Ch 4 Resampling via Monte Carlo

Question 1

WHat is resampling

Answer 1

Resampling: iteratively generate random data and analyse the stats
eg Monte Carlo used to run multiple simulations

Question 2

1. Traditional approach to computing statistics

2. Example: estimate probability S&P returns 5% in a week:

Answer 2

1. Assume a probability distribution and algebraically solve for the statistic in question.
2. Example: estimate probability S&P returns 5% in a week: 𝑅^𝑊𝑒𝑒𝑘=(1+𝑟_𝑀𝑜𝑛𝑑𝑎𝑦 ) (1+𝑟_𝑇𝑢𝑒𝑠𝑑𝑎𝑦 )⋯(1+𝑟_𝐹𝑟𝑖𝑑𝑎𝑦 )−1

Question 3

Why resample?

Use in what circumstances: (3)

Answer 3

1. Particularly powerful when closed form solutions are difficult to compute, such as

a) compute more troublesome ‘statistics’: products, ratios, VaR e.g. geometric returns, sharpe ratios
b) There is time variation in the dataset, e.g. volatility regimes
c) There are data features that are difficult to capture in a model, e.g. serial correlation, heteroskedasticity

Advances in computing power makes sampling more tractable every day

Question 4

1. Define serial correlation
2. Also known as…
3. Serial correlation = 0…
4. Serial correlation = (approaches) 1…

Answer 4

1. Definition Serial correlation: relationship between given variable and itself over various time intervals. Serial correlations are often found in repeating patterns, when the level of a variable effects its future level.
2. AKA “autocorrelation” or “lagged correlation.”
3. serial correlation = 0: there is no correlation, each of the observations are independent of one another.
4. serial correlation skews toward one, it means that the observations are serially correlated

Question 5

1. Define heteroskedasticity
2. two forms
3. mathematically..

Answer 5

1. Define Heteroskedasticity: when the standard deviations of a variable, monitored over a specific amount of time, are nonconstant.
2. 2 forms: conditional and unconditional.
   a) Conditional heteroskedasticity identifies nonconstant volatility when future periods of high and low volatility cannot be identified. Not predictable
   b) Unconditional heteroskedasticity is used when futures periods of high and low volatility can be identified. Predictable in nature - eg seasonality
3. refers to the error variance, or dependence of scatter, within a minimum of one independent variable within a particular sample. These variations can be used to calculate the margin of error between data sets, such as expected results and actual results, as it provides a measure for the deviation of data points from the mean value.

Question 6

Define Resampling

Answer 6

Resampling: repeatedly calling samples from a probability distribution, using tens or hundreds of thousands. The probability distribution may be analytical (e.g. Normal, Student’s t), or it may be empirical (realised)

Question 7

Define Bootstrapping

Answer 7

Bootstrapping – sampling with replacement

Question 8

Define Jackknife

Answer 8

Jackknife: randomly excludes a small number of observations on each iteration. Applies only to a random set of observations. Cannot be used with an assumed analytic distribution

Question 9

Define Monte Carlo

Answer 9

Monte Carlo: Resampling with a known analytic distribution

Question 10

Portfolio Opimisation in Practice: Phillipe Jorion

Answer 10

1. Jorian created 1000 alternative histories using a multivariate Normal distribution using the realised returns and covariances
2. Using each alternative history, he constructed a mean-variance optimal asset-allocation, rebalancing each month for 10 years
3. He tested how each of these alternative asset allocation porftolios performed on the real history

Question 11

Bootstrapping (sampling with replacement)

• SImple method
• Another example

Answer 11

● Eg simple method
● Take original data set
● Resample from that set to form a new sample
● Repeat many times, then for each sample calculate the required stats.
● Create histogram of results
● Another example
● Loop 1000 times. Simulate (10 yrs of monthly returns using these numbers
● Calculate average returns, volatilities and correlations
● Derive optimum portfolio’s AA using MVO and parameters of previous step
● Determine how portfolio wouold have performed using actual historical data
● Plot the means and volatilities
● Plot asset allocations

Question 12

Issues with bootstrapping (4)

Answer 12

● Level of returns
● Heteroskedasticity
● (Standard deviations are non constant)
● Changing Correlations (may need to adjust correlations to better fit expectations of changing correlations in the future)
● Autocorrelation
● Similarity of a variable with itself over a separate block of time. Consider momentum.
● Periods where markets generally go up, followed by periods where they are generally decreasing (serial correlation)

Question 13

Block bootstrapping - why use

Answer 13

● By randomly choosing individual months in the sample we destroy any autocorrelation in original data set – ie it wont be seen in the sample
● Mitigate this by BLOCK BOOTSTRAPPING: rather than choosing 1 month at a time; select a block of months (ie n = number of months selected in sequence)