Financial Mathematics - Loans - Annuity Due

We're looking at different types of annuities. In the previous section, we took a look at annuity immediate loans. In this section we'll be dealing with annuity due loans.

 Annuity Due Loans

While most loans that people encounter are annuity immediate, sometimes there are annuities that are annuity due (and they can be thought of as a kind of loan.)

A familiar example of one of these would be a jackpot lottery winning. Many of these are paid out as annuities and the first payment is made, not one year from now, but right now (at $ t=0 $).

The flow of funds for this will look like this:

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

There are two ways to think about this. We'll derive the formula both ways. We start out with similar 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).

So let's look at the first derivation.

Derivation 1

We'll do it the hard way first. This amounts to treating the problem like we did before and solving it brute force. 

The basic setup is the same but the result is slightly different. We'll look at it for $n$ period case (which can reduce to the annual case for $ n = 1 $.

$$PV = c + \frac{c}{\left(1+\frac{i}{n} \right)^1} + \frac{c}{\left(1+\frac{i}{n}\right)^2} + \frac{c}{\left(1+\frac{i}{n}\right)^3}  + . . . + \frac{c}{\left(1+\frac{i}{n}\right)^{t \times n-1}}$$

We can factor out a $c$ so that we get the geometric series in brackets:
$$PV = c \left[1+ \frac{1}{\left(1+\frac{i}{n} \right)^1} + \frac{1}{\left(1+\frac{i}{n}\right)^2} + \frac{1}{\left(1+\frac{i}{n}\right)^3}  + . . . + \frac{1}{\left(1+\frac{i}{n}\right)^{t \times n-1}}\right]$$
The expression brackets is a geometric series with $ a=1 $ and $ r =  \frac{1}{\left(1+\frac{i}{n} \right)} $. So that expression will be:
$$ \frac{1-\left(1+\frac{i}{n}\right)^{-(tn)}}{1-\left(1+\frac{i}{n}\right)}$$
That can be substituted in to arrive at:
$$ PV = c \frac{1-\left(1+\frac{i}{n}\right)^{-(tn)}}{1-\left(1+\frac{i}{n}\right)^{-1}}$$
To make this formula look prettier, we can multiply the numerator and denominator by $ \left(1+\frac{i}{n}\right) $ and it looks like this:
$$PV = \left(1+\frac{i}{n}\right) nc \frac{1-\left(1+\frac{i}{n}\right)^{-tn}}{i}$$
You'll notice that this is the same result as the Annuity Immediate result except it's multiplied by a factor of $ \left(1+\frac{i}{n}\right) $.

In actuarial notation this will look like (recall that $ d=\frac{i}{1+i} $):
$$\ddot{a}_{\overline{n|}i} = (1+i)\frac{1-v^n}{i} = \frac{1-v^n}{d} = (1+i)  {a}_{\overline{n|}i}$$
Derivation 2

The second derivation is conceptually simpler but the algebra is a bit more difficult. This requires that we look at the flow of funds table. There are basically two differences between the two loan types: annuity immediate and annuity due.

The first difference is in the initial phase of the loan. The initial phase is actually the difference between the loan amount and the first payment: $PV - c$.

The second difference is that in the settlement period. There is one less payment (in other words, $n \times t -1$ instead of $n \times t$). As a result, we can transform the annuity due into a special case of the annuity immediate in which the present value is equal to $PV - c$ and the stream of payments ends at $n \times t -1$ instead of $n \times t$. Got that?

Here's how it works.

Recall that the Annuity Immediate Loan Formula is given by:
\[\begin{equation}
PV = nc \frac{1- \left(1 + \frac{i}{n}\right)^{-t \times n}}{i} \tag{Annuity Immediate Loan Formula}
\end{equation}\]
Now we're going to make two substitutions. We'll replace $PV$ with $PV - c$ and we'll replace $-t \times n$ with $-(t \times n - 1)$.
\[\begin{equation*}
PV - c = nc \frac{1- \left(1 + \frac{i}{n}\right)^{-(t \times n-1)}}{i}
\end{equation*}\]
Next, we'll move the $c$ over to the other side and play some algebra games.
\[\begin{align*}
PV  &= c + cn \frac{1- \left(1 + \frac{i}{n}\right)^{-(t \times n-1)}}{i}\\
&= \frac{cni}{ni} + \frac{cn}{i} \left(1- \left(1 + \frac{i}{n}\right)^{-(t \times n-1)}\right)\\
&= cn \frac{\frac{i}{n}+1- \left(1 + \frac{i}{n}\right)^{-(t \times n-1)}}{i} \\
&= cn \left(1+\frac{i}{n}\right)\frac{1-\left(1 + \frac{i}{n}\right)^{-(t \times n)}}{i}
\end{align*}\]

So the algebra was ugly but we got the same result.

Example 1: The Mega Millions Jackpot

If you decide to play the lottery (because you like paying extra taxes) and you happen to win, you generally have two options: receive all the cash up front (minus taxes - see, even if you win, you pay more taxes!) or get an annuity stream. So what's the effective interest rate on the annuity stream?

In the case of the Mega Millions game, you can either choose the cash option (present value) or you can get the annuity stream which pays 26 equal payments. The first payment you'll get today (so it's an annuity-due, not an annuity-immediate).

It's quite likely that the effective interest rate of the annuity will vary over time (vary with currently available interest rates) so we'll just do one example for illustrative purposes. In general, you need two pieces of information: Cash Value and Total Value (which is the undiscounted stream of payments).

The largest Mega Millions Jackpot had a cash value of $ \$474 $ million with a total value of $ \$656 $ million. Now since the above formulas cannot be algebraically solved for interest we will have to use some calculators.

For the BA II Plus, do the following:

2nd FV (CLR TVM), 2nd PMT (BGN) 2nd ENTER (SET) 2nd CPT (QUIT) to set to BGN, 26 N, 656 / 26 = PMT, 474 +/- PV, and then press CPT I/Y and you should get 2.81 (%)

For EXCEL enter the following command:

 =RATE(26,656/26,-474,0,1)

and you should get 2.81% (you may have to reformat it.) The "1" in the 5th parameter sets it to annuity-due (if you have 0 or omitted it will default to annuity-immediate.)

And just to show you that you can use the annuity immediate formula, let's make the appropriate adjustments. So instead of 26 payments, we will now have 25 payments. And we need to adjust the present value by the payment 656/26 so that it's -474+656/26. Lastly we will need to change that last 1 back to 0 so that it switches it back to annuity immediate. That will look like this:

=RATE(25, 656/26,-474+656/26,0,0)

And shockingly we get 2.81% again.

Example 2: Pay in full or monthly payments?

Many times companies that charge a service fee for a period (say 1 year) will offer the option to either pay for the full year or to make monthly payments. Insurance companies often give this option for making premium payments on a policy. Let's consider the following example.

Suppose that the insurance company offers the following payment options: $ \$1000  $ for the pay in full option or $ \$180 $ per month for 6 months. Which is the better option? Well, to look at that we'd want to see what interest rate they're charging us for this option.

Since the first payment would have to be made immediately, this would fall under the annuity-due annuity. Now we can't analytically solve for the interest rate but we can, like the above, use a calculator to find an approximate answer.

For the BA II Plus, do the following:

2nd FV (CLR TVM), 2nd PMT (BGN) 2nd ENTER (SET) 2nd CPT (QUIT) to set to BGN, 6 N, 180 PMT, 1000 +/- PV, and then press CPT I/Y and you should get 3.19 (%)

For EXCEL, we enter the following command:

=RATE(6,-180,1000,0,1)

and you should get 3.19% (again, you may need to reformat the number of decimal places).

So 3.19% doesn't sound too bad right? But there's a catch here: that's the monthly interest rate. We'll need to find the annual rate. Recall that there are two different annual interest rates quoted: the Annual Percentage Rate (APR) and the Annual Percentage Yield (APY). So let's calculated both.

The APR is pretty simple. We just multiply the monthly rate by 12 months to get the Annual Percentage Rate.
$$APR = 3.19 \% \times 12 = 38.28 \% $$
Note: if you do this with your financial calculator, you may get $ 38.23 \% $, the difference would be a result of rounding.

To calculate Annual Percentage Yield (APY) we need the formula derived here. The APR we derived will be $i$. I'll use the more accurate $ 38.23\% $ value.
\[\begin{align*}
APY = r &= \left(1+\frac{i}{n}\right)^n - 1 \\
&= \left(1+\frac{38.23\%}{12}\right)^{12} - 1 \\
&= 45.69 \%
\end{align*}\]
So paying in full is a huge savings. The only financial reason not to do so would be if you had an investment opportunity that was guaranteed to earn you more than $ 45.69 \% $ per year. And if you find one of those, it's probably a scam.

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