Present Value of Annuity Guide

💰 What is Present Value of Annuity?

Present Value of Annuity (PVA) is the current value of a series of equal future payments, discounted to today's dollars. It answers: "What lump sum today equals receiving $X per month for Y years?"

                Key Point: PVA is the opposite of Future Value of Annuity. Instead of "how much will my savings grow to?", PVA asks "what are future payments worth today?" This is crucial for valuing pensions, lottery winnings, inheritances, and structured settlements.
            

The PVA Formula

PV = PMT × [(1 - (1 + r)^-n) / r]

Where:

PV = Present Value

PMT = Payment per period

r = Interest rate per period

n = Number of periods

Simple Example

You win $10,000/month for 10 years. What's it worth today at 6% interest?

PMT = $10,000/month

r = 6% / 12 = 0.5% = 0.005

n = 10 × 12 = 120 months

PV = $10,000 × [(1 - (1.005)^-120) / 0.005]

PV = $10,000 × [(1 - 0.5496) / 0.005]

PV = $10,000 × 90.07

PV = $900,700

Interpretation: Receiving $10k/month for 10 years (total $1.2M) is worth $900,700 today. You "lose" $299,300 to the time value of money.

Common Applications

Lottery Winnings:

Choose between $5M lump sum today or $300k/year for 25 years? PVA tells you which is better.

Pension Valuation:

What's a $4,000/month pension for 20 years worth today? PVA gives the answer.

Mortgage Calculations:

How much can you borrow with $2,500/month payment capacity? PVA determines loan amount.

Structured Settlements:

Insurance offers $5,000/month for 15 years. What lump sum is equivalent?

Business Valuation:

Company expects $100k/year profit for 10 years. What's it worth today?

Present vs Future Value

Metric	Question Answered	Use Case
Future Value	How much will savings grow to?	Retirement planning, savings goals
Present Value	What are future payments worth today?	Lottery, pension, loan valuation

Impact of Discount Rate

Same annuity: $5,000/month for 20 years

Discount Rate	Present Value	Note
3%	$894,550	Higher PV (money worth more)
5%	$754,140	Mid-range
7%	$637,250	Lower PV
10%	$510,870	Much lower (higher discount)

💡 Why Rate Matters

Higher discount rates reduce present value. At 10%, the $5k/month stream is worth $510k today. At 3%, it's worth $895k. The rate reflects opportunity cost of money and inflation.

Lump Sum vs Annuity Decisions

Lottery Example: Choose One

Option	Description	Present Value at 5%
Option A	$10M lump sum today	$10,000,000
Option B	$600k/year for 25 years	$8,455,620

                Financial Decision: Option A ($10M lump sum) is worth $1.54M more than Option B in present value terms. However, Option B provides guaranteed income and protection from overspending. The choice involves financial AND behavioral considerations.
            

Pension Valuation Example

Scenario: 65-year-old offered pension buyout

Current pension: $3,500/month

Expected life span: 20 more years

Discount rate: 4% (conservative)

PV = $3,500 × [(1 - (1.00333)^-240) / 0.00333]

PV = $577,920

Decision: Company offers $500k buyout. Present value is $578k, so pension stream is worth MORE. Don't take the buyout unless they offer at least $580k+.

⚠️ Assumptions Matter

PVA calculations assume:
- You'll live the full period (mortality risk)
- Payments continue as stated (counterparty risk)
- Discount rate is accurate
- No inflation adjustments (unless specified)
Real decisions require considering these factors beyond just the math.

Ordinary Annuity vs Annuity Due

Ordinary Annuity (Payment at End):

Most common. Receive payment at end of each month/year. Formula shown above.

Annuity Due (Payment at Beginning):

Receive payment at start of each period. Worth slightly more because each payment comes sooner.

PV (Annuity Due) = PV (Ordinary) × (1 + r)

Example: $2,000/month for 15 years at 6%

Ordinary Annuity: $291,530

Annuity Due: $291,530 × 1.005

Annuity Due: $292,988

Difference: $1,458 more

🔢 Calculating Present Value

Example 1: Mortgage Calculation

Question: You can afford $2,500/month payment. How much can you borrow?

Given:

Monthly payment: $2,500

Interest rate: 6.5% annual (0.5417% monthly)

Loan term: 30 years (360 months)

Calculation:

PV = $2,500 × [(1 - (1.005417)^-360) / 0.005417]

PV = $2,500 × 158.21

PV = $395,525

Answer: You can borrow approximately $395,500 with a $2,500/month payment capacity at 6.5% for 30 years.

Example 2: Inheritance Decision

Scenario: Grandmother's will offers choice

Option A: $250,000 lump sum today

Option B: $2,000/month for 15 years

Your discount rate: 5% (what you could earn)

Calculate PV of Option B:

PV = $2,000 × [(1 - (1.004167)^-180) / 0.004167]

PV = $2,000 × 129.30

PV = $258,600

Comparison:

Option	Present Value	Total Received
A: Lump sum	$250,000	$250,000
B: Payments	$258,600	$360,000

Best choice: Option B is worth $8,600 more in present value terms AND provides $110k more total. Option B wins both ways.

Example 3: Pension vs Lump Sum

Employee retiring, offered:

Option	Description
Pension	$4,500/month for life (expect 22 years)
Lump Sum	$850,000 today

Calculate PV of Pension at 4.5%:

PMT = $4,500/month

n = 22 years × 12 = 264 months

r = 4.5% / 12 = 0.375%

PV = $4,500 × [(1 - (1.00375)^-264) / 0.00375]

PV = $834,565

Analysis:

PV of pension: $834,565

Lump sum offered: $850,000

Lump sum is $15,435 MORE

Considerations:

Lump sum is worth more in pure PV terms ($15k advantage)
But pension continues if you live past 22 years (longevity protection)
Lump sum gives flexibility and inheritance to heirs
Pension removes investment risk and overspending risk

Decision depends on health, other income, discipline, and family needs.

Example 4: Car Lease vs Buy

Lease Option: $450/month for 3 years, nothing down

Buy Option: $15,000 purchase price

Calculate PV of Lease at 5%:

PV = $450 × [(1 - (1.004167)^-36) / 0.004167]

PV = $450 × 33.96

PV = $15,282

Comparison:

PV of lease payments: $15,282

Purchase price: $15,000

Lease costs $282 more in PV

But after 3 years:

Lease: You own nothing, paid $16,200 total
Buy: You own car worth ~$9,000, net cost $6,000

Verdict: Buying is financially superior. Lease only if you want newest model every 3 years and don't mind the premium cost.

Using PVA for Loan Affordability

How much home can you afford?

Gross monthly income: $8,500

Maximum 30% for housing: $2,550/month

Available interest rate: 6.8%

Standard 30-year term

PV = $2,550 × [(1 - (1.005667)^-360) / 0.005667]

Maximum loan: $387,600

With 20% deposit, you can afford: $387,600 / 0.8 = $484,500 house

🌍 Real-World PVA Examples

Lottery Winner's Choice

Jane wins $5 million lottery, must choose:

Option	Terms
Option A	$3.2M lump sum today (after tax)
Option B	$200,000/year for 30 years (after tax)

PV Calculation at 6% (her opportunity cost):

PV = $200,000 × [(1 - (1.06)^-30) / 0.06]

PV = $200,000 × 13.765

PV of Option B = $2,753,000

Comparison:

Option A: $3,200,000

Option B: $2,753,000 (in PV)

Option A is $447,000 better!

Jane's decision: She takes Option A ($3.2M lump sum) because:

Worth $447k more in present value
She's young (35) and disciplined with money
Can invest it herself and potentially beat 6%
Has immediate access for opportunities

If she were 65 or lacked financial discipline, the annuity might be wiser despite lower PV.

Court Settlement Structured Payment

Legal settlement offers two options:

Option	Terms
Lump Sum	$500,000 today
Structured	$3,500/month for 20 years

PV of Structured at 5%:

PV = $3,500 × [(1 - (1.004167)^-240) / 0.004167]

PV = $531,947

Analysis:

Metric	Lump Sum	Structured
Present Value	$500,000	$531,947
Total Received	$500,000	$840,000
Immediate Access	Yes	No
Guaranteed Income	No	Yes, 20 years

Best choice: Structured payment worth $31,947 more AND provides long-term security. Winner: Structured, unless you need lump sum for specific purpose (medical bills, house purchase).

Early Retirement Buyout

55-year-old offered early retirement package:

Current Situation:

Planned retirement: age 65

Pension at 65: $5,000/month for life

Expected lifespan: age 85 (30 years of pension)

Early Retirement Offer:

Retire now at 55

Immediate pension: $3,200/month for life

Plus one-time: $150,000 bonus

Calculate PV of Each (4% discount):

Option 1: Wait to 65

Start in 10 years, $5,000/month for 20 years

PV at 55 = $5,000 × 149.27 × 0.6756

= $503,920

Option 2: Retire now at 55

$3,200/month for 30 years

PV = $3,200 × 209.46

= $670,272

Plus bonus: $150,000

Total = $820,272

Winner: Retire now! Worth $316,352 more in present value, plus you get 10 extra years of freedom and $150k bonus.

🎯 Test Your Knowledge

Complete this quiz on Present Value of Annuity

1. Present Value of Annuity tells you:

How much your savings will grow to

What future payments are worth today

Your monthly payment amount

The interest rate earned

2. Higher discount rates will:

Increase present value

Decrease present value

Not affect present value

Double present value

3. Present Value of Annuity is most useful for:

Planning retirement savings

Valuing lottery winnings or pension offers

Calculating investment returns

Estimating stock prices

4. An annuity due is worth _____ than an ordinary annuity:

Less

The same

Half as much

5. You can afford $3,000/month mortgage payment. At 6% for 30 years, you can borrow approximately:

$360,000

$450,000

$500,000

$900,000

6. Receiving $10,000/month for 10 years is worth today (at 6%):

$1,200,000 (total payments)

Less than $1,200,000

More than $1,200,000

Exactly $1,200,000

7. When comparing lump sum vs annuity offers, choose based on:

Total dollars received

Number of payments

Present value AND personal circumstances

Always take lump sum

8. PV of Annuity and FV of Annuity are:

The same calculation

Opposite concepts (present vs future)

Unrelated concepts

Both measure interest rates

9. What does the discount rate represent in PVA calculations?

Bank savings rate

Inflation rate

Opportunity cost or required return

Mortgage rate

10. A $500k lump sum vs $3,000/month for 20 years. Which is usually worth more at 5%?

Lump sum (PV of annuity is ~$452k)

Monthly payments (higher total)

Exactly equal

Cannot determine

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