The present value of an annuity is the current worth of a series of future payments from an annuity, adjusted by a specified rate of return or discount rate. The higher the discount rate, the lower the present value of the annuity.
Present value (PV) relies on the concept of the time value of money, indicating that a dollar today holds more purchasing power than a dollar in the future due to its investment potential.
Key Learnings
- Present value of an annuity represents the amount needed today to fund series of future payments.
- Due to the time value of money, receiving a sum today is more valuable than receiving the same sum in the future.
- Use PV calculations to compare lump sum payments vs annuities spread over time.
Understanding Annuities and Their Present Value
An annuity is a financial product offering regular payments over time. Annuities can be immediate, starting payments right away, or deferred with a set delay.
Given the time value of money, today’s money is worth more because it can be invested to generate returns. For example, $5,000 today is more valuable than receiving $1,000 annually for five years.
Using present value calculations, you can assess whether it’s better to receive a lump sum or annuities. An important consideration in decisions such as selecting a pension payout option.
Moreover, present value helps compare different annuity options based on the payment amount and schedule.
Impact of Discount Rate on Present Value
The discount rate is pivotal in calculating the PV of an annuity. This rate reflects the potential return from investing and represents the time value of money.
A higher discount rate results in a lower present value since future payments are heavily discounted. In contrast, a lower discount rate increases the present value due to less discounting.
Generally, the chosen discount rate should reflect the return one could earn from alternative investments. Lower-bound often uses the risk-free rate, typically the yield from U.S. Treasury bonds.
Calculating the Present Value of an Annuity
The formula for an ordinary annuity, where payments are made at the end of each period, is given by:
P = PMT \times \frac { 1 - \Big ( \frac { 1 }{ ( 1 + r )^n } \Big ) }{ r }
P = Present value of an annuity stream
PMT = Dollar amount of each annuity payment
r = Interest rate (discount rate)
n = Number of periods
Example Calculation
Consider a scenario where a person can receive $50,000 per year for 25 years (ordinary annuity) with a discount rate of 6%. Alternatively, a lump sum of $650,000 is available.
Using the formula, the PV of the annuity is:
PV = £50,000 \times \frac{1 - ( \frac{1}{ (1 + 0.06)^{25}})}{0.06} = £639,168
Thus, the annuity is less beneficial than the lump-sum by $10,832 ($650,000 - $639,168). The lump-sum is a better option financially.
Annuity Due vs. Ordinary Annuity
An ordinary annuity pays at the end of each period, whereas an annuity due pays at the beginning. Given equal parameters, the annuity due holds more present value due to earlier payments.
To calculate for an annuity due, adjust the ordinary annuity formula by (1 + r):
P = PMT \times \frac {1 - ( \frac{1}{(1 + r)^n } }{ r }\times (1 + r)
So, if we take the previous example and assume it’s an annuity due, its value would be:
PV = £50,000 \times \frac{1 - ( \frac{1}{\ (1 + 0.06)^{25}\ )\ }{0.06} \times (1 + 0.06) = £677,518
This means selecting the annuity due yields $27,518 more than the $650,000 lump-sum.
The Importance of Future Value (FV) to Investors
Future value (FV) of a current asset helps investors project potential growth based on assumptions, assisting in informed decisions for future financial needs. External factors like inflation, however, can diminish the future value.
How Do Ordinary Annuity Comparisons Differ from Annuity Due?
An ordinary annuity includes equal payments made after consecutively fixed intervals. The annuity due makes its payments at the start of each period.
Due to timing, these two types differ in present and future value calculations, commonly affecting choices concerning loans or recurrent financial obligations like rent.
The Bottom Line
Present value of an annuity, based on the time value of money principles, helps compare the lump sum versus structured payments, aiding crucial financial decisions. Understand your needs and return expectations to make an informed choice.
Related Terms: time value of money, discount rate, future value, ordinary annuity, annuity due
References
- Cornell Law School, Legal Information Institute. “26 CFR § 25.2512-5 - Valuation of Annuities, Unitrust Interests, Interests for Life or Term of Years, and Remainder or Reversionary Interests”.
- Rice University, OpenStax. “Principles of Finance: 8.2 Annuities”.
- TreasuryDirect. “Treasury Bonds”.
- George Brown College. “Formula Sheet for Financial Mathematics”. Page 2.
- Wai-sum Chan and Yiu-kuen Tse. “Financial Mathematics for Actuaries (Third Edition)”, Pages 40-43. World Scientific Publishing Company, 2021.
- George Brown College. “Formula Sheet for Financial Mathematics”. Page 3.
- Rahman, Mohammad. “Time Value of Money: A Case Study on Its Concept and Its Application in Real Life Problems”. International Journal of Research in Finance and Management, vol. 1, no. 1, 2018, pp. 18-23.
