Financial Mathematics: Loans - Annuity Immediate

We have already started looking at annuities. Today we'll be continuing that discussion. We'll be using two common examples to illustrate annuities that most people are familiar with: Loans and Savings.

This section will be dealing with loans and particularly the annuity immediate loans. In the following sections we will explore annuity due loans as well as savings formulas.

Loans

A fairly standard loan is one in which one party borrows a certain amount of money and then pays it back in a series of equal payments. We'll assume that there is a borrower and a bank lending the money. The flow of funds for this will look like this:

	Initial	Settlement
Bank	- LOAN	+ PAYMENTS
Borrower	+ LOAN	- PAYMENTS

Typically with loans the first payment is not made immediately but is due at the end of the first period. As a result, most loans will be annuity immediate. We'll consider three scenarios, the first two will be annuity immediate. The last one will be annuity due.

Annuity Immediate Loans

Many loans that people deal with are annuity immediate loans. The first payment is not due until the end of the first period (say in 1 month). So we'll start by looking at these.

Assumptions

We will need to make some assumptions here to set this up. Here are the assumptions:

1. All payments are made in equal amounts. 
2. The payments are made at equal intervals (specified as a fraction of a year).
3. There is no possibility to alter either the times or the amounts of payments (e.g. no prepayment).

Scenario 1: Annual Payments

We'll start by looking at annual payments. Recall that the amount of the loan will equal the amount of all of the payments, provided that we take into the time value of money. The relevant expression that we need which we derived here is:
$$a(t) = \frac{1}{(1+i)^t}$$
where $i$ is the interest rate and $t$ is the time of the payment (in years).

If we assume that the loan will be paid off in $t$ years with equal payments $C$, the way this will look will be something like this:
$$\text{LOAN} = \frac{C}{(1+i)} + \frac{C}{(1+i)^2} + \frac{C}{(1+i)^3} + . . . + \frac{C}{(1+i)^t}$$
The loan amount is often referred to as the present value ($ PV $). We'll use that convention.

We can factor out the payment $ C $ and a $ \frac{1}{1+i} $ to be obtain:
\[\begin{align*}
PV &= \frac{C}{1+i}\left[1 + \frac{1}{(1+i)} + \frac{1}{(1+i)^2} + . . . + \frac{1}{(1+i)^{t-1}}\right] \\
&= \sum_{j=0}^{t-1}\frac{C}{1+i} \left( \frac{1}{1+i} \right)^j
\end{align*}\]
So this ends up looking like our friend the geometric series with $ a = \frac{C}{1+i} $ and $ r = \frac{1}{1+i} $. Therefore this reduces to:
$$PV = \frac{C}{1+i}\frac{1-\frac{1}{(1+i)^t}}{1-\frac{1}{1+i}}$$
We can distribute that $ 1+i $ in the denominator and make things look a little prettier:
$$PV= C \frac{1-(1+i)^{-t}}{i}$$
In actuarial notation this will look like this:
$$a_{\overline{n|}i} = \frac{1-v^n}{i}$$
Note: $ a_{\overline{n|}i} $ is a loan where $ \$ 1 $ payments are made. You can multiply this by a constant $C$ to obtain other types of loans.

Just like we did here, there are basically four situations that you would encounter. The first is that you want to know how much loan you can get given a known annual payment, $C$,  known interest rate, $i$, and known number of payments, $t$. This formula is the one we already derived:
$$PV= C \frac{1-(1+i)^{-t}}{i}$$
On ther other hand we may want to know what the annual payment will be given that we know the other variables:
$$C = PV \frac{i}{1-(1+i)^{-t}}$$
The third alternative is we want to know how long it will take to pay off a loan given the loan amount, interest rate and annual payments:
$$t = - \frac{\ln \left[1-\frac{iPV}{C}\right]}{\ln(1+i)}$$
The last scenario is if we want to know the interest rate, given the other variables. Unfortunately, this cannot be solved algebraically. So this would have to be approximated via a fancy guess and check process (a task far more suitable for a computer or someone who has lots of time on his hands.)

Example

Here's a quick example. Suppose that you borrow $ \$1000 $ at an interest rate of $ 6 \% $ for a period of 10 years. How much would your annual payment have to be?
$$C = \$ 1000 \frac{6 \%}{1-(1+6 \%)^{-10}} = \$ 135.87$$
We can solve this using financial calculators as well. 

For Excel, we would type in this formula:

=PMT(6%, 10, -1000) =$135.87

Recall that the negative sign on the $1000 represents the fact that borrowing is the opposite of making payments. If you wanted, you could plug in a positive 1000 and it would show the payments as negative.

For the BAII Plus we would type in 10 "N", 6 "I/Y", 1000 "+/-" "PV and then "CPT" "PMT" to compute the payment. You should get the same $135.87.

Scenario 2: Monthly Payments

We first looked at the annual payments because it's a little easier to work with. Now let's consider a more common example of monthly payments.

With monthly payments, instead of having $t$ payments where $t$ is the number of years for the loan, we now have $ t \times 12 $ payments. But the setup is pretty much the same.

Just as in the last case we will refer to the present value ($ PV $) of the loan. And since $ C $ represented the annual payments in the last case we will consider the payments to be $ c = C/12 $ since they are monthly.

Recall that will need to use the intra-year factor to discount each payment where $ n = 12 $. The setup will then look like this:
$$PV = \frac{c}{\left(1+\frac{i}{12} \right)^1} + \frac{c}{\left(1+\frac{i}{12}\right)^2} + \frac{c}{\left(1+\frac{i}{12}\right)^3} + \frac{c}{\left(1+\frac{i}{12}\right)^4} + . . . + \frac{c}{\left(1+\frac{i}{12}\right)^{t \times 12}}$$
Like we did in the first scenario, we will factor out $ \frac{c}{\left(1+\frac{i}{12} \right)^1} $ to obtain:
\[\begin{align*}
PV &= \frac{c}{\left(1+\frac{i}{12} \right)^1} \left[1+ \frac{1}{\left(1+\frac{i}{12} \right)^1} + \frac{1}{\left(1+\frac{i}{12}\right)^2} + \frac{1}{\left(1+\frac{i}{12}\right)^3} +  . . . + \frac{1}{\left(1+\frac{i}{12}\right)^{t \times 12-1}}\right] \\
&= \sum_{j=0}^{t\times 12 - 1}\frac{c}{1+\frac{i}{12}}\left(\frac{1}{1+\frac{i}{12}}\right)^j
\end{align*}\]
As we did before, we note that this expression is a geometric series with $ a=\frac{c}{1+\frac{i}{12}} $ and $ r = (1+i/12)^{-1} $. We can make this substitution to obtain:
 $$PV = \frac{c}{\left(1+\frac{i}{12} \right)^1}\frac{1- \left(1 + \frac{i}{12}\right)^{-t \times 12}}{1- \left(1 + \frac{i}{12}\right)^{-1}}$$
We can simplify this a bit:
$$PV = 12c \frac{1- \left(1 + \frac{i}{12}\right)^{-t \times 12}}{i}$$
Note: We can generalize this for when the payment interval is $n$ times a year. This is done by substituting $n$ for all the places 12 occurs:
\[\begin{equation}

PV = nc \frac{1- \left(1 + \frac{i}{n}\right)^{-t \times n}}{i} \tag{Annuity Immediate Loan Formula}

\end{equation}\]
In actuarial notation this will look like:
$$a^{(m)}_{\overline{n|}i} = \frac{1-v^n}{i^{(m)}}$$
where the $m$ signifies the number of payments per year.

This formula can actually be used in the annual payments scenario by substituting $ n=1 $.  As a result this is a formula worth noting (it will come up again with bond valuation).

Example

Here's a quick example. Suppose that you can afford a $ \$1000 $ mortgage payment and you can get a mortgage at a $ 5\% $ interest rate. How much house can you afford? We'll assume we're dealing with a standard 30 year mortgage (you can work out a 15 year mortgage on your own). That means that $ c = \$ 1000 $, $ n = 12 $,  $ i = 5 \% $ and $ t = 30 $.

$$ PV = 12 \times \$ 1000 \frac{1- \left(1 + \frac{5 \%}{12}\right)^{-30 \times 12}}{5 \%}= \$186,281.62  $$

To work out this problem using a BA II Plus, do the following:

2nd FV (CLR TVM),  360 N, 5/12 = I/Y, 1000 PMT and then press CPT PV and you should get: -186,281.62.

If you want a positive value, make the 1000 negative (payments and present value have opposite signs).

If the top of your calculator says "BGN", that means the first payment is made at the beginning of the period, not the end. "BGN" is for annuity due.

For Excel you just punch in:

=PV(5%/12, 360, 1000)

Your output will be: ($186,281.62)

If you want a positive value for the present value then the payments have to be negative: "=PV(5%/12, 360, -1000)". This is because they have to have opposite signs.


So that about wraps up our discussion on Annuity Immediate Loans. In the next section, we will be looking at Annuity Due Loans.

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