Ch 6 Statistical Subtleties for Monte Carlo

Question 1

1. Issues with Monte Carlo

2. Solution?

Answer 1

1. Issues with Monte Carlo
   - takes a long time to run to get ‘really’ precise outputs
   - The more simulations the better. Sometimes 100s of thousands are needed
2. SOlution?
   - Can we run fewer simulations and obtain the same level of precision? Yes, in may circumstances; By reducing the variance of our estimates, then fewer simulations can be run

Question 2

Antithetic Variables

- Why

Answer 2

1. Why: Simulations can take a long time to run. Aim to reduce standard error. Observe approximate halving of standard error in text for same number of iterations. Achieve a result for half the original number of data points.

Question 3

Antithetic Variables

• Describe

Answer 3

Describe: Each simulation is based upon a random probability, p, 0 and 1, being translated into a random fund return

• By also using 1-p to generate a random fund return we improve the simulation by using an antithetic variable.
• Correlation is perfectly negative (-1) and covariance is -s^2
• Because.. return 𝑟 and 𝑟 ̃ perfectly offset each other; each monte-carlo run will better span the possible returns and have a more representative set of underlying simulations

Question 4

Control Variables

- Define

Answer 4

Define: Control variables use known information about an average to make adjustments to the simulation

Question 5

Random Numbers

- Understand change in one parameter by…

Answer 5

• The influence of a change is one parameter is best understood by keeping all other parameters constant
• This applies to the random numbers generated for a Monte-Carlo simulation
• To understand the affect of varying retirement age, taxation levels, etc, using the same set of random numbers will reduce the standard errors of the changes

Question 6

Variance Reduction techniques

Answer 6

Variance reduction techniques allow us to get more out of the Monte-Carlo analysis

The notes show the impact of two techniques: Antithetic variables and Control Variables

Other techniques (in Appendix 4) may also be used

Additionally by reusing random numbers better insights can be gained regarding the impact of changes to parameters in isolation

Question 7

Antithetic Variables

Answer 7

Each simulation is based upon a random probability, p, 0 and 1, being translated into a random fund return
By also using 1-p to generate a random fund return we improve the simulation by using an antithetic variable
The reason is, that return r and r ̃ perfectly offset each other. When one is high the other is low, and vice-versa, hence each monte-carlo run will better span the possible returns and have a more representative set of underlying simulations
The aim is to reduce the standard error. And can run the simulation with half the number of random points

Question 8

Antithetic Variables

- if antithetic, correlation between variables will be..

Answer 8

… perfectly negative (ie -1)

Question 9

Using the same random numbers

• Why
• Monte-carlo

Answer 9

WHY: The influence of a change is one parameter is best understood by keeping all other parameters constant

M-C: This applies to the random numbers generated for a Monte-Carlo simulation

● To understand the affect of varying retirement age, taxation levels, etc, using the same set of random numbers will reduce the standard errors of the changes
● Note correlation between columns should be very high if using the same set of random numbers in example; same set of numbers is used for 6 scenarios; changing just the age at retirement