OK, so are you the mutual fund investor who likes to be “aggressive” – a word bandied around often in these circles? Which of the following portfolios do you like, given the annual returns? Assume the costs are the same.

P1. Year 1 – 5%, Year 2 – 5%, Year 3 – 4%
P2. Year 1 – 10%, Year 2 – -8% and Year 3 – 12%

In questions like these, there is more to the eye than mathematics. You could for example look at the first portfolio and say: “What use is this if it cannot beat my FD return in any of three years in a row?” Or you could arrive at conclusions about the capabilities of the fund managers. But for this article, let us put other angles aside and focus on the math.

First off, P1 (portfolio -1) is stable, and P2 is bumpy. Maybe they invest in different spaces (large-cap/small-cap, equities/bonds), but we do not make any assumptions about the underlying investments. If we calculate the sum of returns, it is 14% for both P1 and P2, but does the sum signify anything, we do not know.

Of course, to evaluate the portfolios, we need the total return at the end of three years, which is as follows.

Return R1: (1+0.05)(1+0.05)(1+0.04) =1.1466 = 14.66% return over 3 years
Return R2: (1+0.1)(1-0.08)(1+0.12) = 1.13344 = 13.34% return over 3 years

Now, if you are reasonably savvy about finance where the basic unit is a basis point (1 basis point = 0.01%), then you would be quick to realise that a 1.32% difference (132 basis points) is, for want of a fancy adjective, big. And to opt for an aggressive, high-risk portfolio, only to end up with smaller returns? Something must be wrong – are the numbers fudged? Are they specific to the chosen portfolios? Does this not generalize?

The answer to all these questions is No. Keep this in mind: When you are invested for long, the negative returns hurt you more than the positive returns do good. The problem with negative returns is that they erode your investment in the market. For portfolio P2 above, the amount of money subjected to the good returns in Year 3 is reduced due to the negative returns in Year 2. The bad Year 2 for P2 enables the stable P1 to outdo its peer – much like the hare vs. tortoise in kindergarten stories.

Now, if you are curious and want to understand this further, let us do some basic algebra. Suppose $a$, $b$ and $c$ are the yearly returns, then the overall return is $(1+a)(1+b)(1+c)-1$. To see why, Rs. 1 invested at the start of year 1 will be $1+a$ at start of year 2, which will in turn grow to $(1+a)(1+b)$ at start of year 3, etc. Thus, the return can be expanded as $a+b+c+ab+bc+ac+abc$. If the return for year 2 is negative, then the terms, $b$, $ab$, $bc$, $abc$ all become negative. The effect of $abc$ is typically smaller than the other terms, as it involves multiplication of three fractions (just like $0.1^3=0.001<0.01= 0.1^2$).

Given the return $a+b+c+ab+bc+ac+abc$, it is possible to play with different values of $a$, $b$ and $c$ to convince oneself that the anomaly noted above with P2 is no happenstance. Similar analysis can also be done for more years, but the core idea is the same.

TL, DR: Slow and steady wins the race, oftentimes. The last word is emphasized, as nothing comes without strings attached as far as personal finance is concerned.