Information Ratio Repair

he information ratio (IR) is a measure of volatility-adjusted return. Morningstar does not report this ratio, though Zephyr Associates does in their StyleADVISOR software. As noted by Tom Idzorek at Zephyr Associates (www.styleADVISOR.com) the information ratio is also commonly known as the appraisal ratio.

Technically, the information ratio is excess return divided by tracking error. Excess return is the amount of overperformance (or underperformance) of a given asset relative to a pre-determined benchmark index. Tracking error is the standard deviation of the excess return. The IR is often described as (a) residual return divided by residual risk or (b) alpha divided by residual risk. Whatever the definition, the higher the IR the better.

While the IR is similar to the Sharpe Ratio (SR) there is a significant difference (the Sharpe Ratio incidentally is reported in Morningstar). The SR compares the return of an asset against the return of T-bills, whereas the IR measures "excess return" in comparison to the most relevant equity (or debt) benchmark index. In spite of the title, "excess" return can be negative or positive. In fact, it is precisely because excess return can be negative that this article is being written.

An example will be helpful. Let's analyze USAA Growth & Income. The first step in calculating the IR is to calculate each period's excess return (I used annual periods, Zephyr uses monthly periods). This is accomplished by subtracting the annual return of the appropriate benchmark (in this case, the S&P 500) from the annual return of USAA Growth & Income. In 1999, the excess return was -6.74% (as calculated by 14.30% minus 21.04%). In 2000 it was 12.09%, and so on. (see Figure 1)

The five year annualized return of USAA G&I was 2.37%. The five year annualized return for the S&P 500 was -0.57%, therefore the five year excess return was 2.37% - (-0.57%) = 2.94%.

The standard deviation of the five annual excess return figures was 6.97%. Thus, the five year information ratio for USAA Growth & Income was as follows:

  Excess Return		2.94%
Information Ratio  =   -----------------   =	------	--  =	  0.42
  Tracking Error		6.97%

Figure 1. Inside the IR

	1999	2000	2001	2002	2003	5 Yr Return	Tracking Error[2]	5 year Information Ratio
Return of Standard & Poor's 500	21.04	-9.10	-11.88	-22.09	28.67	-.57
Return of USAA Growth & Income	14.30	2.99	-6.13	-21.27	29.22	2.37
Excess Return (ER) of USAA Growth & Income	-6.74	12.09	5.75	0.82	0.55	2.94	6.97	0.42

The information ratio is best utilized as a ranking criterion for mutual funds of a similar asset and style class. Thus, in the example above, if another equity fund that has the S&P 500 as its best-fit index produced an IR larger than .42 it would be a superior choice on a volatility-adjusted basis.

But, alas life (or at least the IR) is not so simple. There is an anomaly with the information ratio when negative excess returns are encountered. Another example will illustrate the dilemma (see Figure 2).

Figure 2. The Negative Excess Return Problem

	1999	2000	2001	2002	2003	5 Yr Return	Tracking Error	5 year Information Ratio
Return of Standard & Poor's 500	21.04	-9.10	-11.88	-22.09	28.67	-.57
Return of Delaware Devon A	-10.85	-13.72	-10.31	-23.24	27.72	-7.52
Excess Return (ER) of Delaware Devon A	-31.89	-4.62	1.57	-1.15	-.95	-6.96	13.86	-0.50
Return for Safeco Core Equity Inv	9.37	-10.96	-9.72	-26.33	24.68	-4.19
Excess Return (ER) of Safeco Core Equity Inv	-11.67	-1.86	2.16	-4.24	-3.99	-3.62	5.03	-0.72

Notice that Delaware Devon A was assigned a higher IR (i.e. a smaller negative) than Safeco Core Equity. Both had negative 5 year excess returns over the period from 1999-2003. However, Safeco Core Equity had a 5 year return that was 333 basis points higher than Delaware Devon (-4.19% vs. -7.52%) and a standard deviation of annual excess return that was nearly 900 basis points lower (5.03% vs. 13.86%).

In a two-fund world, Safeco Core Equity is clearly the better choice if one accepts the standard assumptions implicit in the IR (higher excess return is better, lower standard deviation of excess return is better). And yet, Safeco Core Equity is assigned a lower IR. This anomaly occurs when negative excess returns are present.

The solution requires a modification to the formula as shown below.

Modified Information Ratio

   ER
--------					     	 					    
   SD(ER/abs ER)	

ER = Excess Return (where:  Excess Return = Asset Return - Benchmark Return)
SD = Standard Deviation of ER
abs = Absolute value

The standard information ratio is modified by adding an exponent to the denominator. The exponent is excess return (or ER) divided by the absolute value of the ER.

When ER is positive, the standard IR is identical to the modified IR. When ER is negative, the modified IR and the standard IR can be very different.

Application of the modified information ratio is demonstrated in Figures 3-5. Figure 3 includes 25 funds with a negative ER over the period from 1999 - 2003. These funds are ranked from best to worst using the standard IR (in yellow). Notice that Delaware Devon is ranked 6th despite having the lowest ER and the largest standard deviation of ER. Safeco Core Equity had a higher 5 year ER and substantially lower tracking error, yet was ranked 12th using the standard IR formula.

Using the modified information ratio (in purple) the ranking of funds in Figure 3 changes dramatically. Delaware Devon is ranked 25th and Safeco Core Equity is ranked 22nd. The ranking of funds based upon the modified information rewards higher ER and lower standard deviation of ER; the same reward protocol when ER is positive.

Figure 4 graphically displays the ER (in blue) and tracking error (in pink) of the 25 funds in Figure 3 according to the standard formula IR ranking (from highest to lowest). Notice the odd result in which funds with the largest tracking error and lowest ER are ranked in the upper half of the group.

Figure 5 displays the ER (in blue) and the tracking error (in pink) of the same 25 funds - but in this case they are ranked from best to worst according to the modified IR formula. As tracking error increases and ER deteriorates, funds systematically receive a lower IR ranking. This graph displays the coherent relationship that should exist between ER, tracking error and IR ranking.

As a final note, there is one odd feature of the modified information ratio: the enormous range in the size of the ratio (or quotient). Delaware Devon's modified IR was -96.39. The largest negative IR (in this sample of funds) using the standard IR formula was -2.98 for Fifth Third LrgCap Core. Thus, the size or magnitude of the modified IR is of lesser value. Its real value is as a ranking criterion. The modified IR correctly ranks funds according to residual return over residual risk -- which is exactly what the IR is supposed to do.

Figure 3. Standard and Modified IR Ranking with Negative ER

Equity Fund Name	5 Year Excess Return (1999-2003)	Tracking Error	Standard Information Ratio	Standard IR Rank	Modified Information Ratio	Modified IR Rank
Hartford Stock HLS IA	-0.74	1.76	-0.42	1	-1.31	4
Evergreen Blue Chip B	-2.10	4.82	-0.44	2	-10.15	15
Dreyfus Growth Opport	-1.85	4.00	-0.46	3	-7.40	10
Frank Russell Tax LgCp S	-1.19	2.49	-0.48	4	-2.96	6
Matterhorn Growth	-3.55	7.35	-0.48	5	-26.09	23
Delaware Devon A	-6.96	13.86	-0.50	6	-96.39	25
McMorgan Eq Investment	-2.51	4.97	-0.50	7	-12.46	18
Rydex Nova Inv	-6.13	10.73	-0.57	8	-65.71	24
DLB Enhanced Idx Core Eq	-1.79	2.95	-0.61	9	-5.28	8
Perform Lg Cap Eq Instl	-2.61	4.13	-0.63	10	-10.78	17
BBH Tax-Efficient Eq N	-3.32	4.71	-0.70	11	-15.64	20
Safeco Core Equity Inv	-3.62	5.03	-0.72	12	-18.21	22
JP Morgan Discipl Eq Ins	-1.34	1.67	-0.80	13	-2.24	5
Wright Major Blue Eqty	-2.95	3.43	-0.86	14	-10.11	14
Security Equity A	-3.97	4.25	-0.93	15	-16.89	21
Fidelity Tax-Managed Stk	-1.14	1.08	-1.06	16	-1.22	3
Lutheran Brother A	-3.26	3.02	-1.08	17	-9.82	13
Wilmington Large Core I	-3.55	2.95	-1.20	18	-10.48	16
Marshall LgCp Gr&Inc Inv	-2.79	2.07	-1.34	19	-5.79	9
Principal Ptr Blue Chip A	-4.44	3.28	-1.35	20	-14.55	19
ING Disc LargeCap fund B	-2.57	1.79	-1.44	21	-4.58	7
Lake Forest Core Equity	-3.97	2.25	-1.76	22	-8.94	11
Van Kampen Pace A	-4.84	1.95	-2.48	23	-9.43	12
Wells Fargo Index All A	-1.10	0.44	-2.49	24	-0.49	1
Fifth Third LgCp Core Ins	-1.88	0.63	-2.98	25	-1.18	2

Figure 4. The Problem

Figure 5. The Solution

____________________________________________________________________________________
Craig L. Israelsen is an Associate Professor in the Department of Consumer and Family Economics at the University of Missouri-Columbia (http://www.missouri.edu/index.cfm) where he teaches courses in Personal Finance and Family Living. He holds a Ph.D. in Family Resource Management from Brigham Young University. He received a B.S. in Agribusiness and a M.S. in Agricultural Economics from Utah State University. Primary among his research interests is the analysis of mutual funds. He writes monthly for Financial Planning magazine.