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# R8.P1.T1.Amenc_Ch4_RISK_MGMT_Topic:INFORMATION_RATIO_RESIDUAL vs_ACTIVE

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @gargi.adhikari It's a good question, I am using Amenc's formula below: IR = t-stat/sqrt(T), such that t-stat = IR*sqrt(T); in the example above, the active IR = 4.49% (displayed as 0.045) such that t-stat = 4.49*5.0 years, where 5.0 years is a simple input. So the 4.49% is just the information ratio although I made a mistake to show it formatted a percentage (%), which probably throws you off. because I do think it's unitless (%/%); i.e., I think its correct to say that IR is a (unitless) percent not a percentage, so it's 0.045 not really 4.5%.

I could instead have asked for confidence as an input; e.g., like Amenc, let 95% determine a 1.96 2-tailed deviate. Then the number of years implied would be (1.96/0.045)^2 = 1,902 years of this sort of performance in order to be deemed statistically significant at 95%, wow! This is always confusing to folks, the key idea is that the information ratio scales with the square root of time (sound familiar?) because IR = α/σ(α) scales with Δt per [α*Δt]/[σ(α)*sqrt(Δt)]; i.e., alpha is linear but standard deviation scales with sqrt(T), such that sustained/scaled IR = [α*Δt]/[σ(α)*sqrt(Δt)] = [α/σ(α)]*[Δt/sqrt(Δt)] = [α/σ(α)]*sqrt(Δt). But IR is itself a t-stat (coefficient divided by its own standard error!), so this is t-stat = IR*sqrt(T); i.e., scaling the annual IR over the number of sustained periods renders it its own t-stat. Please note this notation doesn't distinguish between which alpha; the XLS happens to use the active IR. I hope that explains! ##### Active Member
@David Harper CFA FRM Thanks so much for the in depth explanation and the excerpt from AMENC was perfect !! ... Thanks a ZILLION ! Last edited:

#### cynthiahuang

##### New Member
Hi, would you please take a look at question P1.T1.32, part one (what is the information ratio)?

32.1 Make the following assumptions:
Riskfree rate is 3%
The benchmark is the market (i.e., CAPM) and the benchmark return was 8%
Portfolio beta is 1.2
Portfolio return was 10%
Tracking error was 10%
Minimum acceptable return (MAR) was 2%
Downside deviation was 5%
What is (was) the information ratio?
a) 0.10
b) 0.20
c) 0.30
d) 0.40

32.1. A. 0.10
Alpha = 10% - (1.2 beta * 5% ERP) - 3% riskfree rate = 1%.
IR = alpha/TE = 1%/10% = 0.10
... please note that (B) is tempting because alpha of 10% - 8% is tempting. However, that is
active return not residual return (alpha).

My understanding is Tracking Error is linked to active return, therefore we should use active return/TE to get the information ratio. In the explanation, however, residual return was used, would you please explain the reason behind it?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @cynthiahuang Yes, you are correct. Per the source thread here at https://www.bionicturtle.com/forum/threads/l1-t1-32-tracking-error-sharpe-sortino-amenc.3465/ , this was a question I wrote (based strictly on Amenc) before we explained to GARP the problems created by an ambiguous information ratio. You are correct that "tracking error" connotes active risk (connotes is the verb that I deliberately am using to allow for some author variation, but TE = active return is clearly the default). We subsequently informed GARP's view, which is currently (and has been for years) that the IR can be either: active_return/active_risk, or residual_return/residual_risk. See this for detailed, definitive view: https://trtl.bz/2IXrRN5
... but I do intent to re-write these questions at some point this year, to remedy the lack of rigor to which you refer. Thanks!