When it comes to managing money and making smart financial decisions, understanding future value (FV), present value (PV), and annuities is crucial. Let’s break these concepts down in excellent detail, step by step, using simple language and examples.
Future Value (FV) of a Single Sum
The amount your money will grow to in the future if you invest it today at a specific interest rate is known as Future Value (FV). This is how compound interest works—your money earns interest, and that interest earns interest, creating a snowball effect.
The Formula:
Example:
You invest $200 today in an account that earns 10% interest annually. How much will you have in 2 years?
So, in 2 years, your $200 will grow to $242.
Future value illustrates the potential growth of your investment over time.
Present Value (PV) of a Single Sum
Present Value (PV) is the reverse of FV. It tells you how much money you need to invest today to reach a specific amount in the future. Discounting is the process of determining the current value of future money.
The Formula:
Example:
You want $200 in 2 years, and you can earn 10% interest annually. How much do you need to invest today?
You need to invest $165.29 today to have $200 in 2 years.
Present value tells you the amount of money you need now to achieve a future goal.
Annuities
An annuity is a series of equal cash flows (like payments or deposits) that happen at regular intervals, such as monthly or yearly.
- Ordinary Annuity. Each period ends with payments.
- Annuity Due. We make payments at the start of each period.
Future Value of an Ordinary Annuity
The FV of an ordinary annuity shows how much a series of regular payments will grow to in the future.
Example:
You invest $200 at the end of each year for 3 years in an account earning 10% interest. How much will you have at the end of 3 years?
- Each payment grows for a different number of years:
- First $200 grows for 2 years.
- Second $200 grows for 1 year.
- Last $200 doesn’t grow (paid at the end of the 3rd year).
- Add it all up using the calculator:
So, the total will be $662.
Present Value of an Ordinary Annuity
The PV of an ordinary annuity calculates how much money you need today to fund a series of future payments.
Example:
You want to receive $200 at the end of each year for 3 years, discounted at 10%. How much do you need to invest today?
- We discount each payment to the current value:
- There is a one-year discount on the first $200.
- Second, $200 for 2 years.
- Third, $200 for 3 years.
- Add it all up using the calculator:
So, you need $497.37 today.
Annuity Due
Similar to an ordinary annuity, an annuity due requires payments at the start of each period.
Example:
You invest $200 at the beginning of each year for 3 years at 10% interest. How much will you have at the end of 3 years?
- For an annuity due, the first payment starts earning interest right away.
- Use the same calculation as an ordinary annuity, but multiply the FV by (1 + I/Y):
So, the future value is $728.20.
Annuity due payments grow more than ordinary annuities because the payments are invested earlier.
Perpetuity
A perpetuity is a special type of annuity where payments continue forever. For example, some preferred stocks pay dividends indefinitely.
Formula for PV of Perpetuity:
- PMT = Fixed payment
- I/Y = Interest rate
Example:
A perpetuity pays $100 annually, and the interest rate is 5%. What is its present value?
So, the perpetuity is worth $2,000.
Why These Concepts Matter
- Future value shows how your money can grow with time and interest.
- Present value helps you plan for future financial goals by knowing how much to invest today.
- Annuities allow you to calculate consistent cash flows for regular needs like retirement or loan repayments.
- Perpetuities are excellent for understanding infinite cash flows, like dividends.
Understanding these concepts puts you in control of your financial future, whether it’s saving for retirement, buying a home, or investing wisely.
