The Math of things

2. Inflation and real value
Inflation means a rise in the general level of prices of goods and services in an economy over a period of time. In other word, it indicates a decline in the real value of money.

For example, let's say eggs cost $2 per dozen today. A personal with $10,000 could buy 5000 dozens of eggs. After 10 years, the egg's price become $20. The same $10,000 could only buy 500 dozen's of eggs. Therefore, the real value of the money is only 1/10 of what it was.

Unfortunately, just like interests, inflation also compounds. Therefore, at an annual rate of i, the real value of the money over n years would decrease by

(1+i)^n

For example, at a 3% inflation rate, over 40 years, the ral value of the money would decline

(1+3%)^40 = 3.2620

Giving back to our original example with compound interest. Our impressive $452,592.56 asset would be worth

$452,592.56 / (1+3%)^40 = $452,592.56 / 3.2620 = $138,745.34

in today's dollar.

Therefore, one should never be impressed by the numbers a financial advisor throws out, since it's the purchase power that is important. Just like compound interests can quickly grow our asset, compound inflation can work against us to shrink our purchase power. This is especially important in retirement calculation due the long time frame.

Simple. Look at the cpi going back over the last 50 years and try and fit a curve with an exponential equation. You can't. At best you can fit a straight line. It isn't perfect, but it fits much better than an exponential equation.

Put a dollar in the bank, take it out after a year and you will have it returned with interest. Put a pound of freeze-dried coffee in a safety deposit box and take it out in a year's time. Depending on weather, disease, people's taste, wage settlements in the coffee industry plus a ton of other factors, that pound of coffee may be worth more or less. The dollar will always spin off a positive, the coffee won't always be worth more.

The reason most planners like to view inflation as an exponential is that it makes it much easier to integrate it in financial planning models.

Pick your poison, but get serious... do you really want to plan for a 3% inflation over 40 years?

Thank you all for your comments. I plan to write a bit more on this over the next several days. It will get more interesting once we get into things like mortgage payments.

Steve, the numbers will never 100% conform to the curve. All we can do is to observe a long term average trend. I personally think inflation is compounded because otherwise the percentage does not make sense. For example, let's say the inflation rate is 5% and eggs went from $2 to $2.1 from 2000 to 2001. If it's liner, then 2002 would be $2.2, 2003 would be $2.3. etc... However, that doesn't make sense since what's so special about the 2000 number that we use it as the bases of all other years? If inflation is linear, then it can not be percentage based. Instead, we can only say that eggs inflate by $0.1 per year.

In any case, everybody knows how to calculate linear growth. If people think inflation is linear, it's not hard to do the calculation.

3. Compound growth with fixed contributions.

Finally some more interesting stuff. Very few people invest all their money in a single year. Instead, we earn money every year and put some money into investment. Let's assume we put in a fixed amount at the beginning of each year for n years, say $X. With annual growth rate of r, what would be the future value of the investment?

At the end of year 1, we should have X * (1+r)
At the beginning of year 2, we would have X * (1+r) + X
At the end of year 2, we should have (X * (1+r) + X) * (1+r) = X * (1+r)^2 + X*(1+r)
At the end of year 3, we would have X*(1+r)^3 + X*(1+r)^2 + X*(1+r)
...
At the end of year n, we would have X*(1+r)^n + X*(1+r)^(n-1) +... + X*(1+r)

X is the common multiplier here, so let's get it out first:

X * ((1+r)^n + (1+r)^(n-1) +... + (1+r))

The second term is something called geometric sequence, which means each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio.

Luckily, there's a formula to calculate the sum of a geometric sequence. It's actually derived very cleverly.

Let's assume Y = (1+r)^n + (1+r)^(n-1) +... + (1+r). Then we have

Y * (1+r) = (1+r)^(n+1) + (1+r)^n + (1+r)^(n-1) +.... + (1+r)^2

Then we use Y*(1+r) - Y, and we get

Y * (1+r) - Y = (1+r)^(n+1) - (1+r)

Everything in between is canceled out. Therefore, we can easily get the value of Y.

Y = ((1+r)^(n+1) - (1+r))/r

Therefore, the future value of our investment is:

X * ((1+r)^(n+1) - (1+r))/r

Nice, eh?

Let's take an example. Let's say somebody invest $5000 in TFSA every year for 40 years and the annual return is 10%. We will have

$5,000 * ((1+10%)^41 - (1+10%)) / 10% = $2,434,259.06

As you can see, it's not really all that hard to become a millionaire.

Of course, things get a lot less bright if we are talking about the real value. Again assuming 3% inflation rate, the investment would be worth

$2,434,259.06 / (1+3%)^40 = $746,238.77

Still substantial though.