# The Math of things



## archanfel (Apr 7, 2009)

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


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## archanfel (Apr 7, 2009)

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.


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## Patricia (Apr 3, 2009)

Good post archanfel, thanks.


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## Sampson (Apr 3, 2009)

"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'.


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## steve41 (Apr 18, 2009)

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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## CanadianCapitalist (Mar 31, 2009)

steve41 said:


> 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.

http://www.bankofcanada.ca/en/cpi.html


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## lb71 (Apr 3, 2009)

CanadianCapitalist said:


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


+1


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## steve41 (Apr 18, 2009)

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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## CanadianCapitalist (Mar 31, 2009)

steve41 said:


> 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.


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## steve41 (Apr 18, 2009)

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.


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## archanfel (Apr 7, 2009)

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.


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## steve41 (Apr 18, 2009)

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.


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## archanfel (Apr 7, 2009)

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.


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## steve41 (Apr 18, 2009)

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.


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## archanfel (Apr 7, 2009)

steve41 said:


> 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.


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## steve41 (Apr 18, 2009)

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.


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## archanfel (Apr 7, 2009)

steve41 said:


> 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?


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## steve41 (Apr 18, 2009)

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.


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## Sampson (Apr 3, 2009)

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?


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## archanfel (Apr 7, 2009)

steve41 said:


> 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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## steve41 (Apr 18, 2009)

Exactly my point... the behaviour of the markets and the behaviour of the cpi since 1970 when the financial markets and inflation started to be fiscally managed, just don't relate. They are representative of two different entities.


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## lb71 (Apr 3, 2009)

steve41 said:


> The linear inflation model results in a more conservative planning outcome.


 Actually, the compounding inflation model is more conservative in a financial plan.


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## archanfel (Apr 7, 2009)

4. The peril of MER (Assuming compounded inflation and growth  )

MER is the percentage your fund manager charges you for managing your fund. It's generally higher for actively traded mutual fund because it takes skills to manage the fund (or so they tell me). A typical MER in Canada is 2.5%. In other word, if you invested $10,000, you pay the manager $250 to manage it. Very reasonable indeed. 

Going back to our compound interest rate formula:

FV = X * (1+r)^n

Using our original example, a $100,000 investment, 10% annual return over 40 years would have a future value of 

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

However, with MER rate of m added, the formula becomes:

FV = X * (1+r-m)^n

Thus the same example with a 2.5% MER yield a future value of 

$10,000 * (1+10%-2.5%)^40 = $180,442.39 

In other word, The FV of your total expense is $272,150.17. More than what you would get. Ouch!

How about fixed contribution?

The original formula was 

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

Now it becomes

FV = X * ((1+r-m)^(n+1) - (1+r-m))/(r-m)

The example of $5000/year investment with 10% growth over 40 years, would be

$5,000 * ((1+10%-2.5%)^41 - (1+10%-2.5%)) / (10%-2.5%) = $1,221,503.79 

And you paid the manager $1,212,755.26. Not bad at all, for him. 

Of course, you have to put things in perspective and convert them to today's dollar. However, the fact remains that a 2.5% MER can potentially take away half of your returns.


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## steve41 (Apr 18, 2009)

> Actually, the compounding inflation model is more conservative in a financial plan.


What I am saying is that a planner will tell you you need 1.2M to retire with a $40K lifestyle if he uses a compounding cpi assumption and if he uses a linear cpi he will tell you you need only .911M to achieve the same lifestyle.

Whether you call it conservative, overly cautious or foolhardy is a matter of interpretation.


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## lb71 (Apr 3, 2009)

steve41 said:


> What I am saying is that a planner will tell you you need 1.2M to retire with a $40K lifestyle if he uses a compounding cpi assumption and if he uses a linear cpi he will tell you you need only .911M to achieve the same lifestyle.
> 
> Whether you call it conservative, overly cautious or foolhardy is a matter of interpretation.


OK. But what you said before ("The linear inflation model results in a more conservative planning outcome") is not the same thing as you just said now. Being conservative means planning to retire with 1.2 mn. Planning to retire with 0.9 mn is not conservative. (It may be practical in your eyes, but not conservative.)


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## steve41 (Apr 18, 2009)

I agree. I guess I didn't use the term 'conservative' in the proper context. Suffice it to say that using the linear cpi will result in a smaller amount of capital needed at retirement for the same retirement lifestyle target.

Some planners will use the linear cpi model, some the exponential model. Whether the planner who says that you need 1.2M is conservative or is over-hyping the client, is a matter of interpretation.


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## archanfel (Apr 7, 2009)

It really depends on your math. Without a compound inflation, I doubt your income would have compounded growth either. Therefore, it would be a lot harder to reach that .9 million than the 1.2 million. 

Unfortunately, while it's not hard to compute compounded growth contribution, I am not sure whether a linear growth contribution would be as easy. 

What do you think, Steve? Want to give it a try?


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## steve41 (Apr 18, 2009)

I ran both scenarios, one using compound and one using linear inflation. 5% growth rate, 2% inflation, province.. BC, dying broke at 95, normal CPP/OAS expectation. I assumed savings were RRSP in nature. What's missing?


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## archanfel (Apr 7, 2009)

5. The peril of Taxes 

Just when you thought the MER was bad enough, that's only half of the story. Since active funds are traded, well, actively, a lot of capital gains will be realized even if you don't sell the fund. 

Let's take a look at a passive index fund first. The capital gain tax situation is quite simple. When you sell, you will be charged taxes at your marginal tax rate discounted by 50%. 

Therefore, assume the tax rate is t. The after tax future value would be 

X * (1+r)^n * (1-t/2).

using or original example with a marginal tax rate of 40%, we got

$10,000 * (1+10%)^40 * (1 - 40% * 50%) = $452,592.56 * 80% = $362,074.04 

Unfortunately, for actively traded fund, the tax is charged every year. Therefore, we got:

X * (1 + (r-m)*(1-t/2) )^40 

With our example, we got 

$10,000 * (1+(10%-2.5%)*(1-t/2))^40 = $102,857.18 

Therefore, after the MER takes away 60% of your return, taxes takes away another 40%. Leaving you with only 28% of your returns after tax. 

Double Ouch. Similar computations can be done with the fixed contribution scenario. Without going to details, we will have $1,947,407.24 for passive index fund and $820,238.42 for actively traded mutual fund. Again, big differences.


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## archanfel (Apr 7, 2009)

steve41 said:


> I ran both scenarios, one using compound and one using linear inflation. 5% growth rate, 2% inflation, province.. BC, dying broke at 95, normal CPP/OAS expectation. I assumed savings were RRSP in nature. What's missing?


How much are you contributing each year?


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## steve41 (Apr 18, 2009)

It's not that straightforward. I am using a 'needs-based' model where tax is calculated using the full taxation algorithm rather than a single tax rate.

This is beyond the scope of algebraic formulas and spreadsheeting.... it is a highly recursive, reverse tax computation.


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## archanfel (Apr 7, 2009)

steve41 said:


> It's not that straightforward. I am using a 'needs-based' model where tax is calculated using the full taxation algorithm rather than a single tax rate.
> 
> This is beyond the scope of algebraic formulas and spreadsheeting.... it is a highly recursive, reverse tax computation.


Why would it matter? It's a straightforward question. How much are you contributing each year? I don't really care how you get the number.


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## steve41 (Apr 18, 2009)

Let's see... 7248 the 1st year, 7454, 7661, 7867..... all the way up to 14264 the final year. It's not a matter of a fixed amount every year, that's the nature of the 'needs-based' model.


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## archanfel (Apr 7, 2009)

steve41 said:


> Let's see... 7248 the 1st year, 7454, 7661, 7867..... all the way up to 14264 the final year. It's not a matter of a fixed amount every year, that's the nature of the 'needs-based' model.


Interesting. I think your number is wrong since your second year's contribution is 2.84% higher than the first year. That's very unlikely if you income only went up by 2%, unless BC has a regressive tax system. 

In any case, how would you calculate your future value then, without using spreadsheet and computers.


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## steve41 (Apr 18, 2009)

As I said... it is a very complex recursion. Spreadsheets and compound interest equations can't do this stuff. In a nutshell, the calculation starts with the salary (indexed) and determines what investment schedule (varying over time) is needed to achieve enough rsp capital such that the after tax income from retirement out to age 95 equates to exactly $40K net. The computation delivers a constant after tax pre-retirement income (ATI) of $44,413 in order to achieve the target $40K retirement ATI. 

Remembering that tax is not a simple number or factor, but the entire tax (T1) algorithm.

Spreadsheets cannot even begin to do this.


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## ShowMeTheMoney (Apr 12, 2009)

Just to throw a wrench in this. You're all assuming fixed rates of income, inflation, taxes, income etc... How about adding an expected variance around these mean values? Then you can calculate expected values for your retirement nest egg, but the math can get quite complicated and Monte Carlo simulations are the way to go. But even then you can't predict government policy and taxes 40 years hence. I don't know of any financial planner who will tell you that. No one can predict how many eggs your investments will buy next year (except maybe real-return bonds), much less in 40 years. That said, save, invest to increase your odds of reaching your goal, and then hope for the best.


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## Rickson9 (Apr 9, 2009)

Thank god you don't need to know this stuff to become wealthy or my wife and I would be sunk!


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## steve41 (Apr 18, 2009)

Certainly this stuff can be monte-carloed as well, but one thing seems to hold fairly constant... the nature of the tax calculation. I have taken older tax versions of the model and projected forward 5 or 6 years. The tax calculation is pretty consistent, when you take bracket indexing into account.

Not so with market returns though.

Some individuals are happy with 4% withdrawal rules, simple compound interest tools and average tax rate approximations, others prefer a more tax accurate approach. I will note that when you are trying to make sense of leveraging, tfsa vs rrsp, should I pay down my mortgage or contribute to my rrsp... then income tax and the complex way it interacts with various forms of capital -over time- requires a more rigorous methodology.

As I say... you pick your poison.


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## archanfel (Apr 7, 2009)

steve41 said:


> As I said... it is a very complex recursion. Spreadsheets and compound interest equations can't do this stuff. In a nutshell, the calculation starts with the salary (indexed) and determines what investment schedule (varying over time) is needed to achieve enough rsp capital such that the after tax income from retirement out to age 95 equates to exactly $40K net. The computation delivers a constant after tax pre-retirement income (ATI) of $44,413 in order to achieve the target $40K retirement ATI.
> 
> Remembering that tax is not a simple number or factor, but the entire tax (T1) algorithm.
> 
> Spreadsheets cannot even begin to do this.


I don't think the progressive tax matter all that much since we are assuming average growth (in my case, compounded, in your case linear). Tax code will change over the year (e.g. personal allowance and buckets should also be indexed), therefore we can only assume the tax rate would stay relatively constant. 

The fact if your income went up by 2% and your spending also went up by 2%. The difference can not be more than 2%. 

You seem to care more about how much you need to save. I really don't care about that since 1.2M is better than .9M no matter what's your assumptions. I don't care what FA's motives are, I would still be 300K richer than you. The question is whether you can get there. i.e. if you can only save $5000 a year, where would you be. Save just enough does not make sense since the model will never be perfect. My opinion is one should save as much as he can without affect his living standard. 

Financial opinions aside, no matter how you got those numbers, since they are linear(or assume they are linear), how would you calculate future value of your investment?


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## archanfel (Apr 7, 2009)

Rickson9 said:


> Thank god you don't need to know this stuff to become wealthy or my wife and I would be sunk!


No, you really don't need to know this stuff. This is just for fun. As I said, high school math.


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## archanfel (Apr 7, 2009)

ShowMeTheMoney said:


> Just to throw a wrench in this. You're all assuming fixed rates of income, inflation, taxes, income etc... How about adding an expected variance around these mean values? Then you can calculate expected values for your retirement nest egg, but the math can get quite complicated and Monte Carlo simulations are the way to go. But even then you can't predict government policy and taxes 40 years hence. I don't know of any financial planner who will tell you that. No one can predict how many eggs your investments will buy next year (except maybe real-return bonds), much less in 40 years. That said, save, invest to increase your odds of reaching your goal, and then hope for the best.


Interesting idea. The mean would stay the same I suspect. However, the distribution probably wouldn't. Monte Carlo simulations would work. 

It really is not about save just enough. It's about laying out a road map and check your progress against. The model will never be perfect, that's why it's important to adjust it every year.


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## archanfel (Apr 7, 2009)

6. How to calculate mortgage payment. 

As promised, let's take a look at how to calculate monthly mortgage payment. A mortgage is when you borrow a certain amount Y at a certain interest rate r, you pay back a fixed amount X each month and expect the loan and interests will be paid off over a period of year n. Every time you pay off some principle, that principle stop accumulate interest. 

At the beginning, you own the bank $Y.
After the first month, Y * (1+r/12) - X 
After the second month (Y*(1+r/12) - X)*(1+r/12) - X = Y * (1+r/12)^2 - X*(1+r/12) - X
After the third month (Y * (1+r/12)^2 - X*(1+r/12) - X) * (1+r/12) - X = Y * (1+r/12)^3 - X*(1+r)^2-X*(1+r) - X
After the nth month Y*(1+r/12)^n - X *((1+r/12)^(n-1) + (1+r/12)^(n-2) +... + (1+r/12) + 1)

Simplify the equation we got

Y*(1+r/12)^n - X *(((1+r/12)^n-1)/(r/12))

Of course, we know after nth month, the principle should be 0. 

Therefore, we got:

X *(((1+r/12)^n-1)/(r/12)) = Y*(1+r/12)^n 

There's another way to think about this. Assume you don't pay off the mortgage every month. Instead, you save the monthly payment in a saving account that pay the same interest rate as the mortgage rate. At the end of the amortization period, the saving account should be enough to pay off the mortgage in a single shot. The saving account is the left hand side of our equation and the final mortgage amount is right hand side. 

Note that this is assuming a monthly compound interest rate. Canadian fixed mortgage interest rate usually compound semi-annually. Check with your bank on how do they calculate the interest. It also assumes that the mortgage is paid at the end of the month. If you pay at the beginning of the month, you will of course need to pay less.


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## lb71 (Apr 3, 2009)

archanfel said:


> 5. The peril of Taxes
> 
> Just when you thought the MER was bad enough, that's only half of the story. Since active funds are traded, well, actively, a lot of capital gains will be realized even if you don't sell the fund.
> 
> ...


You forgot to deduct the original pricipal from your 40 year balance in the passive formula for the tax impact. I think it should be:

X *{ [ (1+r)^n - 1 ]* (1-t/2) + 1 }.

Using your example:

$10,000 * { [ (1+10%)^40 - 1] * (1 - 40% * 50%) + 1 }= $364,074.04 

A minor difference in this case, but if you look at a short term projection, your original formula would have ended up with a lower balance. For example, two years:

$10,000 * (1+10%)^2 * (1 - 40% * 50%) = $9,680 

vs

$10,000 * { [ (1+10%)^2 - 1] * (1 - 40% * 50%) + 1 } = $11,680

Unfortunately, the issue with taxes gets much more complicated. Your 10% return could be a combination of dividend returns, capital gains and bond interest, all of which have their own tax treatment. It might be a good idea to come up with an assumed blended tax rate. 

Also, if you invest in a passive fund you can still realize capital gains during the year.


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## archanfel (Apr 7, 2009)

lb71 said:


> You forgot to deduct the original pricipal from your 40 year balance in the passive formula for the tax impact. I think it should be:
> 
> X *{ [ (1+r)^n - 1 ]* (1-t/2) + 1 }.
> 
> ...


Ah, very true. Thanks for pointing that out. 

Yes, taxes become complicated once you add in dividend and interest. I blame the government for having a overly complicate tax scheme. Head tax, anybody?  

Still, actively traded fund are tax inefficient with no guarantee of higher return.


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## lb71 (Apr 3, 2009)

archanfel said:


> Ah, very true. Thanks for pointing that out.
> 
> Yes, taxes become complicated once you add in dividend and interest. I blame the government for having a overly complicate tax scheme. Head tax, anybody?
> 
> Still, actively traded fund are tax inefficient with no guarantee of higher return.


No problem. If it wasn't for complex tax schemes, personal finance wouldn't be as much fun.


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## steve41 (Apr 18, 2009)

OK.... for anyone reading this thread... don't for love of God, assume you have got to master any of the above weird algebra in order to get a handle on cash flow financial planning. This is all available in programs in which the authors have already done all this heavy lifting.

It is as simple as entering your age, how much capital you have saved (of different forms... rrsp, nonreg, equity, tfsa), your salary, pension parameters, retirement age, aftertax income goals and, in the case of tax-based models, your province. The program takes this data and does the number crunching for you. 

I am sure most of you have figured this out by now, but once you have the proper tools, cash flow planning is pretty straightforward. Now, investment planning... picking the right investments... is a whole different ball game.


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## Rickson9 (Apr 9, 2009)

steve41 said:


> OK.... for anyone reading this thread... don't for love of God, assume you have got to master any of the above weird algebra in order to get a handle on cash flow financial planning. This is all available in programs in which the authors have already done all this heavy lifting.
> 
> It is as simple as entering your age, how much capital you have saved (of different forms... rrsp, nonreg, equity, tfsa), your salary, pension parameters, retirement age, aftertax income goals and, in the case of tax-based models, your province. The program takes this data and does the number crunching for you.
> 
> I am sure most of you have figured this out by now, but once you have the proper tools, cash flow planning is pretty straightforward. Now, investment planning... picking the right investments... is a whole different ball game.


I agree. Spending time on the mathematics presented on this thread will NOT help you become wealthy. It might, however, make you a better mathematician (if that is what you're looking for).


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## ethos1 (Apr 4, 2009)

Rickson9 said:


> I agree. Spending time on the mathematics presented on this thread will NOT help you become wealthy. It might, however, make you a better mathematician (if that is what you're looking for).


the days of log tables, slide rules and statistical mathematical analysis are long gone

Financial software programs as well as on-line calculators & spreadsheets are the way to go - even on-line tax return e-filing is childs play


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## archanfel (Apr 7, 2009)

Rickson9 said:


> I agree. Spending time on the mathematics presented on this thread will NOT help you become wealthy. It might, however, make you a better mathematician (if that is what you're looking for).


It will not make you a better mathematicians either, since they are just high school math.  

It's just for fun when you get bored and want's to know what's really behind those online tools. So if all these are giving you a headache, stop reading, you really don't need to know.


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## Ben (Apr 3, 2009)

Archanfel, there is certainly a lot to be said for understanding the math of investing/compounding, etc. I am not a fan of online tools either - like you, I like to see inside the black box.

Your spirited advocation of financial learning is commendable, although for most people, the derivation of the equations will simply make them go blurry-eyed!

If you have the interest, a short post presenting the most commonly used equations might be of value to the community. Or alternatively, a link to a good site that presents the same equations and uses examples to bring them to life.


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## Rickson9 (Apr 9, 2009)

archanfel said:


> It will not make you a better mathematicians either, since they are just high school math.


Like I said, it would make the majority better mathematicians 

We only use arithmetic!


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## kayo (Apr 5, 2009)

Very interesting post! Brings back memories when I am sitting in a classroom trying to crunch numbers like that. 

With that said, I'm glad there are automated tools to help us calculate personal finance easily. I am not sure I will be fond of the idea of sitting at home crunching numbers after a good 8 hr work day.


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## archanfel (Apr 7, 2009)

Ben said:


> Archanfel, there is certainly a lot to be said for understanding the math of investing/compounding, etc. I am not a fan of online tools either - like you, I like to see inside the black box.
> 
> Your spirited advocation of financial learning is commendable, although for most people, the derivation of the equations will simply make them go blurry-eyed!
> 
> If you have the interest, a short post presenting the most commonly used equations might be of value to the community. Or alternatively, a link to a good site that presents the same equations and uses examples to bring them to life.


Unfortunately, I haven't found a good site. Ask Dr. Math (http://mathforum.org/dr.math/) is probably the best one although it's not specific for financial calculations. My math is really rusty after so many years out of school (not that it was good in the first place  ), so I found the site very help especially since you can ask them specific questions.

I actually find the derivation of the equation the fun part. Some of them are very clever. But you are right, they can be boring especially since there's no way to properly format it in a forum post. 

In any case, the next post is probably my last one, until i can think of another interesting equation.


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## steve41 (Apr 18, 2009)

I can always remember my teacher/prof announcing after he had required us to learn the various compound interest derivations.... "Oh, by the way, these are completely impractical in the real world. Even a simple modification such as changing the interest rate part way thru the projection, or adjusting the pmt level by a fixed percentage, (which are two very real life situations) make this type of mathematical approach completely impractical."

It is better to solve PV, FV, etc... the way computers do it: by using iterative-goal seeking techniques.


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## archanfel (Apr 7, 2009)

7. Compound growth with contributions indexed to inflation.

Very few of us contribute to our retirement found exactly the same amount each year. More importantly, the TFSA limit is indexed to inflation. Our retirement spending is also likely to be indexed to inflation. 

Per Ben's suggestion, I am not going to talk about how to derive the equation. Assuming the initial contribution is X, inflation rate of i, growth rate of r and a time period of n, the equation for the future value is:

FV = X * {(1+r)^(n+1) - (1+r)*(1+i)^n} / (r-i)

Therefore, assume you max out your TFSA each year, with a 2% inflation rate and a 10% growth rate, your TFSA after 40 years would be worth: $2,959,771.09 or $1,340,451.96 in today's dollar. Yummy. 

I also did a bit calculation with Steve's linear inflation assumption. Assuming the contribution goes up $Y per year. The equation would be:

FV = X *({(1+r)^(n+1)-(1+r)} /r) +Y*(({(1+r)^(n+1)-(1+r)^2)} / r^2 - (n-1)*(1+r) / r) 

Using the same scenario, Y would be $100 (2% of $5000). The future value would be $2,877,110.87
or $1,598,394.93 in today's dollar.


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## archanfel (Apr 7, 2009)

steve41 said:


> I can always remember my teacher/prof announcing after he had required us to learn the various compound interest derivations.... "Oh, by the way, these are completely impractical in the real world. Even a simple modification such as changing the interest rate part way thru the projection, or adjusting the pmt level by a fixed percentage, (which are two very real life situations) make this type of mathematical approach completely impractical."
> 
> It is better to solve PV, FV, etc... the way computers do it: by using iterative-goal seeking techniques.


If you meant:

1. After 5 years, raise the interest rate from 2% to 5%. 
2. Index the monthly payment by inflation. 

Both are very easy to do.


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## steve41 (Apr 18, 2009)

No... how about an annuity-type derivation where the annuity pmts increase a fixed amount from 55 to 65 to account for cpp/oas (indexed) and the rate takes a 2% dive at 75, say, to account for the subject's increased risk aversion in their later years.

I am sure it is do-able, however, but why bother? All it does is cause the average joe to throw up their hands and go scurrying off to the nearest FA/salesman. Computers are here, in case you haven't noticed. Why not start to use them?


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## archanfel (Apr 7, 2009)

steve41 said:


> No... how about an annuity-type derivation where the annuity pmts increase a fixed amount from 55 to 65 to account for cpp/oas (indexed) and the rate takes a 2% dive at 75, say, to account for the subject's increased risk aversion in their later years.
> 
> I am sure it is do-able, however, but why bother? All it does is cause the average joe to throw up their hands and go scurrying off to the nearest FA/salesman. Computers are here, in case you haven't noticed. Why not start to use them?


Again, those are very easy to do. The really hard part is to change the rate constantly to reflect growing risk aversion over the years. I don't even want to start to think about how to do that. 

As for computers, to be honest, very few people really know how to use them. People generally go to an online calculator and be happy about it. However, online calculators have their limits. For example, do you know an online calculator that would do what you just described? 

Spreadsheet is far more flexible, but you still need to know how to use them. For example, if I want to save an indexed amount every two weeks, with growth rate compound daily, so that I would have enough at 55 to retire and spend $2000/month indexed, until I am 90. And with the two conditions you just mentioned. How would you calculate it in a spreadsheet? 

You can also write a program to do it. But I don't think a lot of people know how to program. Not to mention you really need to know a bit math to write the program efficiently. As fast as computers are today, brutal force can take its toll.


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## steve41 (Apr 18, 2009)

Varied rate from 6% down to 3% to age 83... 3% thereafter. Took me a minute, and included taxation (full tax algorithm) Result.... A $500K RRSP will deliver a constant $36,483 lifestyle til age 95 including cpp/oas.

This is pretty simple stuff. Entering the varying rate was the only fiddly bit. The rest, entering the guy's $500K rrsp and his ages (current-65 and 'die-broke'-95) was the only thing I had to do... 3 seconds of recursive number crunch, and it was done.


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## archanfel (Apr 7, 2009)

steve41 said:


> Varied rate from 6% down to 3% to age 83... 3% thereafter. Took me a minute, and included taxation (full tax algorithm) Result.... A $500K RRSP will deliver a constant $36,483 lifestyle til age 95 including cpp/oas.
> 
> This is pretty simple stuff. Entering the varying rate was the only fiddly bit. The rest, entering the guy's $500K rrsp and his ages (current-65 and 'die-broke'-95) was the only thing I had to do... 3 seconds of recursive number crunch, and it was done.


uh, that's not my question. I believe my question was "if I want to save an indexed amount every two weeks, with growth rate compound daily, so that I would have enough at 55 to retire and spend $2000/month indexed, until I am 90. And with the two conditions you just mentioned. How would you calculate it in a spreadsheet?"


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## steve41 (Apr 18, 2009)

You solve it using goal-seeking. Making the 24000 the target. I use my own program (not a spreadsheet) to do this kind of stuff, but goal-seeking using a spreadsheet, while it can take a few minutes to converge (or it used to years ago when I last played around with one) is still the way you would tackle this kind of problem.

As soon as you introduce another level of complexity (especially the reverse tax calculation) then spreadsheets are just too slow and unsophisticated.


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## steve41 (Apr 18, 2009)

Further..... my methodology is not to set the contribution amount, rather it is to specify the salary and have the program compute what needs to be contributed based on the more important metric... lifestyle. After all, you don't budget based on what you are going to contribute to your RRSP, rather, you budget based on what you need/want for spending on beer, groceries and gas.

So, the question about attaining a $40K lifestyle in retirement needs a salary to be specified in order to make the plan more meaningful.


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## archanfel (Apr 7, 2009)

steve41 said:


> You solve it using goal-seeking. Making the 24000 the target. I use my own program (not a spreadsheet) to do this kind of stuff, but goal-seeking using a spreadsheet, while it can take a few minutes to converge (or it used to years ago when I last played around with one) is still the way you would tackle this kind of problem.
> 
> As soon as you introduce another level of complexity (especially the reverse tax calculation) then spreadsheets are just too slow and unsophisticated.


How many do you think know how to do goal seeking in spreadsheet or write a program? On the other hand, I would think a lot of people know how to use a calculator if you just tell them the formula to use.

I am not discounting the usefulness of the computer. Certain things can only be solved using things like Newton's method. I wrote a program to do sudoko for me although it's not based on brutal force or it would take a long time even with today's computers. I am simply saying that if you are interested in it, here is how it works behind the scene. 

And I would suggest you write one describing how to write that your program. It might be useful to a lot of people.


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## archanfel (Apr 7, 2009)

steve41 said:


> Further..... my methodology is not to set the contribution amount, rather it is to specify the salary and have the program compute what needs to be contributed based on the more important metric... lifestyle. After all, you don't budget based on what you are going to contribute to your RRSP, rather, you budget based on what you need/want for spending on beer, groceries and gas.
> 
> So, the question about attaining a $40K lifestyle in retirement needs a salary to be specified in order to make the plan more meaningful.


Why would I care about salary unless I can't afford the contribution. And my question was how much I NEED to contribute, so I'd say it's precisely based on lifestyle. 

Anyway, you can assume a salary of $5 million/year. Should be enough to remove any constraint on your calculation.


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## steve41 (Apr 18, 2009)

Uh... _after tax _means _after tax_... the size of your salary definitely dictates how your plan will unfold. Unless you you are in the underground economy and don't declare tax.


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## steve41 (Apr 18, 2009)

....and for a salary of $5M, you won't be contributing just to your rrsp, you will have to contribute the excess outside your rrsp. Tax starts to be a real nuisance in this type of calculation.


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## BeautifulAngel (Jun 30, 2017)

Very educational and helpful, thank you for sharing


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