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Last time we were discussing investments.

It is good when one has a decent principal to invest. After that one can do nothing but calculate how much it will be after so many years. However, most people miss this opportunity. They also want to accumulate an amount of money, and the strategy is: invest with small portions, put aside some amounts regularly. That is how Individual Retirement Accounts (IRA), 401(k), and 529 plans for education work. Today we consider these strategies in some details.

Suppose you deposit \( \$ 100 \) monthly into a bank with

an APR of \( 12\% \) ( \(1 \% \) per month)

At the end of the first month you have \(\$ 100 \)on the account.

At the end of the second month you deposit another \( \$ 100\).

But they have already kept your \( \$ 100 \) during a month, so you earn \( 1 \% \) of this sum, which is \( \$ 1 \).

All together, at the end of the second month you have

\( \$ 100 \) which you have brought them last time,
\( \$ 1 \) of interest,
and another \( \$ 100 \) which you have just brought them now.

That is

\( \$ 100 + \$ 1 + \$ 100 = \$ 201 \)

At the end of third month:

\( 201 + 1\% \times 201 + 100 = \$ 303.01 \)

At the end of the 4-th month:

\( 301.01 + 1\% \times 301.01 +100 = \$ 406.04 \)

At the end of the 5-th month:

\(406.04 +1\% \times 406.04 +100= \$ 510.10 \)

At the end of the 6-th month:

\( 510.10 +1\% \times 510.10 +100=\$ 615.20 \)

and so on ...

It is correct but quite awkward to calculate this way

Instead we will use Saving Plan Formula which encodes these calculations.

\( A = PMT \times \frac{ \left[ \left( 1 + \frac{APR}{n} \right)^{(nY)} -1 \right] }{\left( \frac{APR}{n} \right)} \)

where

\(A\) = accumulated savings plan balance (FV -- future value)

\(PMT\) = regular payment (deposit) amount

\(APR\) = annual percentage rate (in decimal)

\(n\) = number of payment periods per year

\(Y\) = number of years

\( A = PMT \times \frac{ \left[ \left( 1 + \frac{APR}{n} \right)^{(nY)} -1 \right] }{\left( \frac{APR}{n} \right)} \)

In the example we started with:

\(PMT= \$ 100\), \(APR = 12 \%\), \(n=12\) for monthly payments

After 6 months ( \(Y=0.5 \) ):

\(
A = 100 \times \frac{ \left[ \left( 1 + \frac{0.12}{12} \right)^{(12 \times 0.5)} -1 \right] }{\left( \frac{0.12}{12} \right)} = \$ 615.20
\)

as we already calculated, while
after 5 years ( \(Y=5 \) ):

\( A = 100 \times \frac{ \left[ \left( 1 + \frac{0.12}{12} \right)^{(12 \times 5)} -1 \right] }{\left( \frac{0.12}{12} \right)} = \$ 8166.97 \)

\( A = PMT \times \frac{ \left[ \left( 1 + \frac{APR}{n} \right)^{(nY)} -1 \right] }{\left( \frac{APR}{n} \right)} \)

Assume that you know the final accumulated balance you need. What should be your regular payment to get there after a certain amount of time?

Savings Plan Formula solved for payments:

\(
PMT= A \times \frac {\left( \frac{APR}{n} \right)} { \left[ \left( 1 + \frac{APR}{n} \right)^{(nY)} -1 \right] }
\)

\( PMT= A \times \frac {\left( \frac{APR}{n} \right)} { \left[ \left( 1 + \frac{APR}{n} \right)^{(nY)} -1 \right] } \)

In our example, \(PMT= \$ 100\), \(APR = 12 \%\), \(n=12\) for monthly payments, an amount of \( \$ 8,166.97 \) has been accumulated within a period of 5 years.

Assume that we want to accumulate \( \$ 10,000 \) under the same conditions within the same period of time. Then instead of \(\$ 100 \), the monthly deposit should be\( PMT= 10000 \times \frac {\left( \frac{0.12}{12} \right)} { \left[ \left( 1 + \frac{0.12}{12} \right)^{(12 \times 5)} -1 \right] } = \$ 122.44 \)

There are various investments.

Clearly, the bigger is the APY, the better an investment looks.

However, in real life, the APR is not constant; it floats. So does APY.

We thus need other quantitative characteristics of investments.

Assume that you deposit \(\$ 1,000 \), and accumulate \(\$ 1,500 \) after 5 years. How good was the investment?

The amount grew up by \( \frac{1500-1000}{1000} = 0.5 = 50\% \)

This is your total return.

In general, total return is the percentage change in the investment value:

\( \text{total return} = \frac{A-P}{P} \times 100 \% \)

where A is accumulated balance, and P is the starting principal.

Another person has got same \(\$ 1,500 \) but after 10 years instead of 5 years

While the total return is the same \(50 \%\), the quality of these two investments is clearly different.

Probably, the APR was different: yours was higher.

But we cannot compare the two APR's directly, because

in real life, APR is not constant, it is floating with time

The annual return is the APY that would give the same overall growth
It is calculated by the formula

\(
\text{annual return} = \left( \frac{A}{P} \right) ^{(1/Y)} -1
\)

where A is accumulated balance, P is the starting principal, and Y is the number of years.

In the examples, when \(A=\$1,500\) comes out \( P=\$ 1,000\) within 5 years,

\( \text{annual return} = \left( \frac{1500}{1000} \right) ^{(1/5)} -1 =0.084 = 8.4 \% \)

while for the same growth within 10 years,

\( \text{annual return} = \left( \frac{1500}{1000} \right) ^{(1/10)} -1 =0.041 = 4.1 \% \)

Today we considered Savings plan formula

Savings Plan Formula solved for payments

\(
PMT= A \times \frac {\left( \frac{APR}{n} \right)} { \left[ \left( 1 + \frac{APR}{n} \right)^{(nY)} -1 \right] }
\)

Characteristics of an investment:

\( \text{total return} = \frac{A-P}{P} \times 100 \% \)

\( \text{annual return} = \left( \frac{A}{P} \right) ^{(1/Y)} -1 \)