# Outperforming Indices in Different Markets

Consider two scenarios in which the stock market index compounds at different rates:

1. robust 10% annual return
2. tepid 5% annual return

Now suppose you had access to a genuinely good asset manager (like Buffett) who guarantees a 1% out-performance of the indices, and charges no fees.

Clearly more is better than less in this case. So you should hire him or her!

But the purely academic question here is the following, “are the effects of outperforming the indices more valuable in robust or tepid markets?”

I was interested in this question, because I hear some murmur which suggests that active management might work better than passive indexing, when the market is expected to compound tepidly.

One can do a quick calculation to find out. Suppose that the holding period is n = 10 years.

At 10% market return, \$1 grows to \$2.59 in 10 years. At 11%, it grows to \$2.84 – a nearly 9.5% out-performance.

At 5% and 6% returns, \$1 grows to \$1.63, and \$1.79, respectively. This is a nearly 10% out-performance.

So technically it is true: out-performance is more valuable in tepid markets.

But in practice, no “alpha”-generating asset manager is going to work for you free. The extra value (note this 0.5% is not 0.5% compounded) is probably not enough to justify switching investing strategies from passive to active simply based on tepid expected forward returns.

## Math Aside

To understand this apparent insensitivity to market return, lets bring out some low-powered calculus.

Let $r$ be the market return, and let $\Delta r$ be the manager out-performance.

After n years, the market compounds \$1 to $f(r) = (1 + r)^{n}$. If $\Delta r \ll r$, we can approximate the out-performance $f(r + \Delta r)$ by its Taylor series:

$f(r + \Delta r) \approx f(r) +\Delta r\ df/dr$

Taking the derivative, and rearranging terms, we get,

$\dfrac{f(r + \Delta r)}{f(r)} \approx 1 + \dfrac{n \Delta r}{1+r}$

The percentage outperformance is controlled by the last term on the RHS. The factor “n” just shows that any incremental annual outperformance is essentially additive. The remaining term $\Delta r/1 + r$ does have a r in it, which means that technically it does matter whether the market is robust or tepid.

However, in practice, $1 + r \approx 1$. It turns out that the percentage $n \Delta r/1 + r \approx n \Delta r$, and the state of the market effectively drops out of the equation.