1.6 The Time Value of Money II: Multiple Deposits

Annuities

From: https://en.wikipedia.org/wiki/Annuity

An annuity is a series of payments made at equal intervals.^[1] Examples of annuities are regular deposits to a savings account, monthly home mortgage payments, monthly insurance payments and pension payments. Annuities can be classified by the frequency of payment dates. The payments (deposits) may be made weekly, monthly, quarterly, yearly, or at any other regular interval of time. Annuities may be calculated by mathematical functions known as “annuity functions”.

An annuity which provides for payments for the remainder of a person’s lifetime is a life annuity.

Payments of an annuity-immediate are made at the end of payment periods, so that interest accrues between the issue of the annuity and the first payment. These are also called ordinary annuities. Payments of an annuity-due are made at the beginning of payment periods, so a payment is made immediately on issueter. In our class we will only use ordinary (annuity-immediate) annuities.

Annuities can be founded in many ways. While I give three options below, we will only use fixed annuities in this class.

• Fixed annuities – These are annuities with fixed payments. If provided by an insurance company, the company guarantees a fixed return on the initial investment. Fixed annuities are not regulated by the Securities and Exchange Commission.
• Variable annuities – Registered products that are regulated by the SEC in the United States of America. They allow direct investment into various funds that are specially created for Variable annuities. Typically, the insurance company guarantees a certain death benefit or lifetime withdrawal benefits.
• Equity-indexed annuities – Annuities with payments linked to an index. Typically, the minimum payment will be 0% and the maximum will be predetermined. The performance of an index determines whether the minimum, the maximum or something in between is credited to the customer.

Future Value of annuities

If the payments are made at the end of the time periods, so that interest is accumulated before the payment, the annuity is called an annuity-immediate, or ordinary annuity. Mortgage payments are annuity-immediate, interest is earned before being paid.

The future value of an annuity is the accumulated amount, including payments and interest, of a stream of payments made to an interest-bearing account. For an annuity-immediate, it is the value immediately after the n-th payment.

Let us begin by working through an example where you put 500 into an account each year for five years. The account pays 5% compounded annually. Because we are only using ordinary annuities, we will see that year 1 is sort of a wash. This is because the contributions are made at the end of the year. The annual breakdown is as follows:

Year	Starting Balance	Interest Earned	Contribution	Ending Balance
1	0.00	0.00	500.00	500.00
2	500.00	25.00	500.00	1,025.00
3	1,025.00	51.25	500.00	1,576.25
4	1,576.25	78.81	500.00	2,155.06
5	2,155.06	107.75	500.00	2,762.82

So we see that we contributed 2,500 over the five years but the account is worth 2,762.82 due to the interest payments. What if we want to do something more complicated? A table quickly becomes far too cumbersome. We need a method which allows us to find the future value using some some of equation(s). Note: For these problems, the payment frequency must match the interest compounding frequency. For instance, we will not do any problems where interest is compounded annually but payments are made monthly.

There are two steps to finding the future value. The first is to calculate the future value annuity factor. Think of this as a type of multiplier that takes into account the interest rate, compounding/payment frequency, and the number of years (or periods) for which payments will be made. The calculation for the FVAF is

We pronounce the term as “s-nigh”.

Then, to calculate the future value, we take the FVAF and multiply it by the regular payment. This gives us

To give you some idea as to why we need these equations and what we can do with them:

Present Value of annuities

The present value of an annuity is the value of a stream of payments, discounted by the interest rate to account for the fact that payments are being made at various moments in the future.

We generally use present value calculations for loans and situations where you owe money and will pay it back. When you take out a loan, you are essentially getting a lump-sum payment upfront and then paying it back over time. The bank pays for your car, house, etc. and then you pay the bank. The methodology is very similar, but slightly different from future value. We will again only use ordinary annuities.

The major difference is that this is a three-step process. The first step is to calculate the FVAF just like we did earlier. Then, we calculate the present value annuity factor or PVAF. Again, this of this as a sort of multiplier. This is calculated as:

This is pronounced “Annie.”

Then, to determine the present value, we use the following formula:

To get a better understanding as to why we need this framework, let us look at a few examples.

Amortization Tables

When discussing loans, we can create a table that shows how payments are being applied each month. If you are paying 300 per month, your loan balance is not being reduced by that amount each month because some of your payment goes towards interest.

So let us consider this example: You borrow 10,000 which you will repay with five annual payments. The interest rate is 3%. Therefore, our calculations are:

Because this is an ordinary annuity, we start the month with a balance, accrue interest on the starting balance, then make a payment and pay-off the interest with the remainder going towards the balance. Let us take a look below.

To help look at the process, let us examine the first row. We begin with 10,000 of debt. We previously found the payment would be 2,183.55. So how will our payment be split between interest and principal? To answer that we need to calculate the interest which we do by multiplying our starting balance for the year by the interest rate. Here we get That means that the remainder of the 2,183.55 payment, or 1,883.55 will go towards paying down the loan balance. When we subtract the 1,883.55 from 10,000, we get a remaining balance of 8,116.45. The process then continues over and over.