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Time Value of Money

Calculate present value, future value, and compound interest

⏱️ 27 min⚡ 9 interactions

1. Why a Dollar Today is Worth More Than Tomorrow

Would you rather have $1,000 today or $1,000 in five years? The answer is obvious: today! But why? The Time Value of Money (TVM) explains that money available now is worth more than the same amount in the future because of its potential earning capacity.

💰 Core Concept

Time Value of Money (TVM) is the idea that money you have now can be invested to earn returns, making it worth more than the same amount in the future. This principle underlies all financial decisions: investing, borrowing, saving, and business valuation.

🎯 Why Money Has Time Value: The Three Pillars

1. Opportunity Cost: Money Can Earn Returns

The primary reason money today is worth more than tomorrow is opportunity cost. Money received today can be invested immediately to generate returns. Money received later loses this earning potential during the waiting period.

Simple Example:
✓ Receive $1,000 Today
Invest at 8% annual return
After 1 year: $1,080
After 5 years: $1,469
After 10 years: $2,159
✗ Receive $1,000 in 5 Years
Cannot invest yet - no returns
After 1 year: $0
After 5 years: $1,000
After 10 years: $1,469
Opportunity cost of waiting 5 years: You miss out on $469 in growth! The earlier you receive money, the more time it has to compound.

2. Inflation: Purchasing Power Erosion

Even if you don't invest, inflation makes future money less valuable. As prices rise over time, each dollar buys fewer goods and services. $100 today might only have the purchasing power of $75 in 10 years at 3% inflation.

Real-World Impact (3% Annual Inflation):
Today
$100
buys
100 items
5 years
$100
buys
86 items
10 years
$100
buys
74 items
20 years
$100
buys
55 items
Historical US inflation: 2-3% annually. Hyperinflation events (Zimbabwe 2008, Venezuela 2018) can exceed 1000%/year!

3. Risk & Uncertainty: Bird in Hand Worth Two in Bush

Future payments carry risk: the payer might default, die, or go bankrupt. Economic conditions could change. Immediate payment eliminates this uncertainty. This is why bonds with longer maturities offer higher yields - investors demand compensation for bearing time-related risks.

Risk Factors Affecting Future Value:
• Credit risk: Borrower may not repay (corporate bond default rate: 1-5% for investment grade, 20-40% for junk bonds)
• Economic risk: Recession, market crash, currency devaluation
• Liquidity risk: Might need money urgently before payment date (medical emergency, job loss)
• Opportunity risk: Better investments might appear while waiting
Safety premium: US Treasury bonds (nearly risk-free) yield 3-4%. Corporate bonds yield 5-8%. The difference (1-4%) compensates for additional risk of waiting for corporate payment.

The Fundamental TVM Equation

All time value calculations stem from the future value formula. It's the mathematical expression of compound growth over time.

FV = PV × (1 + r)ⁿ
FV = Future Value (what you'll have)
PV = Present Value (what you have now)
r = Interest rate per period (as decimal: 5% = 0.05)
n = Number of compounding periods
Example Calculation:
PV = $10,000
r = 7% = 0.07
n = 10 years
FV = $10,000 × (1.07)¹⁰
FV = $19,672
Rearranging for Present Value (Discounting):
PV = FV / (1 + r)ⁿ
This is the discounting formula - it tells you what future money is worth today. Essential for evaluating investments, loans, and business decisions.

Applications Across Finance

TVM isn't just theory - it's the foundation of virtually every financial calculation. Understanding it unlocks rational decision-making in personal and business finance.

Personal Finance
• Retirement planning (how much to save)
• Mortgage decisions (rent vs buy)
• Student loans (is degree worth it?)
• Car leases vs purchases
• Insurance decisions (term vs whole life)
Corporate Finance
• Capital budgeting (which projects to fund)
• Bond pricing and yield curves
• Stock valuation (DCF models)
• Mergers & acquisitions
• Lease vs buy equipment decisions
Investing
• Asset allocation over time
• Dollar-cost averaging benefits
• Real estate investment returns
• Annuity vs lump sum decisions
• Tax-deferred account advantages

📈 Interactive: Future Value Calculator

$1,000
5%
10 years
Future Value
$1,629
Your $1,000 grows to $1,629 in 10 years
Total gain: $629(62.9% growth)
Formula: FV = PV × (1 + r)^n
Where: PV = Present Value, r = interest rate, n = number of periods

2. The Power of Compounding

🚀 Understanding Compound Interest vs Simple Interest

Simple Interest: Linear Growth

Simple interest means earning interest only on the original principal, not on accumulated interest. Growth is linear - you earn the same dollar amount each period. This is rare in modern finance but useful for understanding the baseline.

FVsimple = PV × (1 + r × n)
Example: $1,000 at 10% for 5 years
Year 1: $1,000 + $100 = $1,100
Year 2: $1,100 + $100 = $1,200
Year 3: $1,200 + $100 = $1,300
Year 4: $1,300 + $100 = $1,400
Year 5: $1,400 + $100 = $1,500
Total interest: $500 (same $100 every year)
📈
Linear growth trajectory
No acceleration over time

Compound Interest: Exponential Growth

Compound interest means earning interest on your interest. Each period, your earnings grow because the base keeps increasing. This creates exponential growth - the hallmark of wealth building. Albert Einstein allegedly called it "the eighth wonder of the world."

FVcompound = PV × (1 + r)ⁿ
Same: $1,000 at 10% for 5 years
Year 1: $1,000 × 1.10 = $1,100 (+$100)
Year 2: $1,100 × 1.10 = $1,210 (+$110)
Year 3: $1,210 × 1.10 = $1,331 (+$121)
Year 4: $1,331 × 1.10 = $1,464 (+$133)
Year 5: $1,464 × 1.10 = $1,611 (+$147)
Total interest: $611
Bonus from compounding: $111 (22% more!)
🚀
Exponential growth trajectory
Accelerates over time
Interest earns interest!

The Power of Time: Long-Term Compounding Effects

Compounding's true magic appears over long time horizons. The gap between simple and compound interest widens exponentially. This is why starting early matters more than contributing large amounts later.

$10,000 invested at 8% annual return:
YearsSimple InterestCompound InterestDifferenceGap %
5$14,000$14,693+$6935%
10$18,000$21,589+$3,58920%
20$26,000$46,610+$20,61079%
30$34,000$100,627+$66,627196%
40$42,000$217,245+$175,245417%
Key insight: After 40 years, compound interest gives you 5× more than simple interest!
This demonstrates why time in the market beats timing the market. An early start with modest contributions outperforms late start with larger contributions.

Compounding Frequency: More Often = More Growth

Interest can compound at different frequencies: annually, semi-annually, quarterly, monthly, daily, or continuously. More frequent compounding means interest is added to principal more often, creating slightly higher returns even with the same stated annual rate.

FV = PV × (1 + r/m)^(m×n)
where m = compounding periods per year
$10,000 at 12% annual rate for 10 years:
Annual (m=1):
$31,058
Semi-annual (m=2):
$32,071 (+$1,013)
Quarterly (m=4):
$32,620 (+$1,562)
Monthly (m=12):
$33,004 (+$1,946)
Daily (m=365):
$33,194 (+$2,136)
Continuous (m=∞):
$33,201 (+$2,143)
Diminishing returns: Going from annual to daily adds $2,136 (6.9%). Daily to continuous adds only $7 (0.02%).
Continuous compounding formula: FV = PV × e^(rt), where e ≈ 2.71828 (Euler's number)

The Rule of 72: Mental Math for Doubling Time

The Rule of 72 is a quick approximation: divide 72 by your annual return percentage to estimate how many years it takes to double your money. Remarkably accurate for rates between 6-10%.

Years to Double ≈ 72 / Annual Rate (%)
3%
72 / 3 =
24 years
6%
72 / 6 =
12 years
9%
72 / 9 =
8 years
12%
72 / 12 =
6 years
Real-world examples:
• S&P 500 historical average (~10% annually): Money doubles every ~7.2 years
• High-yield savings (4% today): Money doubles every 18 years
• Inflation (3% average): Purchasing power halves every 24 years!
Accuracy check: At 8%, Rule of 72 says 9 years. Actual: 9.01 years. Impressively close!

Real-World Impact: Warren Buffett's Wealth

Warren Buffett's fortune demonstrates compounding's power. He began investing at age 11. Of his ~$100 billion net worth, 99% was accumulated after his 50th birthday. Not because he got better at investing, but because exponential growth accelerates over time.

Early Years (Ages 11-30):
• Started with $114,000 (in 2023 dollars)
• By age 30: ~$1 million
• Doubling time: ~3.5 years
21.5% annual return
Later Years (Ages 50-93):
• Age 50 wealth: ~$376 million
• Age 93 wealth: ~$100 billion
• Same 21.5% annual return
266× growth (vs 9× earlier)
The lesson: Time is the most valuable asset in investing. Even modest returns compound into extraordinary wealth given enough time. This is why starting in your 20s with $500/month beats starting in your 40s with $1,500/month.

🔄 Interactive: Compounding Frequency

With annual compounding
$1,629
Extra earnings vs annual
$0
Rule of 72: Years to double
14.4 years
💡 More frequent compounding = higher returns! Daily compounding earns more than annual because interest is calculated more often, and you earn "interest on interest" sooner.

💵 Interactive: Regular Savings Plan

$1,000
$200
Total Invested
$25,000
Future Value
$32,703
Total Earnings
$7,703
3%
73%
24%
Initial: $1,000Contributions: $24,000Earnings: $7,703

3. Present Value: What's Future Money Worth Today?

⏮️ Discounting: Bringing Future Value Back to Today

What is Discounting?

Discounting is the inverse of compounding. While compounding projects today's money forward to find future value, discounting brings future money back to the present to find what it's worth today. This is essential for comparing cash flows across different time periods.

Compounding (Forward)
FV = PV × (1 + r)ⁿ
• Starts with Present Value
• Projects into the future
• Adds growth (interest)
• Result: Future Value
Discounting (Backward)
PV = FV / (1 + r)ⁿ
• Starts with Future Value
• Brings back to present
• Removes growth (discount)
• Result: Present Value
Key concept: $(1 + r)ⁿ$ in the denominator means future money is divided by compound growth factors, making it smaller. The further in the future, the less it's worth today.

Why Discount Future Cash Flows?

Future payments must be discounted to reflect their true value today for several reasons: opportunity cost, inflation, risk, and time preference. Without discounting, you can't fairly compare cash flows occurring at different times.

1. Opportunity Cost
Money received today can earn returns elsewhere. $1,000 in 5 years isn't worth $1,000 today because you miss 5 years of investment returns. If you could earn 8% annually, that future $1,000 is only worth $681 today (what you'd need to invest now to reach $1,000 in 5 years).
2. Inflation Protection
With 3% inflation, $1,000 in 10 years only buys what $744 buys today. Discounting accounts for purchasing power erosion. Nominal cash flows must be adjusted to real (inflation-adjusted) values.
3. Risk Premium
Future payments are uncertain (default risk, market volatility, economic changes). The discount rate includes a risk premium - riskier cash flows need higher discount rates. US Treasury (safe): 4%, Corporate bonds (riskier): 7%, Startup equity (very risky): 25%+.
4. Time Preference
People prefer money sooner (immediate gratification, flexibility, life uncertainty). This psychological factor is why lotteries offer lump sum (immediate) or annuity (spread over 30 years) - most choose lump sum even though annuity totals more.

Discount Rate Selection: The Critical Choice

The discount rate (also called required return or hurdle rate) is arguably the most important input in financial analysis. Small changes dramatically affect valuations. A project worth $1 million at 8% might be worthless at 12%.

How to Choose the Right Discount Rate:
ContextDiscount Rate ApproachTypical Range
Personal FinanceExpected return on alternative investments (savings rate, index funds, etc.)3-10%
Corporate ProjectsWeighted Average Cost of Capital (WACC) - blend of debt & equity costs6-15%
Stock ValuationCost of Equity (CAPM: risk-free rate + beta × market risk premium)8-20%
Government ProjectsSocial discount rate (reflects society's time preference)2-7%
Venture CapitalHigh risk premium reflecting startup failure rates25-40%
Rule of thumb: Higher risk = Higher discount rate
• US Treasury: 3-4% (nearly risk-free baseline)
• S&P 500: 10% (historical equity market return)
• Real estate: 8-12% (property type & location dependent)
• Emerging markets: 15-25% (country & currency risk)

Net Present Value (NPV): The Decision Rule

Net Present Value sums all cash flows (inflows and outflows) after discounting them to present value. It's the gold standard for investment decisions because it accounts for timing, magnitude, and risk of all cash flows.

NPV = Σ [CFₜ / (1 + r)ᵗ] - Initial Investment
Where CFₜ = cash flow in period t, r = discount rate, t = time period
NPV > 0
✓
Accept the project
Investment creates value
Returns exceed required rate
Increases shareholder wealth
NPV = 0
≈
Neutral decision
Exactly breaks even
Returns equal required rate
No value created/destroyed
NPV < 0
✗
Reject the project
Investment destroys value
Returns below required rate
Better to invest elsewhere
Simple Example: Should you buy a rental property?
• Initial investment: -$200,000 (down payment + closing)
• Year 1-5 net cash flow: +$10,000/year (rent - expenses)
• Year 5 sale proceeds: +$250,000 (property value increased)
• Discount rate: 8% (your required return)
NPV = -$200,000 + $10,000/(1.08)¹ + ... + ($10,000+$250,000)/(1.08)⁵
NPV = +$16,847
Decision: Buy the property! Creates $16,847 of value today.

Discount Rate Sensitivity: Small Changes, Big Impact

Present values are extremely sensitive to discount rate changes, especially for cash flows far in the future. This is why tech stocks (valued on distant profits) swing wildly when interest rates change.

PV of $100,000 received in 30 years:
5%
discount rate
$23,138
8%
discount rate
$9,938
12%
discount rate
$3,338
15%
discount rate
$1,510
Going from 5% to 15% reduces present value by 93%!
This explains why:
• Rising interest rates crush growth stock valuations
• Long-term bonds are more price-sensitive than short-term bonds
• Climate change policies use low discount rates (1-3%) to value distant future benefits
• Pension funds struggle when discount rates fall (liabilities increase)

Real-World Application: Bond Pricing

Bond prices perfectly demonstrate discounting in action. A bond's price is simply the present value of all future coupon payments plus the face value at maturity, discounted at the market interest rate (yield).

Example: 10-year bond with 5% coupon, $1,000 face value
• Pays $50/year for 10 years (5% × $1,000)
• Returns $1,000 principal at maturity
• Market yield changes from 5% to 7% (rates rose)
When issued (5% yield):
Price = PV of coupons + PV of principal
Price = $50×[PVA,5%,10yr] + $1,000/(1.05)¹⁰
Price = $1,000 (par value)
After rate increase (7% yield):
Price = PV of coupons + PV of principal
Price = $50×[PVA,7%,10yr] + $1,000/(1.07)¹⁰
Price = $859 (14.1% loss!)
Key lesson: When discount rates (market yields) rise, present values fall. This is why bonds lose value when interest rates increase, and why the Fed's rate decisions immediately impact all asset prices.

⏪ Interactive: Discount Rate Impact

8%
$10,000 in 1 year
$9,259
today
$10,000 in 5 years
$6,806
today
$10,000 in 10 years
$4,632
today
$10,000 in 20 years
$2,145
today
💡 Higher discount rate = lower present value. If you need 8% returns, $10,000 in 10 years is only worth $4,632 today.

📊 Interactive: Net Present Value (NPV)

NPV helps decide if an investment is worthwhile by calculating today's value of future cash flows.

8%
Future Cash Flows (5 years):
Year 1
Year 2
Year 3
Year 4
Year 5
Net Present Value
$5,830
✓ Accept Investment

4. Real-World Scenarios

🌍 TVM in Action: Life's Major Financial Decisions

Retirement Planning: Building Your Nest Egg

Retirement planning is perhaps TVM's most important personal application. You need to accumulate enough wealth (future value) through regular contributions to sustain decades of withdrawals. The key questions: How much do I need? How much should I save monthly? When can I retire?

The 4% Rule & How TVM Validates It:
The famous "4% rule" says you can safely withdraw 4% of your retirement portfolio annually (adjusted for inflation) without running out of money over a 30-year retirement.
Why it works (TVM math):
• Portfolio returns ~7% annually (historical stock/bond mix)
• Inflation ~3% annually (reduces real return to ~4%)
• Withdraw 4% annually → portfolio maintains purchasing power
• Present value of 30 years of 4% withdrawals ≈ 100% of portfolio
Accumulation Phase (Working Years):
Goal: $1 million at retirement
Time horizon: 30 years
Expected return: 8%/year
Current savings: $0
Monthly contribution needed: $670
Total invested: $241,200
Growth from compounding: $758,800!
Distribution Phase (Retirement):
Portfolio: $1 million
Withdrawal rate: 4%
Annual income: $40,000
Adjusted for inflation: 3%/year
Year 1: $40,000
Year 10: $52,190
Year 30: $97,092
Purchasing power maintained!
Key insight: Starting early is crucial. $670/month for 30 years beats $2,000/month for 10 years ($241K vs $240K invested, but first scenario yields $1M vs $366K due to compounding time).

College Savings: The 529 Plan Strategy

College costs grow faster than inflation (~5% annually vs 3% general inflation). TVM helps determine how much to save monthly in a 529 plan to cover future tuition. Tax-advantaged growth makes these plans powerful compounding vehicles.

Real Scenario: Baby born today, private university in 18 years
The Challenge:
• Today's cost: $60,000/year × 4 years = $240,000
• Inflation: 5% annually for 18 years
• Future cost: $240,000 × (1.05)^18 = $578,330!
College costs more than double in 18 years
The Solution:
Target: $578,330 in 18 years
529 plan return: 7%/year
Time to save: 18 years (216 months)
Monthly savings needed: $1,390
Total invested: $300,240
Growth from compounding: $278,090
Tax savings: ~$55,000
Alternative strategies:
• Lump sum today: Invest $177,090 now at 7% → $578,330 in 18 years (if you have it)
• Delay start: Wait 8 years, need $2,700/month for remaining 10 years (costs way more!)
• State university: $25K/year today → $241K future → $580/month savings

Business Valuation: Discounted Cash Flow (DCF) Models

DCF valuation is how Wall Street prices stocks and companies. The principle: a business is worth the present value of all future free cash flows it will generate. This is pure TVM applied to corporate finance.

DCF Model Structure:
Enterprise Value = Σ [FCFₜ / (1 + WACC)ᵗ] + Terminal Value / (1 + WACC)ⁿ
FCF = Free Cash Flow (cash after all expenses & reinvestment)
WACC = Weighted Average Cost of Capital (discount rate, typically 8-12%)
t = Year (usually forecast 5-10 years)
Terminal Value = Value beyond forecast period (often 60-80% of total)
Example: Valuing a SaaS Startup
YearFCFDiscount (10%)Present Value
1$5M1.100$4.55M
2$8M1.210$6.61M
3$12M1.331$9.02M
4$18M1.464$12.30M
5$25M1.611$15.52M
Terminal$625M1.611$388.00M
Total Enterprise Value:$436M
Note how terminal value dominates (89% of total value)
• Change WACC from 10% → 12%: Value drops to $369M (-15%)
• Change terminal growth from 3% → 4%: Value rises to $485M (+11%)
This sensitivity is why analysts argue endlessly about discount rate assumptions!

Personal Finance Decisions: Buy vs Lease, Mortgage Choices

Everyday financial choices involve TVM: car leasing, mortgage terms, credit card payments, rent vs buy decisions. Without understanding present value, it's easy to make costly mistakes by focusing only on monthly payments.

Case Study 1: 15-Year vs 30-Year Mortgage
30-Year Mortgage:
Loan amount: $400,000
Interest rate: 7.0%
Monthly payment: $2,661
Total paid: $958,000
Total interest: $558,000
Lower monthly payment, but pay 140% more in interest!
15-Year Mortgage:
Loan amount: $400,000
Interest rate: 6.5%
Monthly payment: $3,484
Total paid: $627,000
Total interest: $227,000
Save $331,000 in interest - but can you afford $3,484/month?
But wait... TVM says it's more nuanced!
The $331K savings is nominal (future) dollars. What's the present value of this advantage?
• Monthly difference: $3,484 - $2,661 = $823 more/month for 15-year
• If you invested that $823/month at 8% return instead...
• After 15 years: $288,000 investment account
• After 30 years (investing for next 15 too): $916,000!
Conclusion: If you can earn > 7% investing, mathematically better to take 30-year mortgage and invest the difference. But behavioral finance says most people won't actually invest it!
Case Study 2: Car Lease vs Buy
Lease Option:
Monthly payment: $450
Term: 36 months
Down payment: $2,000
End-of-lease value: $0
Total cost: $18,200
Buy Option:
Purchase price: $35,000
Down payment: $7,000
Finance: $28,000 at 5%, 36 mo
Monthly payment: $841
Resale after 3 years: $20,000
Total paid: $37,276
Net cost: $17,276
PV analysis: Lease looks cheaper ($18,200 vs $17,276 - virtually equal). But buying gives you $20,000 asset! If you keep car 10 years, ownership wins dramatically. Leasing only makes sense if you value always having new car > $924/year cost difference.

Capital Budgeting: How Companies Decide Which Projects to Fund

Capital budgeting is how businesses allocate scarce capital across competing investment opportunities. NPV analysis determines which projects create shareholder value. This is TVM driving billion-dollar decisions.

Real Example: Manufacturing Company Choosing Between Two Projects
Project A: New Factory
Initial investment: -$50M
Year 1-10 cash flow: +$8M/year
Salvage value (year 10): +$10M
WACC: 10%
Total cash inflows: $90M
NPV @ 10%: +$3.14M
IRR: 11.8%
Payback: 6.25 years
Project B: Technology Upgrade
Initial investment: -$30M
Year 1-5 cash flow: +$10M/year
Year 6-10 cash flow: +$5M/year
WACC: 10%
Total cash inflows: $75M
NPV @ 10%: +$13.58M
IRR: 18.9%
Payback: 3.0 years
Decision Analysis:
• Project B wins! Higher NPV ($13.58M vs $3.14M) creates more shareholder value
• Project B also has higher IRR (18.9% vs 11.8%) and faster payback (3 vs 6.25 years)
• If company has $50M to invest: Do Project B ($30M) + smaller projects with remaining $20M
• Project A only attractive if WACC drops below ~12% or strategic value beyond financials
NPV rule: Accept all projects with NPV > 0, rank by NPV when capital constrained. This maximizes firm value.
Why TVM matters here:
• Without discounting, both projects appear profitable (cash inflows > initial investment)
• TVM reveals timing differences: Project B front-loads cash flows (more valuable)
• Higher discount rates favor projects with quicker payoffs (less risk)
Real companies run sensitivity analysis: test NPV at discount rates from 8-15% to assess risk

🎯 Interactive: Life Scenarios

Retirement Planning:
At 5% annual return:
$32,703
Total invested:
$25,000

📉 Interactive: Inflation Impact

3%
Nominal Future Value
$1,629
Real Future Value (Inflation-Adjusted)
$1,212
Purchasing Power Loss
25.6%
⚠️ Inflation erodes purchasing power! Even though you'll have $1,629, it will only buy what $1,212 buys today.

💰 Interactive: Lump Sum vs Annuity

You won $1,000,000! Take it all now or receive $50,000/year for 30 years?

Lump Sum Value
$1,000,000
Annuity PV @ 8%
$562,889
Better Choice
Lump Sum
At 8% discount rate, the annuity payments are worth $562,889 today. Take the lump sum and invest it!

⏰ Interactive: Time Horizon Matters

See how different time horizons affect your wealth at 5% annual return

5 Years$1,276
10 Years$1,629
20 Years$2,653
30 Years$4,322
40 Years$7,040
⚡ Time is your greatest asset! Starting 10 years earlier can double or triple your final wealth due to compound growth.

5. Key Takeaways

💰

Money Today > Money Tomorrow

A dollar today is always worth more than a dollar tomorrow because you can invest it and earn returns. This is the foundation of all financial decisions.

🔄

Compound Interest is Magic

Einstein called it the "eighth wonder of the world." More frequent compounding and longer time horizons dramatically increase wealth through exponential growth.

⏪

Discounting Finds True Value

Present Value (PV) tells you what future cash flows are worth today. Use it to compare investment opportunities, lottery winnings, or salary offers with different payment structures.

📉

Don't Forget Inflation

Your nominal returns mean nothing if inflation erodes purchasing power. Always calculate real returns (nominal - inflation) to understand true wealth growth.

⏰

Start Early, Save Consistently

Time is the most powerful variable in the TVM equation. Starting 10 years earlier with regular contributions can result in 2-3x more wealth at retirement.

🎯

Use NPV for Decisions

Net Present Value (NPV) is the gold standard for evaluating investments, projects, and business decisions. Positive NPV = create value. Negative NPV = destroy value.