Corporate Finance Fundamentals
Module 3: Time Value of Money
Module Overview
Welcome to Module 3—arguably the most important module in the entire course. The time value of money (TVM) is the foundation upon which all of corporate finance is built. Master these concepts, and everything else will fall into place.
Don't worry about the math. We'll take it step by step, building from simple to complex. By the end, you'll be calculating present values, future values, and solving real financial problems with confidence.
Learning Objectives:
By the end of this module, you will be able to:
- Calculate future value using compound interest
- Calculate present value by discounting future cash flows
- Work with different compounding frequencies
- Value annuities (regular payment streams)
- Value perpetuities (infinite payment streams)
- Solve for interest rates and time periods
- Apply TVM to real-world financial decisions
- Use Excel for complex TVM calculations
Estimated Time: 5-6 hours
What You'll Need:
- Calculator (financial calculator ideal, but regular calculator works)
- Excel or Google Sheets
- Patience and practice
3.1 The Foundation: Why Time Value Matters
The Core Principle (Review and Expansion)
A dollar today is worth more than a dollar tomorrow.
We introduced this in Module 1. Now let's understand it deeply.
The Three Reasons (Detailed)
1. Opportunity Cost (Investment Opportunity)
Money can be invested to earn a return. If you have $100 today:
- Invest at 10% per year → $110 in one year
- Invest at 10% per year → $121 in two years
- The opportunity to invest is why today's money is more valuable
Example: You're offered:
- Option A: $1,000 today
- Option B: $1,000 in one year
If you can earn 8% on investments:
- Take Option A: $1,000 today → invest → $1,080 in one year
- Take Option B: $1,000 in one year → $1,000 in one year
Option A is worth $80 more because of investment opportunity.
2. Inflation (Purchasing Power)
Money loses purchasing power over time as prices rise.
Example: With 3% annual inflation:
- $1,000 today buys 100 units of goods
- $1,000 in 10 years buys only 74 units of goods
Your money can buy 26% less due to inflation!
3. Risk and Uncertainty
Future payments are uncertain. The further in the future, the less certain.
Example: Would you rather have:
- $10,000 guaranteed today
- $10,000 promised in 20 years from a friend
The promise in 20 years involves risks:
- Your friend might not be able to pay
- Your friend might not exist in 20 years
- Circumstances might change
Present money is certain; future money is not.
The Discount Rate
To compare money across time, we need a discount rate (also called the interest rate, required return, or opportunity cost).
The discount rate reflects:
- What you could earn on alternative investments (opportunity cost)
- Expected inflation
- Risk of the cash flow
- Your time preference
Higher discount rate = Future money is worth less today Lower discount rate = Future money is worth more today
This single concept—discounting—is the key to all of finance.
3.2 Future Value: Compounding Forward
Simple Interest vs. Compound Interest
Simple Interest: Interest calculated only on the principal
Simple Interest = Principal × Rate × Time
Example: $1,000 at 10% simple interest for 3 years:
- Year 1: $1,000 + $100 = $1,100
- Year 2: $1,100 + $100 = $1,200
- Year 3: $1,200 + $100 = $1,300
You earn $100 each year on the original $1,000.
Compound Interest: Interest calculated on principal plus accumulated interest
FV = PV × (1 + r)^n
Example: $1,000 at 10% compound interest for 3 years:
- Year 1: $1,000 × 1.10 = $1,100
- Year 2: $1,100 × 1.10 = $1,210
- Year 3: $1,210 × 1.10 = $1,331
You earn interest on interest! This is compounding.
Difference: $1,331 (compound) vs. $1,300 (simple) = $31 extra from compounding
Over long periods, compounding creates enormous differences.
The Future Value Formula
FV = PV × (1 + r)^n
Where:
- FV = Future Value (what money grows to)
- PV = Present Value (starting amount)
- r = Interest rate per period (as a decimal)
- n = Number of periods
Example 1: Basic Future Value
You invest $5,000 at 8% per year for 10 years. What's the future value?
FV = $5,000 × (1 + 0.08)^10
FV = $5,000 × (1.08)^10
FV = $5,000 × 2.1589
FV = $10,794.50
Your $5,000 more than doubles in 10 years at 8%.
Example 2: The Power of Time
$10,000 invested at 10% per year:
- After 10 years: $10,000 × (1.10)^10 = $25,937
- After 20 years: $10,000 × (1.10)^20 = $67,275
- After 30 years: $10,000 × (1.10)^30 = $174,494
Notice how growth accelerates over time—that's compounding!
Example 3: The Power of Rate
$10,000 invested for 20 years:
- At 5%: $10,000 × (1.05)^20 = $26,533
- At 10%: $10,000 × (1.10)^20 = $67,275
- At 15%: $10,000 × (1.15)^20 = $163,665
Doubling the rate more than doubles the result over time.
The Rule of 72
A quick way to estimate doubling time:
Years to Double ≈ 72 / Interest Rate
Examples:
- At 6%: 72 / 6 = 12 years to double
- At 9%: 72 / 9 = 8 years to double
- At 12%: 72 / 12 = 6 years to double
Reverse calculation: If you want to double your money in 10 years:
Required Rate = 72 / 10 = 7.2%
Compounding Frequency
So far, we've assumed annual compounding. But interest can compound more frequently.
Different Compounding Frequencies:
- Annual: Once per year
- Semi-annual: Twice per year
- Quarterly: Four times per year
- Monthly: Twelve times per year
- Daily: 365 times per year
- Continuous: Infinitely often
Formula for Different Frequencies:
FV = PV × (1 + r/m)^(n×m)
Where:
- m = compounding periods per year
Example: $10,000 at 12% for 5 years
Annual compounding (m=1):
FV = $10,000 × (1 + 0.12/1)^(5×1)
FV = $10,000 × (1.12)^5
FV = $17,623
Quarterly compounding (m=4):
FV = $10,000 × (1 + 0.12/4)^(5×4)
FV = $10,000 × (1.03)^20
FV = $18,061
Monthly compounding (m=12):
FV = $10,000 × (1 + 0.12/12)^(5×12)
FV = $10,000 × (1.01)^60
FV = $18,167
Daily compounding (m=365):
FV = $10,000 × (1 + 0.12/365)^(5×365)
FV = $10,000 × (1.000329)^1825
FV = $18,221
Key Insight: More frequent compounding increases returns, but the effect diminishes as frequency increases. Going from annual to monthly matters more than going from monthly to daily.
Effective Annual Rate (EAR)
When interest compounds more than once per year, the effective annual rate is higher than the stated (nominal) rate.
EAR = (1 + r/m)^m - 1
Example: 12% nominal rate, monthly compounding
EAR = (1 + 0.12/12)^12 - 1
EAR = (1.01)^12 - 1
EAR = 1.1268 - 1
EAR = 0.1268 = 12.68%
The effective annual rate is 12.68%, even though the stated rate is 12%.
Why it matters: Use EAR to compare investments with different compounding frequencies.
Example comparison:
- Investment A: 12% annual compounding → EAR = 12%
- Investment B: 11.8% monthly compounding → EAR = 12.46%
Investment B is actually better despite the lower nominal rate!
3.3 Present Value: Discounting Backward
The Flip Side: What's Future Money Worth Today?
If we can compound forward, we can discount backward.
Question: What's $10,000 received in 5 years worth today at 10% discount rate?
Answer: The amount that, if invested today at 10%, would grow to $10,000 in 5 years.
The Present Value Formula
PV = FV / (1 + r)^n
Or equivalently:
PV = FV × (1 + r)^-n
Where:
- PV = Present Value (value today)
- FV = Future Value (future amount)
- r = Discount rate per period
- n = Number of periods
Notice: This is just the future value formula rearranged!
Present Value Examples
Example 1: Single Future Amount
You'll receive $20,000 in 7 years. What's it worth today at 9% discount rate?
PV = $20,000 / (1.09)^7
PV = $20,000 / 1.8280
PV = $10,942
Interpretation: $10,942 today is equivalent to $20,000 in 7 years at 9% discount rate.
Example 2: Multiple Discount Rates
$50,000 received in 10 years is worth:
At 5% discount rate:
PV = $50,000 / (1.05)^10 = $30,696
At 10% discount rate:
PV = $50,000 / (1.10)^10 = $19,277
At 15% discount rate:
PV = $50,000 / (1.15)^10 = $12,360
Key Insight: Higher discount rates make future money worth less today. This makes intuitive sense—if you can earn a higher return elsewhere, future money is less attractive.
Example 3: Near vs. Distant Future
$1,000 at 10% discount rate:
Received in 1 year:
PV = $1,000 / (1.10)^1 = $909
Received in 5 years:
PV = $1,000 / (1.10)^5 = $621
Received in 10 years:
PV = $1,000 / (1.10)^10 = $386
Received in 30 years:
PV = $1,000 / (1.10)^30 = $57
Key Insight: Distant cash flows are worth very little today. The farther the future, the steeper the discount.
Present Value of Multiple Cash Flows
Real-world situations usually involve multiple cash flows at different times.
Rule: Calculate the present value of each cash flow separately, then add them up.
Total PV = PV₁ + PV₂ + PV₃ + ... + PVₙ
Example: Investment Opportunity
You're considering an investment that will pay:
- $5,000 in Year 1
- $8,000 in Year 2
- $10,000 in Year 3
Your required return is 12%. What's the maximum you should pay?
Calculate each PV:
PV₁ = $5,000 / (1.12)^1 = $4,464
PV₂ = $8,000 / (1.12)^2 = $6,378
PV₃ = $10,000 / (1.12)^3 = $7,118
Total PV:
Total = $4,464 + $6,378 + $7,118 = $17,960
Conclusion: The investment is worth $17,960 today. Don't pay more than this!
Example: Lottery Winnings
You win the lottery! Two payout options:
- Option A: $5 million today
- Option B: $1 million per year for 6 years
Which is better at 8% discount rate?
Option A: $5,000,000 (obviously)
Option B:
Year 1: $1M / (1.08)^1 = $925,926
Year 2: $1M / (1.08)^2 = $857,339
Year 3: $1M / (1.08)^3 = $793,832
Year 4: $1M / (1.08)^4 = $735,030
Year 5: $1M / (1.08)^5 = $680,583
Year 6: $1M / (1.08)^6 = $630,170
----------
Total PV: $4,622,880
Conclusion: Take Option A! The lump sum of $5M is worth $377,120 more than the installment payments.
Present Value and Decision Making
Present value allows us to compare alternatives that involve cash flows at different times.
General Decision Rule:
- Accept investments where PV of benefits > PV of costs
- Choose the option with the highest net present value (NPV)
- NPV = PV of benefits - PV of costs
Example: Should you get an MBA?
Costs:
- Tuition: $100,000 (paid upfront)
- Lost income: $80,000/year for 2 years
Benefits:
- Higher salary: $30,000/year extra for 35 years (assume you work until 65)
Discount rate: 7%
PV of Costs:
Tuition: $100,000 (Year 0)
Lost income Year 1: $80,000 / (1.07)^1 = $74,766
Lost income Year 2: $80,000 / (1.07)^2 = $69,875
Total Cost PV = $244,641
PV of Benefits: Extra $30,000/year for years 3-37 (35 years of work)
This requires annuity formula (coming next section)
PV ≈ $370,000 (approximate)
NPV:
NPV = $370,000 - $244,641 = $125,359
Conclusion: Positive NPV of $125,359 suggests the MBA is a good investment!
(Note: This is simplified. Real analysis would include career growth, networking benefits, etc.)
3.4 Annuities: Valuing Regular Payment Streams
What is an Annuity?
An annuity is a series of equal payments made at regular intervals.
Examples:
- Mortgage payments: $2,000/month for 30 years
- Car loan payments: $400/month for 5 years
- Retirement savings: $500/month for 40 years
- Bond interest: $50/year for 20 years
- Lottery winnings: $1M/year for 20 years
Annuities are everywhere in finance!
Types of Annuities
1. Ordinary Annuity (Annuity in Arrears)
- Payments at the END of each period
- Most common type
- Example: Loan payments at month-end
2. Annuity Due
- Payments at the BEGINNING of each period
- Example: Rent payments, lease payments
We'll focus on ordinary annuities (the standard).
Present Value of an Annuity
Instead of calculating PV for each payment separately, we can use a formula:
PV = PMT × [(1 - (1 + r)^-n) / r]
Or equivalently:
PV = PMT × [1 - 1/(1 + r)^n] / r
Where:
- PV = Present value of the annuity
- PMT = Payment per period
- r = Interest rate per period
- n = Number of periods
The bracketed part is called the Present Value Interest Factor for an Annuity (PVIFA).
Example 1: Lottery Winnings
You win $50,000 per year for 10 years. What's the present value at 8%?
PV = $50,000 × [(1 - (1.08)^-10) / 0.08]
PV = $50,000 × [(1 - 0.4632) / 0.08]
PV = $50,000 × [0.5368 / 0.08]
PV = $50,000 × 6.710
PV = $335,500
Interpretation: Receiving $50,000/year for 10 years is equivalent to receiving $335,500 today at 8% discount rate.
Example 2: Retirement Planning
You want to withdraw $60,000 per year for 25 years in retirement. How much do you need at 6% annual return?
PV = $60,000 × [(1 - (1.06)^-25) / 0.06]
PV = $60,000 × [(1 - 0.2330) / 0.06]
PV = $60,000 × [0.7670 / 0.06]
PV = $60,000 × 12.783
PV = $766,980
Answer: You need about $767,000 saved to withdraw $60,000/year for 25 years.
Example 3: Car Loan
You're offered a car loan: $500/month for 60 months. Interest rate is 6% annual (0.5% monthly). What's the maximum car price this can finance?
r = 0.06 / 12 = 0.005 (monthly rate)
n = 60 months
PV = $500 × [(1 - (1.005)^-60) / 0.005]
PV = $500 × [(1 - 0.7414) / 0.005]
PV = $500 × [0.2586 / 0.005]
PV = $500 × 51.726
PV = $25,863
Answer: $500/month for 5 years at 6% can finance a car worth $25,863.
Future Value of an Annuity
Sometimes we want to know what regular savings will grow to.
FV = PMT × [((1 + r)^n - 1) / r]
The bracketed part is the Future Value Interest Factor for an Annuity (FVIFA).
Example 1: Retirement Savings
You save $500/month for 30 years at 8% annual return (0.667% monthly). How much will you have?
r = 0.08 / 12 = 0.00667
n = 30 × 12 = 360 months
FV = $500 × [((1.00667)^360 - 1) / 0.00667]
FV = $500 × [(11.024 - 1) / 0.00667]
FV = $500 × [10.024 / 0.00667]
FV = $500 × 1,502.28
FV = $751,140
Wow! $500/month becomes $751,140 in 30 years!
You contributed: $500 × 360 = $180,000 Investment growth: $751,140 - $180,000 = $571,140
Compounding turned $180,000 into $751,140!
Example 2: College Savings
You save $300/month for 18 years for your child's college at 7% annual return.
r = 0.07 / 12 = 0.00583
n = 18 × 12 = 216 months
FV = $300 × [((1.00583)^216 - 1) / 0.00583]
FV = $300 × [(3.480 - 1) / 0.00583]
FV = $300 × 425.4
FV = $127,620
Result: $300/month for 18 years grows to $127,620.
Solving for Payment (PMT)
Often we know the loan amount (PV) and need to find the payment.
Rearrange the PV annuity formula:
PMT = PV × [r / (1 - (1 + r)^-n)]
Example: Mortgage Payment
You buy a house with a $300,000 mortgage at 5% for 30 years. What's the monthly payment?
PV = $300,000
r = 0.05 / 12 = 0.00417
n = 30 × 12 = 360 months
PMT = $300,000 × [0.00417 / (1 - (1.00417)^-360)]
PMT = $300,000 × [0.00417 / (1 - 0.2314)]
PMT = $300,000 × [0.00417 / 0.7686]
PMT = $300,000 × 0.005423
PMT = $1,610
Answer: Monthly payment is $1,610.
Total paid: $1,610 × 360 = $579,600 Interest paid: $579,600 - $300,000 = $279,600
You pay almost as much in interest as in principal!
Solving for Interest Rate (r)
Sometimes we know PV, PMT, and n, but need to find the rate. This requires trial and error or a financial calculator/Excel.
Example: Investment Return
You invest $10,000 today and receive $2,000/year for 6 years. What's your return?
We need to solve:
$10,000 = $2,000 × [(1 - (1 + r)^-6) / r]
This requires iteration or financial calculator. The answer is approximately 5.47%.
Using Excel: =RATE(6, -2000, 10000) returns 5.47%
Solving for Number of Periods (n)
Example: How long to pay off a loan?
You have a $15,000 loan at 8% annual (0.667% monthly) and can pay $300/month. How long to pay it off?
$15,000 = $300 × [(1 - (1.00667)^-n) / 0.00667]
Solving (using logarithms or Excel): n ≈ 60 months = 5 years
Using Excel: =NPER(0.00667, -300, 15000) returns 60 months
3.5 Perpetuities: Forever Cash Flows
What is a Perpetuity?
A perpetuity is an annuity that continues forever—infinite payments.
Examples:
- Certain types of preferred stock (fixed dividend forever)
- British consols (government bonds with no maturity)
- Endowments (spend interest forever, preserve principal)
Present Value of a Perpetuity
The formula is surprisingly simple:
PV = PMT / r
Where:
- PV = Present value
- PMT = Payment per period
- r = Discount rate per period
Why so simple? As n approaches infinity in the annuity formula, (1 + r)^-n approaches zero, leaving just PMT/r.
Example 1: Preferred Stock
A preferred stock pays $8 per share annually forever. If your required return is 10%, what's it worth?
PV = $8 / 0.10 = $80 per share
Example 2: Endowment
You want to endow a scholarship that pays $50,000/year forever. If the endowment earns 5% per year, how much do you need to donate?
PV = $50,000 / 0.05 = $1,000,000
You need a $1 million endowment.
How it works:
- $1,000,000 × 5% = $50,000 per year
- Pay out $50,000 annually
- Principal remains $1,000,000 forever
Example 3: Business Valuation
A business generates $200,000 per year in perpetuity. What's it worth at 12% discount rate?
PV = $200,000 / 0.12 = $1,667,000
Growing Perpetuity
What if payments grow at a constant rate forever?
PV = PMT / (r - g)
Where:
- g = Growth rate per period
- r must be greater than g
Example: Growing Dividend
A stock pays a $5 dividend this year, growing at 3% annually forever. Required return is 9%. What's the stock worth?
PV = $5 / (0.09 - 0.03)
PV = $5 / 0.06
PV = $83.33
This is the Gordon Growth Model, used for stock valuation (we'll see it again in Module 9).
3.6 Mixed Cash Flow Streams
Real-world situations often combine different types of cash flows.
Strategy: Break Into Components
- Identify each component (single amount, annuity, perpetuity)
- Calculate PV of each component separately
- Add them up
Example 1: Bond Valuation
A bond pays:
- $50 interest every year for 10 years
- $1,000 principal at maturity (Year 10)
Required return: 6%
Component 1: Interest payments (annuity)
PV₁ = $50 × [(1 - (1.06)^-10) / 0.06]
PV₁ = $50 × 7.360
PV₁ = $368
Component 2: Principal (single amount)
PV₂ = $1,000 / (1.06)^10
PV₂ = $1,000 / 1.791
PV₂ = $558
Total Bond Value:
PV = $368 + $558 = $926
Example 2: Uneven Cash Flows
An investment pays:
- Year 1: $1,000
- Year 2: $2,000
- Years 3-7: $3,000 per year (annuity)
- Year 8: $5,000
Discount rate: 10%
Individual components:
PV₁ = $1,000 / (1.10)^1 = $909
PV₂ = $2,000 / (1.10)^2 = $1,653
PV₃₋₇ = $3,000 × [(1-(1.10)^-5) / 0.10] / (1.10)^2 = $9,427
PV₈ = $5,000 / (1.10)^8 = $2,331
Total:
PV = $909 + $1,653 + $9,427 + $2,331 = $14,320
Note: PV₃₋₇ needs extra discounting because the annuity starts in Year 3, not Year 1.
Deferred Annuities
Sometimes annuities don't start immediately.
Example: Deferred Retirement
You'll receive $40,000/year for 20 years, but the first payment is in Year 6 (not Year 1). Discount rate: 8%
Step 1: Find PV as of Year 5 (one period before first payment)
PV at Year 5 = $40,000 × [(1 - (1.08)^-20) / 0.08]
PV at Year 5 = $40,000 × 9.818
PV at Year 5 = $392,720
Step 2: Discount back to today
PV today = $392,720 / (1.08)^5
PV today = $392,720 / 1.469
PV today = $267,250
3.7 Using Excel for TVM Calculations
Excel has built-in financial functions that make TVM calculations easy.
Key Excel Functions
1. FV (Future Value)
=FV(rate, nper, pmt, [pv], [type])
- rate: Interest rate per period
- nper: Number of periods
- pmt: Payment per period (enter as negative for outflows)
- pv: Present value (optional, enter as negative for outflows)
- type: 0 for end of period (default), 1 for beginning
Example:
=FV(0.08, 10, -500, -1000)
Returns $16,865 (FV of $1,000 today plus $500/year for 10 years at 8%)
2. PV (Present Value)
=PV(rate, nper, pmt, [fv], [type])
Example:
=PV(0.06, 20, -1000, -10000)
Returns $16,095 (PV of $1,000/year for 20 years plus $10,000 at end, at 6%)
3. PMT (Payment)
=PMT(rate, nper, pv, [fv], [type])
Example (Mortgage):
=PMT(0.05/12, 360, 300000)
Returns -$1,610 (monthly payment on $300k loan at 5% for 30 years)
4. RATE (Interest Rate)
=RATE(nper, pmt, pv, [fv], [type])
Example:
=RATE(5, -2000, 8000)
Returns 7.93% (return on investing $8,000 to get $2,000/year for 5 years)
5. NPER (Number of Periods)
=NPER(rate, pmt, pv, [fv], [type])
Example:
=NPER(0.07, -1000, 50000)
Returns 87 periods (time to pay off $50k at $1,000/period with 7% rate)
6. NPV (Net Present Value)
=NPV(rate, value1, [value2], ...)
Note: NPV assumes first cash flow is at Year 1, not Year 0!
Example:
=NPV(0.10, 1000, 2000, 3000)
Returns $4,815 (PV of $1k, $2k, $3k at years 1, 2, 3)
If Year 0 cash flow exists (like initial investment), add it separately:
=-10000 + NPV(0.10, 3000, 4000, 5000)
Important Excel Conventions
Sign Convention:
- Negative = Cash outflow (you pay)
- Positive = Cash inflow (you receive)
Be consistent! If you enter PV as negative (you invest), PMT and FV should be positive (you receive).
Building a TVM Calculator in Excel
Create a simple template:
Input Section:
Present Value (PV): [enter amount]
Future Value (FV): [enter amount]
Payment (PMT): [enter amount]
Interest Rate (r): [enter %]
Number of Periods (n): [enter number]
Calculation Section:
To find FV: =FV(r, n, PMT, -PV)
To find PV: =PV(r, n, PMT, -FV)
To find PMT: =PMT(r, n, -PV, FV)
To find Rate: =RATE(n, PMT, -PV, FV)
To find n: =NPER(r, PMT, -PV, FV)
3.8 Real-World Applications
Application 1: Comparing Job Offers
Offer A:
- Salary: $80,000/year for 5 years
- No bonus
Offer B:
- Salary: $75,000/year for 5 years
- Signing bonus: $15,000 (today)
Which is better at 8% discount rate?
Offer A:
PV = $80,000 × [(1 - (1.08)^-5) / 0.08]
PV = $80,000 × 3.993
PV = $319,440
Offer B:
PV of salary = $75,000 × 3.993 = $299,475
PV of bonus = $15,000 (today)
Total = $314,475
Conclusion: Offer A is worth $4,965 more in present value terms.
Application 2: Lease vs. Buy Decision
Lease: $400/month for 36 months Buy: $12,000 today
Discount rate: 6% annual (0.5% monthly)
PV of Lease:
PV = $400 × [(1 - (1.005)^-36) / 0.005]
PV = $400 × 32.87
PV = $13,148
Conclusion: Buying for $12,000 saves $1,148 in present value.
Application 3: Settlement Decisions
You're injured and offered two settlement options:
Option A: $500,000 today Option B: $100,000/year for 6 years
Discount rate: 7%
PV of Option B:
PV = $100,000 × [(1 - (1.07)^-6) / 0.07]
PV = $100,000 × 4.767
PV = $476,700
Conclusion: Take Option A—worth $23,300 more.
Application 4: Home Refinancing
Current mortgage:
- Balance: $250,000
- Rate: 6%
- 20 years remaining
- Payment: $1,791/month
Refinance offer:
- Rate: 4%
- 20 years
- Cost: $5,000
Should you refinance?
New payment at 4%:
PMT = $250,000 × [0.00333 / (1 - (1.00333)^-240)]
PMT = $1,515/month
Savings: $1,791 - $1,515 = $276/month
PV of savings:
PV = $276 × [(1 - (1.00333)^-240) / 0.00333]
PV = $276 × 165.02
PV = $45,545
NPV:
NPV = $45,545 - $5,000 = $40,545
Conclusion: Refinance! NPV is positive by $40,545.
Application 5: Equipment Purchase
A machine costs $100,000 and will:
- Save $25,000/year in labor costs for 6 years
- Have $10,000 resale value at Year 6
Required return: 12%. Should you buy?
PV of savings:
PV = $25,000 × [(1 - (1.12)^-6) / 0.12]
PV = $25,000 × 4.111
PV = $102,775
PV of resale:
PV = $10,000 / (1.12)^6 = $5,066
Total PV of benefits: $102,775 + $5,066 = $107,841
NPV:
NPV = $107,841 - $100,000 = $7,841
Conclusion: Buy the machine! Positive NPV of $7,841.
3.9 Advanced Topics
Continuous Compounding
For continuous compounding (infinitely frequent):
FV = PV × e^(r×n)
PV = FV × e^(-r×n)
Where e ≈ 2.71828 (Euler's number)
Example: $10,000 at 8% continuously compounded for 5 years:
FV = $10,000 × e^(0.08×5)
FV = $10,000 × e^0.4
FV = $10,000 × 1.4918
FV = $14,918
Compare to annual compounding: $14,693
Continuous compounding gives slightly more.
Inflation-Adjusted Returns
Nominal Rate: The stated interest rate Real Rate: Return after adjusting for inflation
Fisher Equation:
(1 + nominal rate) = (1 + real rate) × (1 + inflation rate)
Approximation:
Real Rate ≈ Nominal Rate - Inflation Rate
Example: Investment returns 10% nominal, inflation is 3%
Exact real rate:
1.10 = (1 + r_real) × 1.03
1 + r_real = 1.10 / 1.03 = 1.0680
r_real = 6.80%
Approximation:
r_real ≈ 10% - 3% = 7%
Close enough for most purposes!
Multiple Cash Flows in Excel (NPV and IRR)
Example Project:
- Initial cost: $100,000 (Year 0)
- Returns: $30,000, $40,000, $50,000, $40,000 (Years 1-4)
Set up in Excel:
Year Cash Flow
0 -100000
1 30000
2 40000
3 50000
4 40000
Calculate NPV at 10%:
=NPV(0.10, B2:B5) + B1
(Remember: NPV function assumes first value is Year 1, so add Year 0 separately)
Result: NPV = $27,259
Calculate IRR (Internal Rate of Return):
=IRR(B1:B5)
Result: IRR = 20.4%
We'll explore NPV and IRR deeply in Module 4!
Module 3 Practice Problems
Problem Set 1: Future Value
-
You invest $8,000 at 7% annual interest for 12 years. What's the future value?
-
You want to have $1 million in 25 years. How much must you invest today at 9% annual return?
-
Which is better:
- 6% annual compounding
- 5.9% quarterly compounding Calculate the EAR for each.
-
Using the Rule of 72, approximately how long will it take to: a. Double your money at 8% b. Triple your money at 12%
Problem Set 2: Present Value
-
You'll receive $75,000 in 8 years. What's it worth today at: a. 6% discount rate b. 10% discount rate c. 14% discount rate
-
An investment will pay:
- $10,000 in Year 3
- $15,000 in Year 5
- $20,000 in Year 7
What's the total present value at 9% discount rate?
-
You're offered $50,000 today or $80,000 in 6 years. At what discount rate are these equivalent?
Problem Set 3: Annuities
-
You save $600/month for 35 years at 8% annual return (0.667% monthly). How much will you have?
-
You want to withdraw $75,000/year for 30 years in retirement. At 7% return, how much do you need saved?
-
You take a $25,000 car loan at 5% annual (monthly payments) for 5 years. What's the monthly payment?
-
You can afford $1,200/month for a mortgage at 6% for 30 years. What home price can you afford (assume 20% down payment)?
Problem Set 4: Mixed Cash Flows
-
A project requires $50,000 investment today and will pay:
- $15,000/year for Years 1-3
- $25,000 in Year 4
At 11% discount rate, what's the NPV? Should you invest?
-
A bond pays:
- $80 interest per year for 15 years
- $1,000 principal at Year 15
What's the bond worth at 7% discount rate?
Problem Set 5: Real-World Applications
-
Retirement Planning: You're 30 years old and want to retire at 65 with $2 million. You can earn 8% annually. a. If you invest a lump sum today, how much do you need? b. If you save monthly, how much per month?
-
Education Funding: Your child will start college in 15 years. You estimate you'll need $40,000/year for 4 years (starting in Year 15). At 6% return: a. What's the PV today of all four payments? b. How much must you save monthly to reach this goal?
-
Business Valuation: A business generates $150,000/year in profit. You expect this to:
- Grow at 5% per year for 10 years
- Then stay constant forever
At 12% discount rate, what's the business worth? (Hint: Calculate PV of growing 10-year annuity + PV of perpetuity starting in Year 11)
Problem Set 6: Excel Practice
-
Create an Excel amortization table for a $200,000 mortgage at 5% for 30 years showing:
- Monthly payment
- For each month: Beginning balance, payment, interest, principal, ending balance
- Total interest paid over life of loan
-
Build a retirement calculator in Excel that allows users to input:
- Current age
- Retirement age
- Current savings
- Monthly contribution
- Expected return
- Desired retirement income
Output should show:
- Amount at retirement
- Whether goal is achievable
Additional Resources
Excel Templates
Download practice files:
- TVM Calculator Template
- Loan Amortization Template
- Retirement Planning Template
- Investment Comparison Template
Online Calculators
For checking your work:
- Bankrate.com (mortgage, loan calculators)
- Calculator.net (financial calculators)
- Investor.gov (compound interest calculator)
Further Reading
- "The Millionaire Next Door" by Thomas Stanley - Real-world wealth building through compounding
- "A Random Walk Down Wall Street" by Burton Malkiel - Investment applications of TVM
- Online tutorials: Khan Academy, Corporate Finance Institute
Financial Calculator
If you're serious about finance, consider purchasing:
- HP 12C (classic financial calculator)
- TI BA II Plus (popular for CFA/CFP exams)
- Or simply master Excel—equally powerful!
Looking Ahead to Module 4
You've now mastered the most fundamental skill in finance: the time value of money. You can:
- Compound forward and discount backward
- Value streams of cash flows
- Make time-adjusted comparisons
- Solve real financial problems
In Module 4, we'll put TVM to work evaluating investment decisions. You'll learn:
- Net Present Value (NPV)
- Internal Rate of Return (IRR)
- Payback Period
- Profitability Index
- How companies decide which projects to pursue
Every capital budgeting technique builds on the TVM concepts you just mastered. You're ready!
Prepare for Module 4 by:
- Being comfortable with all TVM formulas
- Practicing Excel financial functions
- Understanding the NPV concept intuitively
Summary
Congratulations on completing Module 3—the mathematical heart of corporate finance!
You now understand:
✓ Why money has time value ✓ How to calculate future values with compound interest ✓ How to calculate present values by discounting ✓ How to work with different compounding frequencies ✓ How to value annuities (regular payment streams) ✓ How to value perpetuities (infinite payments) ✓ How to solve for any variable (PV, FV, PMT, r, n) ✓ How to use Excel for financial calculations ✓ How to apply TVM to real decisions
The time value of money is not just a formula—it's a way of thinking about financial decisions.
Every investment, loan, salary, or business decision involves comparing values across time. You now have the tools to make these comparisons rationally.
Practice these concepts until they become second nature. The better you understand TVM, the easier everything else in finance becomes.
Ready to apply your TVM skills? Proceed to Module 4: Capital Budgeting to learn how companies evaluate investment opportunities.
"Compound interest is the eighth wonder of the world. He who understands it, earns it; he who doesn't, pays it." — Albert Einstein (attributed)
"The most powerful force in the universe is compound interest." — Also attributed to Einstein
Whether Einstein actually said these things is debatable, but the wisdom is undeniable. You now understand this powerful force.
See you in Module 4!