Auto-Regressive (AR) Time Series Models This type of time series model utilizes a time period lagged observation as the independent variable to predict the dependent variable, which is the value in the next time period. xt = b0 + b1xt-1 + εt There can be more than one time period lag independent variable. Valid statistical […]

# Financial Mathematics

## Time Series Analysis: Simple and Log-linear Trend Models

Simple Time Series Models This is basic trend modeling. A simple trend model can be expressed as follows: yt = b0 + b1t+ εt b0 = the y-intercept; where t = 0. b1 = the slope coefficient of the time trend. t = the time period. ŷt = the estimated value for time t based […]

## Fcalc – the Global Test for Regression Significance

A statistically significant Fcalc (i.e. one that passes the Fcritical threshold, based on your degrees of freedom) can indicate that your model as a whole is meaningful. This test is really applicable for multiple regressions, where there is more than one slope coefficient (b1, b2, b3 … bi), as a t-test will not work for […]

## Dynamic Present Value

In this lecture we move from present values to dynamic present values. If interest rates evolve along the forward curve, then the present value of the remaining cash flows of any instrument will evolve in a predictable trajectory. The fastest way to compute these is by backward induction. Dynamic present values help us understand the […]

## Probability of Attaining a Return Goal

Earlier we looked at calculating the probability of beating a fixed target. Now we will look at calculating the probability of beating a benchmark which is itself stochastic. Let us consider two assets A and B with the following details: Mean Standard Deviation Correlation A B We have a total of $10 million to invest. […]

## Probability of One Portfolio Outperforming Another Portfolio

Let us consider two assets A and B with the following details: Mean Standard Deviation Correlation A B We have a total of $10 million to invest. Our objective is to reach a target return of $5 million. Let us look at the following three options and find out the probability of reaching our target […]

## Value at Risk (VaR) of a Portfolio

Value-at- Risk (VaR) is a general measure of risk developed to equate risk across products and to aggregate risk on a portfolio basis. VaR is defined as the predicted worst-case loss with a specific confidence level (for example, 95%) over a period of time (for example, 1 day). For example, every afternoon, J.P. Morgan takes […]

## Diversification and Portfolio Risk

## What is Serial Correlation (Autocorrelation)?

Correlation is a familiar concept used to describe the strength of the relationship between variables. Serial correlation (also known as autocorrelation) is the term used to describe the relationship between observations on the same variable over independent periods of time. If the serial correlation of observations is zero, observations are said to be independent. However, […]

## Minimum Variance Hedge Ratio

One problem with using futures contracts to hedge a portfolio of spot assets, is that a perfect futures contracts may not exist, that is, a perfect hedge cannot be achieved. For example, if an airline wishes to hedge its exposure to variation in jet fuel prices, it will find that there is no jet fuel […]