and variance.
• Ask Julius to describe the distribution of ending account balances and to produce a boxplot and histogram.

Monte-Carlo Option Valuation

• Tell Julius you want to value a European call option by Monte Carlo. Ask Julius what information you need to provide and provide it.
• Ask Julius to value the same call option using Black-Scholes.
• Ask Julius to value a put option both ways.

Two-Stage Growth Model

• Give Julius the following data (from Applied). Then ask what the share price should be.
  • A firm with no debt will have free cash flow of 100M next year.
  • The cash flow will grow by 12% per year for years 2 through 5. Then, it will grow by 3% per year forever.
  • The firm’s cost of capital is 10%, and there are 44.75M shares outstanding.

Check for Errors

• It is quite possible that Julius - like any assistant - might misunderstand the timing you want.
• Quickest way to fix mistakes: edit the code.

Check the following:

• There should be five cash flows in the first stage.
• The first cash flow should be 100M (no growth).
• The terminal value should be 100M with 4 years of growth at 12% and 1 year of growth at 3% divided by (10% - 3%).
• The terminal value should be discounted back 5 years.

Sensitivity Analysis

Ask Julius to vary the first stage growth rate between 6% and 18% and to plot the share price as a function of the growth rate.

Monte-Carlo Valuation

• Now ask Julius to simulate the first-stage growth rate from a normal distribution with a mean of 12% and a standard deviation of 3%.
  
• Ask Julius to compute the share price in each simulation and to describe the share price distribution.
• Ask Julius to produce a histogram of the share price distribution.

A Second Source of Uncertainty

• Tell Julius to model the first-stage growth rate as x + y where
  • x is drawn from a normal distribution with a mean of 12% and a standard deviation of 3%,
  • y is a Bernoulli random variable which is 12% with 10% probability and is 0 with 90% probability.
• Ask Julius to compute the share price in each simulation, to describe the share price distribution, and to produce a histogram.