June 29, 2003 money

# The“Average Return” Myth

Let’s say that you have \$1000 to invest. The first year, you invest it and get a 25% return, so you leave your money invested. The next year, the market doesn’t do as well and your return is -15%. What’s the average rate of return over the two years? You way think that it’s 5%, that is, (25% + (-15%))/2. Let’s do the math for 25% and -15%:

• Using simple interest, after 1 year, \$1000 + \$1000 * 25% = \$1250
• After 2 years, \$1250 + \$1250 * -15% = \$1062.50

Here we’re using Interest = Principle * Rate * Time calculation for yearly aka simple interest (I = PRT and Time is 1 year). Taking the numbers the other way, i.e. -15% the first year and 25% the next year, yields the same result:

• After 1 year, \$1000 + \$1000 * -15% = \$850
• After 2 years, \$850 + \$850 * 25% = \$1062.50

In fact, the result is the same no matter in which order that the rates come or how many there are:

Table 1: From Good to Bad

 year return total 0 0 \$1,000.00 1 25% \$1,250.00 2 15% \$1,437.50 3 5% \$1,509.38 4 -5% \$1,433.91 5 -15% \$1,218.82

Table 2: Starting Bad to Good

 year return total 0 0 \$1,000.00 1 -15% \$   850.00 2 -5% \$   807.50 3 5% \$   847.88 4 15% \$   975.06 5 25% \$1,218.82

Table 3: A Mixed Bag

 year return total 0 0 \$1,000.00 1 25% \$1,250.00 2 -15% \$1,062.50 3 5% \$1,115.63 4 15% \$1,282.97 5 -5% \$1,218.82

This result surprised me. I found it unintuitive that no matter how the rates vary over time, it doesn’t matter if they come first, last or in between. I expected large losses up front to swamp later gains or early gains to make up for late losses, but the change of the underlying principle amount evens things out, e.g. a smaller percentage drop later is against a larger principle if there have been early gains.

When you figure it as a single formula, Future Value = Principle * (1+Rate1) * (1+Rate2) * (1+Rate3) * (1+Rate4) * (1+Rate5) or F = P*(1+R1)*(1+R2)*(1+R3)*(1+R4)*(1+R5), the independence of the order makes more sense, since multiplication is commutative, i.e. it doesn’t matter in what order you do it:

```f = p*    (1+r1)* (1+r2)* (1+r3)*(1+r4)* (1+r5)
f = \$1000*(1+25%)*(1+15%)*(1+5%)*(1-5%)* (1-15%) = \$1218.82
f = \$1000*(1-15%)*(1-5%)* (1+5%)*(1+15%)*(1+25%) = \$1218.82
f = \$1000*(1+25%)*(1-15%)*(1+5%)*(1+15%)*(1-5%)  = \$1218.82```

Having varied rates like in Tables 1-3 in a stock or stock mutual fund investment isn’t uncommon (as we’ve just seen during and after the Internet bubble). On the other hand, if you compare this a fixed yield (like a bond) with our average rate of return” of 5%, you’ll see a different result:

Table 4: Small But Fixed Rate of Return

 year return total 0 0 \$1,000.00 1 5% \$1,050.00 2 5% \$1,102.50 3 5% \$1,157.63 4 5% \$1,215.51 5 5% \$1,276.28

With a fixed interest rate, we can simply our calculations somewhat using Future value = Principle * (1 + Rate)^Number of compounds. So, \$1000 at 5% for 5 years is:

```f = p*(1+r)^n
f = \$1000 * (1 + 5%)^5
f = \$1000 * (1.05)^5
f = \$1276.28```

Any way you calculate it, not only does the boring, fixed interest rate bond out-perform the variable rate even for the same average rate of return, but clearly our average rate of return calculation isn’t very useful. We’re not really getting 5% year to year on our varied stock rates of return, or they’d show the same results as the bond. Instead, if you reverse the formula for Rate, we get:

```r = (f/p)^(1/n) - 1
r = (\$1218.82/\$1000)^(1/5) -1
r = 1.21882^(1/5) -1
r = 4.037%```

This gives us a annualized rate of return:

Table 5: Annualized Rate of Return

 year return total 0 0 \$1,000.00 1 4.037% \$1,040.37 2 4.037% \$1,082.37 3 4.037% \$1,126.07 4 4.037% \$1,171.52 5 4.037% \$1,218.82

Taking this further, because the future value of an investment is the same whether you consider a fixed rate of return or a variable rate of return, for any principle, you can calculate the fixed rate of return by deriving from this formula (assuming r0 is the fixed rate of return and r1-r5 are the variable rates of return):

`p*(1+r0)^n = p*(1+r1)*(1+r2)*(1+r3)*(1+r4)*(1+r5)`

Further, because principle plays the same role on each side of the equation, you can remove it:

`(1+r0)^n = (1+r1)*(1+r2)*(1+r3)*(1+r4)*(1+r5)`

Solving for the annualized rate of return from the variable rates of return gives you this:

`r0 = ((1+r1)*(1+r2)*(1+r3)*(1+r4)*(1+r5))^(1/n) - 1`

Applying it in our example:

```r0 = ((1+25%)*(1+15%)*(1+5%)*(1-5%)*(1-15%))^(1/5) - 1
r0 = (1.25*1.15*1.05*0.95*0.85)^(1/5) - 1
r0 = 4.037%```

So what happens when we increase the variability, but leave the average rate of return the same? The variability adjusted return gets smaller:

Table 6: Extended Variability

 year return total 0 0% \$1,000.00 1 25% \$1,250.00 2 -15% \$1,062.50 3 5% \$1,115.63 4 15% \$1,282.97 5 -5% \$1,218.82 6 5% \$1,279.76 7 -25% \$   959.82 8 40% \$1,343.75 9 -30% \$   940.62 10 35% \$1,269.84

Notice that we’re still got an average rate of return of 5%, but increasing the variability gives us an annualized rate of return of 2.42%.

On the other hand, extending the same average without increasing the variability looks like this:

Table 7: Extended Time, Variability Unchanged

 year return total 0 0% \$1,000.00 1 25% \$1,250.00 2 -15% \$1,062.50 3 5% \$1,115.63 4 15% \$1,282.97 5 -5% \$1,218.82 6 25% \$1,523.53 7 -15% \$1,295.00 8 5% \$1,359.75 9 15% \$1,563.71 10 -5% \$1,485.52

In this case, when the variability remains unchanged, the annualized rate of return remains unchanged at 4.037%. In other words, as the variability increases, the annualized rate of return gets further away and lower than the simple average rate. On the other hand, as the variability decreases, the annualized rate approaches the maximum value of a fixed rate of return, i.e. zero variability.

So, while it’s comforting that so long as your investment doesn’t go to zero, it doesn’t matter when the highs and lows come, it’s somewhat unintuitive that the average rate of return is not what you want to use to calculate the rate of return that you’re actually getting. In fact, the annualized rate of return will always be lower than the average rate as variability increases.