9.3: Lognormal Distribution, Simulations Flashcards
Explain what shortfall risk is.
Shortfall risk is the probability that a portfolio value or return will fall below a particular target or return over a given time period.
What is Roy’s safety-first criterion? How is it symbolically stated?
Roy’s safety-first criterion states that the optimal portfolio minimizes the probability that the return of the portfolio falls below some minimum acceptable level (this being the shortfall risk). The minimum acceptable level is called the threshold level.
Roy's safety-first criterion is symbolically stated as: minimize P (portfolio return < threshold level return)
What is SFR? How are optimal portfolios selected using SFR?
SFR is the number of standard deviations below the mean. The portfolio with the larger SFR therefore has the lower probability of returns below the threshold return.
- Calculate SFRatio = (expected value of portfolio return - threshold level return) / standard deviation of portfolio
- Choose the portfolio with the highest SFRatio.
What is lognormal distribution? What are its 2 characteristics?
Lognormal distribution is generated by the function e^x, where x is normally distributed.
Characteristics:
- Skewed to the right (longer right tail)
- Lognormal distribution is bounded from below by zero so that it is useful for modelling asset prices which never take negative values.
When is lognormal distribution used? How is it calculated?
Price relative is used as the up-move and down-move multiplier terms in constructing a binomial tree for stock price changes over a number of periods.
It is measured as the end-of-period price of the asset divided by the beginning price and is equal to 1 + holding period return.
Price relative is (S1/S0) = 1 + HPR
where S1 = price relative x S0
Remember that since a lognormal distribution takes a minimum value of zero, S1 cannot be less than 0.
What is discretely compounded returns?
Discretely compounded returns are compound returns, given discrete compounding period like semiannual or quarterly.
What is continuous compounding? How is it expressed mathematically?
A) Given the continuously compounded rate, what should be inputed on the TI to get the effective annual rate?
B) Given the effective annual rate, what should be inputed on the TI to get the continuously compounded rate?
Continuous compounding is calculated as:
Effective annual rate = e^Rcc - 1
where Rcc is the stated annual rate
A) Input on TI: insert value of continuously compounded rate [2nd][e^x] =
B) Input on TI: insert value of effective annual rate [LN] =
Explain the Monte Carlo simulation.
Monte Carlo simulation is a technique based on the repeated generation of one or more risk factors that affect security values, in order to generate a distribution of security values.
For each of the risk factors, the analyst must specify the parameters of the probability distribution that the risk factor is assumed to follow. A computer is then used to generate random values for each risk factor based on its assumed probability distributions. Each set of randomly generated risk factors is used with a pricing model to value the security. Procedure is repeated and the distribution of simulated asset values is used to draw inferences about the expected mean value of the security and variance.
What is Monte Carlo simulation used for (5)?
Used for:
- Valuing complex securities
- Simulating the profits/losses from a trading strategy
- Calculating estimates of value at risk VaR to determine the riskiness of a portfolio of assets and liabilities
- Simulating pension fund assets and liabilities over time to examine the variability of the difference between the two.
- Valuing portfolios of assets that have non-normal returns distributions.
What are the limitations of Monte Carlo simulation?
The limitations of Monte Carlo simulation are:
- Complex and will provide answers that are no better than the assumptions
- Simulation is a statistical method which cannot provide insights like analytics
What is historical simulation?
A) How does it differ from Monte Carlo simulation?
B) What is the advantage to using historical simulation?
C) What are the disadvantages (2) to using historical simulation?
Historical simulation is based on actual changes in value or actual changes in risk factors over some prior period.
A) Historical simulation vs. Monte Carlo:
- Historical simulation uses the set of all changes in the relevant risk factors over some prior period (actual risk factors).
- Monte Carlo simulation models the distribution of risk factors.
B) Advantage
No estimation of risk factors in historical simulation (Monte Carlo uses an estimation)
C) Disadvantages
- Past changes in risk factors may not be a good indication of future changes. Infrequent events cannot be reflected unless the events occurred during the period that is drawn from.
- Cannot address “what if” questions that Monte Carlo simulation can.
How is continuously compounded rate of return used to generate a particular HPR?
Calculation is based on price relative calculation. Continuously compounded rate of return is:
In(S1/S0) = ln(1 + HPR) = Rcc
Recall that price relative = (S1/S0) = 1 + HPR
How is continuously compounded rates of return calculated for multiple periods? Provide an example.
Continuously compounded rates of return is additive for multiple periods.
HPR = e^(Rcc)x(T) - 1
For instance, HPR over 2 years is calculated by doubling the continuously compounded annual rate.
Rcc = 10%
so effective holding period over 2 years is e^(0.10)(2) - 1 j
