5.1 Adding to a Savings Account
Learning Outcomes
You make a series of equal-sized payments (deposits), at regular intervals over the course of a fixed time period. When you make these regular deposits, you are adding to the money in the account and building up a positive account balance. Therefore, both PV (your initial deposit) and PMT (your subsequent regular deposits) must have the same sign (positive). The future value (FV) must have the opposite sign (negative)[1]:
| PV | Interest | PMT | FV |
|---|---|---|---|
| Initial Deposit | + % Gain | + Regular Deposits | = Final Withdrawal |
| 0 or + | + | + | − |
To make sense of FV (your final withdrawal) being negative, you could imagine that we are closing the account at the end of the investment period. When we close it, we withdraw all of the funds, making the final/future value opposite in sign from the regular deposits (PMT) and the initial amount deposited (PV).
See the sections below for key formulas, tips and examples related to calculations when adding to a savings account.
Let us start by calculating the ending account balance (FV) when regular deposits (PMT) are made into an account. For this scenario, it is possible that there is an initial balance (PV) or no money in the account (PV = 0). Let’s see an example where you start saving for your retirement and we’ll examine both scenarios — when you have no money in the account at the start and when you have an initial balance.
EXAMPLE 5.1.1— SAVING FOR RETIREMENT
For the next ten years, you deposit $400 each month into a retirement savings account. You deposit the first payment one month from now. The account will pay 4.5%, compounded monthly. You start with $0 in the account at the start of the ten years. How much money will you have in the account at the end of ten years?
| B/E | P/Y | C/Y | N | I/Y | PV | PMT | FV |
|---|---|---|---|---|---|---|---|
| END | 12 | 12 | 10×12=120 | 4.5 | 0 | +400 | CPT -60,479.23 |
Why is B/E set to END? This is because the first deposit is being made in one month, at the end of the first payment interval. When regular payments or deposits are made at the end of the payment interval, we call this an ordinary annuity. We do not need to do anything in the calculator (it is set to END by default). We will talk more about changing this setting in the later sections.
Why are P/Y and C/Y equal to 12? P/Y equals the number of payments or deposits per year. C/Y equals the number of times the interest rate compounds per year and the interest. Since both the compounding and deposits are monthly, both P/Y and C/Y equal 12.
Why does N equal 120? N equals 120 because you will make monthly deposits for 10 years. This will give you a total of 10×12 deposits.
Note: N = The total number of payments or deposits = Number of years × P/Y
Why does PV equal 0? PV is set to 0 because there is no money in the account at the start. PV is the initial (starting) balance in the account.
Finally, why does PMT equal +400? PMT equals to +400 because we make regular deposits of $400 into our savings account. We treat all deposits as positive because it is money going into a savings account[2].
Now we can calculate the amount saved at the end of 10 years (FV). Notice that the BAII Plus will give us a negative value for the future value. This minus sign (negative sign) indicates that the future value (FV) is going in the opposite direction of the deposits (PMT and PV). Don’t include the minus sign (negative sign) in your final answer.
Conclusion: You will have $60,479.23 in your savings account at the end of 10 years.
Check Your Knowledge 5.1.1
For all investments, loans, etc., we can consider the following formula to be true when calculating the interest earned or charged on our investment or loan:
Interest = Money Out – Money In = $ OUT – $ IN
In the case of regular deposits into a savings account, in this textbook, we will consider the initial deposit, PV (if there is one) to be “money in” since we are depositing this money INTO the account.[3] We consider the regular deposits, PMT, to be “money in” since they are also deposits going INTO the savings account. Finally, we consider FV to be “money out” since it is being withdrawn from the account when we close it:
| PV | Interest | PMT | FV |
|---|---|---|---|
| Initial Deposit | + % Gain | + Regular Deposits | = Final Withdrawal |
| $ IN | $ IN | $ IN | $ OUT |
This gives us the following equation for the interest earned:
Check Your Knowledge 5.1.2
Key Takeaways: Regular Deposits into Savings Accounts
- Are there any notes you want to take from this section? Is there anything you’d like to copy and paste below?
- These notes are for you only (they will not be stored anywhere)
- Make sure to download them at the end to use as a reference
- Other texts might treat PV and PMT as negative and FV as positive. What is most important is that we are aware that PV and PMT must have the same sign and FV must be opposite in sign. ↵
- Some texts might treat deposits into a savings account as negative. If they do, the initial deposit, PV (if non-zero) will also be negative and FV (the ending balance) will be positive. ↵
- Other texts or instructors might treat signs differently than we do here. What is most important is that we are aware that money flows in two different directions (in and out). It could be possible to consider the regular deposits as negative and the ending balance (FV) as positive. You will still get the correct answer because the deposits and final balance are opposite in sign. ↵
An annuity where the payments occur at the end of each payment interval.
