While making an investment decision, it is important to assess the risk/return profile of any investment. The relation between risk and return raises three basic questions: How do I estimate the percentage return that I will receive on an investment? How much risk does an asset add to a portfolio? What can I do to […]

# Understanding Portfolio Math

## Constructing an Efficient Frontier

The concept of Efficient Frontier was first introduced by Harry Markowitz in his paper on Portfolio Selection (1952 Journal of Finance). The portfolio theory considers a universe of risky investments and explores these possible investments in order to find the optimum portfolio. So, for a given amount of risk, MTP explains how select a portfolio […]

## 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 […]

## 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, […]

## Diversification and Portfolio Risk

## 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 […]

## 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 […]

## 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. […]