What Are Time Diagrams for Annuities?

A time diagram is a visual representation of the cash flows in an annuity. It shows how payments and interest occur over time on a timeline. These diagrams are helpful for understanding the timing and value of payments and for solving problems related to annuities.

Key Components of a Time  Diagram

1. Timeline: A straight line marked with intervals (e.g., months, years) to represent the time period.
2. Payments: To show regular payments (outflows) or deposits (inflows). The amount is written next to each period tick.
3. Future or Present Values: Arrows connect payments with future values at a given time period under the corresponding point on the timeline, or with the present values.

Example: Time Diagram for a Simple Annuity

Suppose you deposit $100 every year for 3 years into an account earning 5% annual interest. The future value (FV) is calculated at the end of 3 years.

• Year 0: Start with no money (0).
• Year 1: Deposit $100.$
• Year 2: Deposit 100 and the first deposit earns interest.
• Year 3: Deposit $100; previous deposits earn more interest.

The diagram looks like this:

At the end of 3 years, the future value (FV) includes all the payments plus the interest.

Example: Time Diagram for a General Annuity

Suppose you deposit $200 every 6 months for 2 years into an account earning 8% annual interest, compounded quarterly. The future value (FV) is calculated at the end of 2 years.

To create the time diagram in this case, you need to first determine the effective interest rate per a payment period. In other words, the effective interest rate for each 6 month period.

Calculation

Imagine you have an initial deposit $A$. In this example it experiences compound interest of $\frac{8}{4}\%=2\%$ every quarter, which is every 3 months. There are $2$ 3-month periods in 6 months. So the final amount after 6 months is given by

Under the effective interest rate ($i$), the same final amount is reached after one 6-month period. So

This means

so, canceling the $A$ terms we are left with

So the effective interest per each 6-month period, $i$, is $4.04\%$.

Time Diagram

We can now create the time diagram:

The diagram allows us to systematically calculate the value after 2 years, which is $840.80.

Why Use Time Diagrams?

• Visual Aid: Helps you see how payments and interest grow over time.
• Simplifies Problems: Makes it easier to solve annuity problems, especially for understanding the timing of payments.
• Tool for Formulas: Connects the timeline to formulas for calculating present or future value.