## Saturday, September 21, 2013

### My 104% Per Year Investment!

OK, so some of you may recall my blog entitled My 1348% Per Year Investment and are thinking, is this going to be another one of those?

And the short answer to that is "yes". But there will be a bit more to this. As this is a quasi-promotion for a quasi-textbook that I (a quasi-person) am writing. That text is on Financial Mathematics. You can also find it via the huge banner at the top.

In the most recent section, one of the examples I used was whether to pay an insurance premium in full or to make monthly payments. So here our the scenarios my most recent automobile insurance quoted of me (I've scaled these so that they're $\$100 $... I don't really get car insurance that cheap.) Option 1: Pay in Full for$ \$100$
Option 2: Pay $\$19.25$now and$ \$17.66$ for the next 5 months for a total cost of $\$107.55$via electronic fund transfer (EFT). Option 3: Pay$ \$19.25$ now and $\$19.24$for the next 5 months for a total cost of$ \$115.45$ (non EFT payments).

One of the key assumptions in the formulas I derived (and it's important to pay attention to assumptions... you cant use a result in a particular case unless the assumptions are satisfied... or at least approximately true) is that the payments must be equal. Since the first payment is not equal to the other 5 payments, that presents us with a problem.

But wait! I noted that an annuity due can be thought of as a special kind of annuity immediate in which you adjust the present value by the initial payment and then treat this as if we have 5 payments instead of 6 (total payments).

As a result we can plug this into Excel pretty easily. (If you don't have Excel, there are other financial calculators you can use to do the same thing.)

The flow of funds table will look like this:

InitialSettlement
Option 1- PAYMENT IN FULL
+INSURANCE POLICY

Option 2/3-FIRST PAYMENT
+INSURANCE POLICY
- PAYMENTS

This gives us two equations:
\begin{align} \text{INSURANCE POLICY} - \text{PAYMENT IN FULL} &= 0 \\ \text{INSURANCE POLICY} - \text{FIRST PAYMENT} - PV(\text{PAYMENTS}) &= 0 \end{align}
We're going to subtract equation (1) from equation (2) to obtain:
$$\text{PAYMENT IN FULL} - \text{FIRST PAYMENT} = PV(\text{PAYMENTS})$$
The value of the insurance policy is the same in both cases so they will cancel out. So ultimately we'll be comparing paying in full versus paying an initial first payment and a series of payments afterward.

First off, let's compare Option 1 versus Option 2.

To do that in Excel, we need the present value which will be the first payment - the payment in full. That will be $\$19.25 - \$100 = -\$80.75$. Then we need the 5 payment amounts which will be$ \$17.66$ each. That will look like this:

=RATE(5.17.66.-80.75)

And if you adjust the formatting, you should get 3.06%. But recall that's 3.06% per month. If we want to annualize that, we can look at two different figures: APR or APY.

Using the relevant formulas. . .

To calculate the Annual Percentage Rate, we just multiply the monthly rate by the number of months in a year.
\begin{align*} APR &= n \times \frac{i}{n} \\ &= 12 \times 3.06\% \\ &= 36.72\% \end{align*}
To calculate the Annual Percentage Yield, we need that fancier formula:
\begin{align*} APY &= \left(1+\frac{i}{n}\right)^n - 1 \\ &= \left(1+3.06\%\right)^12 -1 \\ &= 43.58\% \end{align*}
If we wanted to compare Option 1 versus Option 2, the damage is even worse. Plug this into Excel:

=RATE(5,19.24,-80.75)
And you get a monthly rate of 6.13%. That comes out to an APR of 73.56% and an APY of 104.20%.

So my effective annualized yield for choosing option 1 over option 3 is 104.20%!

The obvious moral to this story is to pay in full (unless you can find an investment that offers a better guaranteed rate of return). But there's another consideration.

Pay in Full with a Credit Card???

What if you can't pay in full. Many people live paycheck to paycheck and may not be able to come up with the full amount. (The insurance company is counting on this.) What other options do you have?

One option is a credit card. Think about it, the credit card may charge you an interest rate of, say, 25% (APR). But that's still a whole lot cheaper than 36.72% or 73.56%.

So suppose you decide to pay in full with the credit card. That means that you'll pay the $\$19.25$up front but the rest ($ \$80.75$) with the credit card. How much would your monthly payment have to be to pay off a credit card at 25%?

We can plug all of that into Excel too:

=PMT(25%/12, 5, -80.75)

And that will give you a required payment of $\$17.17$. So even that 25% credit card rate is cheaper than the insurance company's offer (which will require either$ \$17.66$ or $\$19.24\$ for options 2 or 3, respectively.)