Case 042Factor investing and quantHard

A multi-factor fund combines a value sleeve and a momentum sleeve, each with an information ratio of 0.4, at equal risk. What is the combined information ratio if their active returns correlate minus 0.5, and if they correlate plus 0.3?

The situation

Dwandva Multi-Factor Fund runs two sleeves against the same equity benchmark. The value sleeve buys cheap stocks; the momentum sleeve buys recent winners. Each targets 4% tracking error and is expected to earn 1.6% a year of active return, an information ratio of 0.4. The fund puts half its active risk budget in each.

Over the last decade the two sleeves' active returns correlated at about minus 0.5. The risk team's stressed estimate, for a market where both factors are crowded, is plus 0.3. The fund's total tracking error budget is 4%.

Your task

What is the combined information ratio at each correlation, what does it mean for sizing within the 4% budget, and which correlation should the fund plan on?

Quick check

At a correlation of minus 0.5, what is the combined information ratio?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

The combined information ratio is 0.80 at a correlation of minus 0.5 and about 0.50 at plus 0.3. The half and half mix keeps 1.6% of active return while its tracking error falls to 2.0% or 3.22%. All the improvement comes from the correlation, so the fund should size on a stressed figure: scaled to 4% risk on the historical minus 0.5, a move to plus 0.3 would push tracking error to 6.4%.

Step 1Why do two average sleeves make a better fund?

Two vendors on a beach, one selling umbrellas and one selling ice cream, each have lumpy months: one wins when it rains, the other when it is sunny. A shop that sells both earns the same average but swings far less. Active returns add up in a straight line, but their risks add up through the correlation, so the mix keeps the return and sheds risk. Value buys stocks that have fallen and become cheap; momentum buys stocks that have risen. They often disagree about the same stock, which is why their active returnsThe return of a portfolio minus the return of its benchmark. have tended to move in opposite directions.

The relationship

IR_{c} = \frac{\tfrac12\alpha_1 + \tfrac12\alpha_2}{\sqrt{\tfrac14\sigma_1^2 + \tfrac14\sigma_2^2 + 2\cdot\tfrac14\rho\,\sigma_1\sigma_2}} = \frac{2 \times 0.4}{\sqrt{2(1+\rho)}}

alpha	active return of each sleeve, 1.6%
sigma	tracking error of each sleeve, 4%
rho	correlation of the two sleeves' active returns

What it says in wordsWith equal sleeves the combined ratio is 0.8 divided by the square root of 2 times one plus the correlation: 0.80 at minus 0.5, 0.57 at zero and 0.50 at plus 0.3.

Do the minus 0.5 case in your head. Each half carries 2% of risk. The variance is 4 + 4 + 2 x (-0.5) x 2 x 2, which is 4, so tracking error is 2%. The mix earns 1.6% on 2.0% of risk, an information ratio of 0.80, double either sleeve. At plus 0.3 the variance is 4 + 4 + 2.4 = 10.4, tracking error 3.22%, and the ratio 0.50: still better than one sleeve, but most of the gain has gone.

The combined information ratio of two 0.4 sleeves falls from 0.80 at a correlation of minus 0.5 through 0.57 at zero to 0.50 at plus 0.3, and reaches 0.40, no better than one sleeve, only at a correlation of one.

Step 2What does the better ratio buy inside a 4% budget?

The mix uses only part of the risk budget, so the fund can scale both sleeves up. At minus 0.5 it can double each sleeve, multiplying by 2.00, and earn 3.2% of active return on the full 4%; at plus 0.3 it can only scale by 1.24, for 1.98%. The ratio also decides how long investors wait before the record proves anything. To reach a t-statistic of 2 takes (2 / IR) squared years: 25 years for one sleeve, about 16 at plus 0.3 and about 6 at minus 0.5.

Each sleeve earns 1.6% on 4% of risk; mixed half and half they earn the same 1.6% on 2.0% at a correlation of minus 0.5 or 3.22% at plus 0.3, and scaling the minus 0.5 mix back up to 4% risk doubles the active return to 3.2%.

Portfolio	Information ratio	Scale to fill 4%	Active return at 4%	Years to t = 2
One sleeve	0.40	1.00	1.6%	25
Mix at +0.3	0.50	1.24	1.98%	16.2
Mix at 0	0.57	1.41	2.26%	12.5
Mix at -0.5	0.80	2.00	3.2%	6.25

A higher combined ratio means more active return from the same 4% of risk and a far shorter wait to statistical proof: about 6 years at minus 0.5 against 25 years for a single 0.4 sleeve.

Step 3Which correlation should the fund plan on?

Not the flattering one. If the fund doubles both sleeves on the historical minus 0.5 and the correlation then moves to plus 0.3, tracking error jumps from 4% to 6.4%, a breach of the budget by more than half. Correlations between factors drift, and in sharp market reversals momentum can lose heavily while value fails to offset it. A defensible middle course sizes for 4% at a correlation of zero, a scale of 1.41: tracking error is then 2.8% if the history holds and 4.6% under stress, and expected active return is 2.26%. Say also what the formula leaves out: each sleeve's 0.4 is itself an estimate, and momentum trades far more, so costs eat into its share of the return.

Where candidates lose it

The usual slip is averaging the information ratios and answering 0.4, or adding them and answering 0.8 whatever the correlation. Ratios do not average; returns add, and risk combines through the correlation.

The second is quoting the minus 0.5 answer as the plan. The correlation is the whole source of the gain and the least stable input, so a candidate who sizes on it without a stress case has built a risk breach into the fund.

What the interviewer asks next

The momentum sleeve has an information ratio of 0.3 and value 0.5. Are equal risk weights still right?
How does combining the signals at the stock level differ from running two separate sleeves?
What would you watch to detect that the correlation is rising?

Company names and figures are illustrative.