# Financial Math Geekery: Annuities

Be warned, this article is very technical and I cannot promise that it will be without jargon or can be easily grasped. Nonetheless, I will try my best to explain the mathematical concepts involved in annuities.

Know that for this article I round all values to the 2^{nd} decimal place. (0.01 ← This much) So the numbers involved may not be 100% accurate, at the expense of not taking up the entire webpage! The differences will be negligible for these examples however.

**Present and Future Value**

The most basic equation we are basing these mathematics on is the Present Value equation, which determines how much a future sum of money is worth today. After all, a $100 today can be worth far more than $101 tomorrow, if you know what you’re doing with that money. The equation is as follows:

PV = FV / (1 + r/m) ^ (m*T)

PV = Present Value, how much the money is worth *now*

FV = Future Value, how much money is involved in the future

r = Annual interest rate, how much the money grows per year

T = Total number of years involved

m = Number of compounding periods in a year, having your money grow at a monthly rate can result in more income than having that same cash grow at an annual rate

To find the Future Value of a present sum of cash, you simply “move” the bottom of the equation next to the PV value. So it would look like:

FV = PV * (1 + r/m) ^ (m*T)

To give a basic idea of how these equations work, say you want to have $10,000 after 5 years after investing that money for a 3% return, how much money do you need to start?

$10,000 / (1 + 0.03/1) ^ (1*5)

= $10,000 / (1.03)^5

= $10,000 / (1.16)

= $8620 = Present Value today, invested at 3% interest for 5 years, to have about $10,000 in 5 years.

**Why does this work?**

This equation is generally true when we break it down.

In year 1 we have $8,620. It grows by 3% and in year 2 it becomes $8878.60. The interest rate however applies to the new value, not the original one, so in year 3 it is $9144.96, the money is growing faster over time as more money is affected by the interest rate.

This is the power of compound interest, while the rates and numbers involved may start out small, over time they can grow large if they are undisturbed. Even though all you’re doing is just multiplying in an interest rate each year.

So $8620 * 1.03 * 1.03 * 1.03 * 1.03 * 1.03 = $10,000

To make this easier to look at, we use exponents, therefore:

$8620 * 1.03^5 = $10,000

Present Value * (1 + interest rate per period) ^ (number of periods) = Future Value

**Note**: Because I am simplifying the math by only going to 2 decimal places, the actual answer of this particular equation is about $7 off. This goes to show just how sensitive interest rates are on the value of money, fractions of fractions can be worth a lot more than just a cup of coffee!

Also, the reason we +1 the interest rate, is because multiplying by decimals otherwise causes numbers to shrink which is clearly not happened. A value of 1.03 indicates that the value increases by 3%, but 0.03 would tell us to shrink the value to 3% exactly!

**What if Money is Coming In/Out Regularly?**

This math situation is called an annuity, given that it is the most common time this situation happens! In this case:

Annual Payment = FV / { [(1 + r) ^ (T * m) – 1] / r }

FV = Annual Payment * { [(1 + r) ^ (T * m) – 1] / r }

Back to our 3% interest, 5 years to $10,000 example, how much would need to be spent as an annual payment to reach this goal?

Annual Payments = $10,000 / { [(1 + 0.03) ^ (5 * 1) – 1] / 0.03 }

= $10,000 / { [(1.03)^5 – 1] / 0.03}

= $10,000 / { [1.16 – 1] / 0.03 }

=$10,000 / (0.16/0.03)

= $10,000 / 5.34

= $1875 = Annual Payments over 5 years, invested at 3%, to reach $10,000

**Why Does This Work?**

The reason payments is more complicated of an equation than a single sum investment, is because of the fact there are multiple amounts of money experiencing different things.

That first $1875 is going to have more time to grow at 3% than the second, which will have more time to grow than the 3^{rd}, and so forth. Rather than try to calculate each of these individually in a wild goose chase to find a correct answer, we come to find the correct payment value by finding the total interest rate change and divide that by the originally stated rate.

Or to put it simply, we find out what would be the total increase of 3% interest over 5 years (about equal to 16% interest in a single year), and then divide that number by the original 3%. This gives us a value we can use to divide into the original target of $10,000, telling us how to evenly “split” that future value over 5 years. Even though $1875 * 5 = $9375, the 3% interest will cover the difference.

**What About Paying At the Start of a Period?**

The difference between money exchanging at the start of a period, versus the end of a period, is that the money in question will be affected by interest longer. Although, this difference is not usually enough to be the same as having the money in for an extra period, but it **is** enough to be worth calculating.

How we find the difference is to simply multiply that final “divisor” by the interest, and this “shifts” all of the interest periods to account for that extra bit of time each one has. To see it as a formula:

Annual Payment = FV / ( { [(1 + r) ^ (T * m) – 1] / r } * [1+r])

FV = Annual Payment * ( { [(1 + r) ^ (T * m) – 1] / r } * (1+r) )

**What If I Paid A Sum, and THEN Put in Regular Payments?**

In this case, you would add both equations. So say you have $5000 today, but still want to reach $10,000 in 5 years at 3% interest, assuming funds are invested at the end of each period. How would that look like?

FV = $5000 * (1+0.03)^5

= $5000 * 1.16

= $5796.37 = Value of that $5000, 5 years in the future at 3% interest

Now we would need to subtract that value from the total we want to have after 5 years.

$10,000 - $5,796.37 = $4203.63 unfunded future cash, let’s fund it with annual payments.

Payments = $4203.63 / { [(1.03)^5 – 1] / 0.03}

= $4203.62 / (0.16/0.03)

= $4203.62 / 5.34

= $787.19 per year, at 3% interest, in addition to $5,000 initial deposit, will achieve $10,000 in 5 years.

**Why This Matters, and a Word of Caution**

It is important to understand how these maths work because they can guide us to make prudent investment decisions, and to not misunderstand your current financial situation.

However, bear in mind that real world situations can be much more complicated. Whether it be fee payments in addition to interest gains, inflation, inconsistent gains in your finances, or other risks such as emergencies in your personal life or geo-political events.

These equations are to financial goals and economic understanding, as a highschool equation of gravity is to physics, and cannot be considered a substitute for true expertise or understanding of any given financial situation.

If you would like to bring 30 years of financial professionalism to your situation give me a call at 203-956-0289. You can also send me an e-mail to wward@1stallied.com. Hope to hear from you soon!