The Information Ratio Vs. The Sharpe Ratio

The information ratio and the Sharpe ratio are two different methods of measuring a portfolio’s risk adjusted rate of return.

The Sharpe ratio (SR) is calculated as excess return per unit of risk (standard deviation):

An important attribute of the Sharpe ratio is that it is unaffected by the addition of cash or leverage in the portfolio. A 50% allocation to the risk free asset would reduce both the excess return and standard deviation of returns by half.

The information ratio (IR) is the ratio of the active return to the standard deviation of active returns, which is known as active risk or benchmark tracking risk:

Some Important Points

• In this topic review, the information ratio that we are considering is usually the ex- ante information ratio (i.e., the information ratio based on expectations). The ex ante information ratio is generally positive (otherwise active management is not worth pursuing), while ex post information ratios will often turn out to be negative. A closet index fund is a fund that is purported to be actively managed but in reality closely tracks the underlying benchmark index.
• These funds will have a Sharpe ratio similar to that of the benchmark index, a very low information ratio, and little active risk. After fees, the information ratio of a closet index fund is often negative.
• A fund with zero systematic risk (e.g., a market neutral long short equity fund) that uses the risk free rate as its benchmark would have an information ratio that is equal to its Sharpe ratio. This is because active return will be equal to the portfolio’s return minus the risk free rate, and active risk will be equal to total risk. Unlike the Sharpe ratio, the information ratio will change with the addition of cash or the use of leverage.
• The numerator (active return) of the information ratio is measured relative to a noncash benchmark. Adding cash to a portfolio is likely to lower active return, while active risk (i.e., volatility of active return) should not change much, meaning that the addition of cash is most likely to decrease the information ratio.
• The information ratio of an unconstrained portfolio is unaffected by the aggressiveness of the active weights. If the active weights of a portfolio are tripled, the active return and the active risk both triple, leaving the information ratio unchanged.
• If we combine an actively managed portfolio with an allocation to the benchmark portfolio, the resulting blended portfolio will have the same information ratio as the original actively managed portfolio. As we increase the weight of the benchmark portfolio, the active return and active risk decrease proportionately, leaving the information ratio unchanged.
• Investors can select an appropriate amount of active risk by investing a portion of their assets in the active portfolio and the remaining portion in the benchmark. For example, if the active risk of a fund is 10%, an investor seeking to limit active risk to 6% can do so by investing 60% in the active portfolio and the remaining 40% in the benchmark portfolio.

For an unconstrained active portfolio, the optimal amount of active risk is the level of active risk that maximizes the portfolio’s Sharpe ratio. This optimal active risk can be calculated as:

The Sharpe ratio of a portfolio with optimal level of active risk can be calculated as:

Furthermore, the total risk of the portfolio is given by: