# Information Ratio

The information ratio, also known as the appraisal ratio, is used to measure the risk-adjusted return of a financial asset portfolio (collection of assets).

On occasion, it may measure the risk-adjusted return of a single asset, or naturally do so in cases of a single-asset portfolio.

## Mathematical Definition

The mathematical definition of the information ratio is the expected value of the return of an asset or portfolio minus the expected return (“E”) of a benchmark divided by the standard deviation of this difference. The numerator value is often referred to as the “active return” while the denominator is referred to as the “tracking error”.

Information Ratio = E [Portfolio return – Benchmark return] / √var(Portfolio return – Benchmark return)

The information ratio is often used to determine the skill of asset managers or traders in adding value in excess of a benchmark return.

If their portfolio return is in excess of the benchmark, they are considered to add value over the considered time period and the information ratio will be positive. A negative information ratio will denote a portfolio manager or trader who has lost value relative to the benchmark.

## What Benchmark to Use

Benchmarks should be chosen relative to the focus of the manager.

For most, this will mean a stock index, such as the S&P 500.

This is generally true not only for stock traders, but also global macro managers, systematic or statistical arbitrage strategies, currency traders, among others.

This is largely because the S&P 500 is a liquid, cheap index one can invest in passively through such products as exchange-traded funds (ETFs).

It is therefore one of the most prominent financial returns benchmarks, not only in the United States but globally.

For a bond trader that takes a more conservative approach, however, it might be a different benchmark that better accords with the risk characteristics of the portfolio.

For example, for a bond trader that invests in mostly investment-grade debt, this might be some similar index.

Whatever the case, the numerator value is simply designed to show whether value was created over a specified time period.

It is not risk-adjusted excess return, but rather excess return itself.

When Jensen’s alpha is used in the numerator, then the information ratio is typically known as the “appraisal ratio”.

The denominator, which takes into account the volatility in the expected return, is intended to determine the return in risk-adjusted terms.

The higher this value, the better the manager, given the individual is producing higher return per unit of risk.

Information ratios of 0.5 are considered firmly above average.

Information ratios of 1.0 or better are considered very good and generally reflect top decile performance.

Some asset managers will rely on the information ratio in calculating a performance fee.

## Limitations

Nonetheless, the information ratio ignores the influence of leverage on portfolio returns.

Thus, even if a portfolio produces better risk-adjusted returns than the index, the information ratio can come back negative.

For example, the S&P 500 is expected to return about 7% annualized over the long-run at 15% annualized volatility.

A trader that returns 3% per year at 4% volatility is doing a better job in risk-adjusted returns, but the information ratio would be negative if the benchmark is the S&P.

If one were to leverage the 3% per year trader’s portfolio up to match the volatility of the 7% return/15% volatility benchmark he would be producing 4.25% in alpha per year before accounting for any leverage costs.

In these cases, it is better to use the geometric information ratio.

## Calculation of the Geometric Information Ratio

To do this, you’re much better off using a computer program or spreadsheet because it’s a mess to compute by hand.

1. First compute the manager’s beta, which is the covariance between the return of the portfolio and the return of the benchmark, then divide it by the variance of the return of the benchmark:

ß = Cov (Return portfolio, Return benchmark) / Var (Return benchmark)

1. Returns over a specified period of time must be normalized using beta.These returns can be either weekly, monthly, quarterly, or some other period.

3.Calculate alpha (excess return over benchmark) by taking the return of the portfolio and subtracting the product of beta and the return of the benchmark.

Alpha = Return portfolio – ß * Return portfolio

1. Sum up the alphas and calculate its geometric mean – i.e., how much alpha is earned per time period, such as one year.

Alpha (geometric mean) = ∑ (1 + alpha)^[1/(number of periods)] – 1

1. Calculate the standard deviation in alpha by taking the square root of the variance of all alphas:

√var(∑ Alpha)

1. Take the value obtained in step four and divide it by the value obtained in step five.

Ensure that the time periods are compatible.