What is the time-value of money? It is an estimate of what particular sum of money will be worth within a particular time-frame. From this you can see what needs to be done to achieve your goals.

There are two categories to this:

1. Future Value of Money: How much a particular present amount will be worth after a particular time at a particular interest rate.
2. Present Value of Money: How much an amount in a particular period is worth today at a particular interest rate.

Each category can be further broken down into an annuity or a single amount. An annuity is an amount paid in at the end of each compounding period. This makes comparing single amounts and and annuities difficult to compare with conventional equations.

The traditional equation for the future value of an annuity is , it can be converted from end of period payment to start of period payment by multiplying by , resulting in . In order to compare these, we also need the equation for the single amount, which is .

If you want to put away a certain total principle, it is far more powerful to take that amount and deposit it all at once, rather than as an annuity over the same period of savings. For example (see a chart here), an annuity of $1 per year for 40 years ($40 total) at 6% APR will grow to $165, but if you take that $40 and deposit all at once, at the same rate and period, it will grow to $411. That’s 2.5 times more than as an annuity and over 1000% growth compared to 400% for the annuity. This is not to say that annuities are bad, but shows how time can have a significant impact on a total particular principle.

Monday I discussed the future value of money, given a known principal, interest and time.  This is all well and good, but what happens if you want to have a certain amount within a specified time at a known interest?

For today, we’ll say that you want to have a $5,000 emergency fund in 18 months and you will be putting the money into a savings account earning 5%.

Now, just like Monday, you have the same two options, either put the initial principal in once, or add it as an annuity.

For a single deposit, use this equation:

For an annuity, use this equation:

So, how much do we have to put in to achieve our goal?

For a single deposit, we need to deposit only $4639.44, but for an annuity, we’d have to deposit $268.07 per month for a grand total of $4,825.25.

At this point, we’ve investigated the future value of money in terms of a fixed total going in, and a fixed future value.  It would appear from these two cases that annuities are a weaker choice given a fixed period and rate for both a fixed input and a fixed output, but what about when period or rate becomes the variable?  

We shall see what happens with these two as variables in the next two posts (tomorrow and Friday), finishing up the week (Saturday) with a complex example.

You all have seen the savings of cutting out that one cup of Starbucks per day, but I haven’t seen anyone plug this savings into an annuity equation, so I give you here what could happen.

Here are the assumptions:
$5 per cup of coffee (Latte with flavor),

Purchases are made only Monday through Friday, at 1 cup per day.
5% APY (I’m assuming a high yield savings account).

An estimate of the future value (fv) of money can be easily calculated in two ways:

For a single deposit, earning interest at a constant rate, use this equation:

For an annuity, that is a fixed amount deposited at regular intervals (derived from annual, but can be used for smaller intervals), can be calculated with the equation:

So, what do those letters mean?  Well, P is your deposit or principal, R is the rate of return and N is the number of times it’s compounded.

In this case, the total principal to be deposited is $5/day * 5 days/week * 52 weeks = $1300, an average of $108.33 per month.  The rate is about 0.42%… Wait I said the APY is 5%, so where’d this 0.42% come from… It’s the approximate monthly rate or MPY.  And of course, since we’re working with months and want to see the results from 1 year, the number of times compounded is 12.

So, with all this in hand, how much are we REALLY saving?

Let’s say for the sake of argument that the full $1300 was available at the beginning of the year and we deposited it all at once, then we’d have about $1366.51 (give or take a few cents, since these equations are only estimates).

Realistically, those of us trying to get out of debt can’t really afford to dump the full $1300 in at the beginning of the year, so we’d probably budget the $108.33 per month as an annuity for the year, netting $1330.21 by year end.

Converting either the annuity or single amount for partial or multiple years is as easy as changing N, but what about putting $1300 in once per year for X years?  Or just putting the $108.33 in for one year and then letting it sit?  Well, there is no simple equation, though you could plug the result of the annuity into the single time equation.  But converting the $1300 into an annual annuity with monthly compounding, not so easy.  To solve these and other problems, I WILL work on deriving equations that will make it possible, though it may take me some time and they won’t necessarily be pretty.

So, the moral of this exercise, don’t just cut the coffee, put the money into your high-yield savings account or for better potential returns, invest it.