Forecasting Active Return Against a Fixed Benchmark
Summary
The document explains how to calculate the distribution of active return when a security is compared with a benchmark that has a fixed return and no volatility. It expresses each investment’s value after a year as a random variable, then subtracts the benchmark value to obtain the active-return distribution. When the security’s expected return is assumed to be zero and its volatility is six percent, its expected value is below that of a benchmark returning five percent. The resulting probability of underperformance is therefore much higher than the probability implied by looking at the security’s volatility alone.
To get a 15.8% chance of underperforming, the security’s expected return must equal the benchmark’s return, while its volatility remains unchanged. In that case, the active return is centered on zero, and one standard deviation below the mean corresponds to underperformance by one volatility unit. The example assumes a normal distribution and a deterministic benchmark; it does not address non-normal returns, uncertainty in the benchmark, or compounding conventions.
Key ideas
- Active return is the security’s value minus the benchmark’s value at the same horizon.
- A benchmark with a fixed positive return shifts the mean of active return relative to a security assumed to have zero expected return.
- The underperformance probability depends on both the expected return difference and the security’s volatility.
- If expected returns match and the benchmark is deterministic, active return has the security’s volatility and is centered on zero.
- The calculation relies on a normal-distribution assumption.
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Full text
# Using Normal Distribution to forecast active return
# Using Normal Distribution to forecast active return
I wanted advice on how to go about forecasting active return via a standard normal distribution,
The asset is a security with annual volatility of 6%. The benchmark is a 5% annual return with 0% volatility (basically a straight line)
Without a benchmark, i would just use the annual volatility and state that over a year, there is a 15.8% chance of this security returning <-6%.
But how would I approach this with a benchmark that has 0% volatility.
I want to be able to state that: there is an 15.8% chance this security will underperform the benchmark by X%. What would 1 std be in active return terms? -1%?
## Answer by Attack68 (score 3, accepted)
https://quant.stackexchange.com/a/40046
If your benchmark has a volatility of zero and return 5% then for a benchmark investment of 1.0 after 1 year it is guaranteed to be worth 1.05, or $$X_b = 1.05$$
On the other hand your second investment needs to have an expected return, your calculations have assumed mean of 0% and vol of 6%, so the value at 1Y is given by $$X \sim \mathcal{N}(1.00, 0.06) \;,$$ which is why you have a 15.8% chance the investment is worth less than 0.94.
If you want to look at the outperformance, $Y$, then:
$$Y = X - X_b \sim \mathcal{N}(-0.05, 0.06)$$
The probability of $Y$ being less than zero, (i.e. underperforming) the benchmark is $$P(Y<0) = \Phi(0) = 79.8\%$$, and this makes much sense since the expectation of your investment is much less than the non volatile benchmark.
If you want there to be a 15.8% chance $X$ underperforms $X_b$ then you must assume that $X$ is drawn from $X \sim \mathcal{N}(1.05,0.06)$, i.e. the expected return of second investment is same as benchmark, so that $Y \sim \mathcal{N}(0,0.06)$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.