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#### archanfel

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While talking with a financial advisor, she insisted that a regular person could not calculate things like future value without a spreadsheet designed by the bank's actuary. While that's true for complex insurance estimations and market forecasts, basic calculations in personal finance rarely require more than high school math. Unfortunately, in today's computer age, people are used to online calculators, spreadsheet, etc... and such basic math skills are largely overlooked. Let's talk about some of the most common calculation done in personal finance.

1. compound growth
We all know the power of compound interest and time. For example, a \$10,000 investment grow at 10% annual return over 40 years might sound like it can grow to \$50,000. In reality, it would actually grow to \$\$452,592.56

Let's assume our principle is X, and our annual return rate is r

After 1 year, we would have X * (1+r)
After 2 years, we would have X * (1+r) * (1+r)
After 3 years, we would have X * (1+r) * (1+r) * (1+r)

We can already see the pattern here. Therefore,
After n years, we would have X * (1+r)^n

Going back to our example, \$10,000 investment grow at 10% annually over 40 years would be worth

\$10,000 * (1+10%)^40 = \$452,592.56

It's interesting to note that the original investment is actually the least important variable in the equation since it's a linear component. For example, half the investment will result in exactly half the future value.

\$5,000 * (1+10%)^40 = \$226,296.28

The interest rate and time, on the other hand, can causes much bigger changes. For example, half the interest rate would lower the future value to only about 7 times.

\$10,000 * (1+5%)^40 = \$70,399.89

Similarly, half the investment time would lower the future value even more

\$10,000 * (1+10%)^20 = \$67,275.00

#### archanfel

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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.

#### Patricia

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Good post archanfel, thanks.

#### Sampson

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"While talking with a financial advisor, she insisted that a regular person could not calculate things like future value without a spreadsheet designed by the bank's actuary."

Ridiculous!

I find that my experience with people in the industry is that THEY are the ones who don't know how to do rudimentary calculations. They are so dependent upon theses 'complex' software models where they input a single value and get an out put answer of 'buy this from us'.

#### steve41

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The problem with saying that inflation compounds, is that when you examine the behaviour of inflation (the cpi), the curve (historical track) of the consumer price index doesn't follow an exponential trend... at best, is linear in nature. Many planners (not too many) opine that the equation is (1+n*i) rather than (1+i)^n It makes for a much more realistic inflation rate especially over long time periods. 3% over 40 years gives a very unlikely projection using (1+i)^n

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Many planners (not too many) opine that the equation is (1+n*i) rather than (1+i)^n It makes for a much more realistic inflation rate especially over long time periods. 3% over 40 years gives a very unlikely projection using (1+i)^n
I disagree and I'd like to know the rationale for not compounding inflation. CPI is calculated by Statistics Canada by pricing a basket of goods and services in nominal dollars. The inflation rate is derived from the changes in these prices and expressed as a percentage change with the year ago period. Therefore, inflation should be compounded, not calculated as simple interest.

#### lb71

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Therefore, inflation should be compounded, not calculated as simple interest.
+1

#### steve41

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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?

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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.
Are you sure the chart is on a linear scale and not a log scale? In fact, if you have "Stocks for the Long Run" look at the logarithmic chart on Page 6. The CPI line (like the bond and stock line) is roughly a straight line.

#### steve41

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I am not an economist, however, intuitively when I invest every year in a 1 year GIC, I know that at the end of the year my capital will have grown. Individual commodities don't behave that way. I have seen what a lot of individual financial planners do, and some (most) prefer the exponential cpi formula. Others, the linear. There is an obvious bias at work as well. Planners, as most in the financial services industry prefer the gloom and doom (a 3% cpi over 40 years) approach because it scares clients into 'saving to achieve \$2M' in their retirement nest egg. The linear inflation model results in a more conservative planning outcome.

#### archanfel

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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.

#### steve41

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Plot the consumer price index going back 20 years. Now, take the current CPI and extend that curve out for the next 50 years at 3% compounding. Look at the overall curve and report back.

#### archanfel

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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.

#### steve41

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Here's an example of the 'advisor bias' regarding the exponential cpi....

Two different plans... the individual is 30, grosses \$65,000 and plans to retire at age 65. His goal is to acheive a retirement lifestyle (after tax, after inflation) of \$40,000.

Using a conservative 5% market rate and 2% inflation, if we assume inflation behaves as an exponential, then he needs to acheive a nest egg of \$1.2 Million. If we assume that inflation behaves in a linear manner, then that required nest egg need only be \$911K. Using a 3% inflation assumption, the different would be even more graphic.

#### archanfel

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Here's an example of the 'advisor bias' regarding the exponential cpi....

Two different plans... the individual is 30, grosses \$65,000 and plans to retire at age 65. His goal is to acheive a retirement lifestyle (after tax, after inflation) of \$40,000.

Using a conservative 5% market rate and 2% inflation, if we assume inflation behaves as an exponential, then he needs to acheive a nest egg of \$1.2 Million. If we assume that inflation behaves in a linear manner, then that required nest egg need only be \$911K. Using a 3% inflation assumption, the different would be even more graphic.
Steve,

A linear inflation is mathematically impossible because if you go back in time, sooner or later, the price would be zero or even negative. You can say it's not exponential, but it's certainly not linear. In any case, I am only exploring the math behind the calculations. Linear computations are easy, so not much point talking about it.

#### steve41

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Look... being able to derive compound interest formulae is easy... any grade 12 kid can do it. Just because a linear representation of inflation is "easy" doesn't make it any less valid. It fits the data better, the exponential cpi doesn't.

Look at it this way... have you ever heard of a negative bank rate? no. Have you ever heard of negative inflation? yes. They are fundamentally different concepts.

#### archanfel

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Look... being able to derive compound interest formulae is easy... any grade 12 kid can do it. Just because a linear representation of inflation is "easy" doesn't make it any less valid. It fits the data better, the exponential cpi doesn't.

Look at it this way... have you ever heard of a negative bank rate? no. Have you ever heard of negative inflation? yes. They are fundamentally different concepts.
Sigh... Here. happy now?

#### steve41

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So... my take on that is that for the last 40 years, the cpi is linear. The big shift back prior to that signalled the era of 'runaway inflation' Since then, gov'ts have made it a priority to actively manage inflation, and so far they have succeeded.

'Runaway inflation' is a euphemism for exponential inflation, IMHO.

#### Sampson

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Should we also not plot market returns as a linear relationship also?
S&P 500

From the Mid 1980's looks pretty linear to me considering we got down to 700 this year. Planners also always assume all returns are reinvested (for myself, sometimes I draw on dividends and distributions).

So if we apply linear models for both market returns and inflation, then wouldn't the \$2.5M mark be once again accurate?

#### archanfel

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So... my take on that is that for the last 40 years, the cpi is linear. The big shift back prior to that signalled the era of 'runaway inflation' Since then, gov'ts have made it a priority to actively manage inflation, and so far they have succeeded.

'Runaway inflation' is a euphemism for exponential inflation, IMHO.
Whatever. You can make any assumption you want. It's your money.

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