how portfolio volatility kills your compounding gains
Average annual return of Nifty 50 index (using nav of NIFTBEES ETF) for the last 15 years (from 1st Jan 2007 to 31st Dec 2021) is 15.1%. So, if you would have invested in Nifty 50 index on 1st Jan 2007, you would expect your money to compound by roughly 15%.
Surprisingly, compounded returns for the 15-year period (CAGR) is just 10.8%. Another interesting example is Gold. Average annual return of Gold (in INR) during the same period was 12%. That is almost 3% less than Nifty 50. But during the same period, Gold has delivered a CAGR of 11.1%, 0.3% higher than Nifty 50.
What’s going on here?
This is a classic example of a phenomena called “variance drain” – how volatility drains your compounded returns.
Before we discuss variance drain, it is important to understand the concepts of “arithmetic mean” and “geometric mean”.
In our Nifty 50 example above, average annual return of 15.1% is the arithmetic average of annual returns in the last 15 years – a simple average of 15 yearly returns (2007 to 2021).
Arithmetic average is ubiquitous in finance. Anytime someone pitches you an investing strategy or an instrument, they generally present the arithmetic average – “this strategy has an average annual return of 15%”.
Most of the time, arithmetic average is typically extrapolated as the expected compounded returns – if Nifty 50 has an average annual return of 15%, then if I hold it for a long enough horizon, I can expect a CAGR of 15%.
Turns out that this is not the case!
As investors, we are more interested in the rate at which our money is compounding, i.e., compounded annual growth rate (CAGR). CAGR is more important for us as it measures the final wealth that we will have at the end of our investing horizon.
CAGR is nothing but “geometric” average of returns. Below we provide a mathematical equations for arithmetic and geometric averages.
Suppose that we have N years (historical) and return for each year is denoted by Ri. Then the arithmetic mean (M) and CAGR (G) are defined as follows:
M = (R1 + R2 + ….. + Ri + …. + RN)/N
G = [(1+R1) x (1+R2) x …. x (1+Ri) x …. x (1+RN)](1/N) – 1
We have already seen that CAGR (G) tends to be less than arithmetic mean (M). This happens because returns exhibit volatility – they tend to fluctuate around M.
Let’s illustrate this with a simple 3-year example. Suppose that annual returns were constant (no volatility) – something like fixed deposits. Let’s say annual returns were 5% for each of the 3 years. In that case, using the definition of M and G above, we can easily verify that both M and G are 5%. This shows that if there is no volatility in returns, CAGR is same as arithmetic mean.
Now consider an asset that exhibits volatility in returns. Let’s assume that returns are 10%, -10% and 15% respectively for the 3 years. Arithmetic mean in this case is still 5%. However, CAGR is just 4.4%. This is “variance drain”.
A more intuitive (and rather extreme) example would be am asset that goes down 50% in the first year and then goes up 100% in the next year. Average annual return is 25%. But CAGR is 0% (your wealth remains unchanged).
Simply put, variance drain is nothing but volatility bringing down your CAGR from expected average return. Higher the volatility, higher will be the drain (we will show this in next section).
Note that variance drain is not a financial phenomenon but a mathematical one (albeit with huge financial implications). It’s a manifestation of how arithmetic and geometric mean are connected by volatility. Below we show an approximate mathematical relationship between M and G, as connected by volatility (V) [1].
G =~ M – V2/2 + (G2/2 – M2/2)
And for G and M much less than 100% (which happens to be the case in reality), the equation further simplifies to
G =~ M – V2/2
=~ is used to signify approximate equality
As can be seen, G is always less than M and higher the volatility V, greater is the difference between G and M. Note how this formula very closely approximates our Nifty 50 example.
In our example, average yearly return of Nifty 50 was 15.1% and volatility of returns was roughly 30%. Equation highlighted above approximates the actual CAGR quite well.
We also note that since we are using yearly returns on 15-years of data, we only have 15 data points. This makes sample average and standard deviation highly unstable. Alternatively, we can use daily returns but it doesn’t change the broader result. For the sake of simplicity, we will continue to work with yearly returns.
The above equation can be very useful for investors to choose between various investment options.
As an illustration, suppose we have 2 investment options – one with an average return of 10% and volatility of 12% and the other with an average return of 12% and volatility of 24%.
A risk-seeking investor might just pick the second option. However, if she were to apply the above equation, she would figure out that expected CAGR is higher in the first option – 9.28% as against 9.12%. So even an infinitely risk-seeking investor will pick the first option.
In this section, we run some monte carlo simulations to illustrate the impact of variance drain. The experiment is set up as follows:
We consider 6 different investment options (or strategies) with expected average annual return ranging from 10% to 15%. We assume a constant Sharpe ratio of 0.6 (assuming risk free rate of 0%). This gives us the expected volatility for each option. For each option, we simulate 1000 paths of 50 years (assuming normal distribution). Each path gives us a 50-year CAGR, which is then averaged to yield the CAGR for that particular option.
We repeat the same experiment, but this time with decreasing sharpe ratio as we climb up the risk ladder.
Exhibits below present the results:
Exhibit 1: Constant sharpe ratio of 0.6
Exhibit 2: Sharpe ratio decreases from 0.6 to 0.5
In the tables above, Variance Drain column is the difference between Mean return and CAGR. The last column shows variance drain as percentage of mean return.
Some interesting observations:
We also note that the results above are based on assumption of normally distributed returns. If we assume a fat-tailed distribution like student-t, impact of variance drain is even more magnified.
What can an investor do about variance drain? As we have noted above, variance drain is a mathematical phenomenon and not a financial one.
However, investors can use the knowledge of variance drain to make some informed choices. Remember, as an investor, what matters to you is the CAGR.
Don’t fall into the trap of “arbitrarily” chasing risk in search of higher returns. It is quite possible that what appears to be a high return portfolio could end up yielding low CAGR due to variance drain.
As you climb up the risk ladder (seek higher risk), the reward (risk premium) must be sufficient to compensate for the variance drain. Typically, in investments, there is a decreasing marginal utility of risk. In other words, the extra return you get for taking an additional unit of risk decreases as risk increases. Therefore, blindly chasing returns could be self-defeating beyond a point.
The best way to control variance drain is to control the volatility in your portfolio. And the easiest way to achieve that is through diversification and rebalancing which we will discuss in our next blog.
[1] Variance Drain, The Journal of Portfolio Management 1995, Messmore, Thomas E