Unit 5 is the most mathematically foundational unit in the entire curriculum. Every concept that follows—bond pricing, stock valuation, capital budgeting, retirement planning—is built directly on Time Value of Money. The core principle: a dollar today is worth more than a dollar in the future because today's dollar can be invested to earn a return. Master TVM and you can evaluate any financial decision—from a personal student loan to The Swanson Initiative endowment—with the same tools.
Part 1 — Core Topics Explained
Every major concept tested on the Unit 5 assessment — the foundation for all remaining units
📋 Learning Objectives
- Explain the time value of money principle and its three underlying reasons
- Calculate FV and PV of a lump sum at any interest rate and time period
- Calculate FV and PV of ordinary annuities and annuities due—and explain the timing difference
- Calculate the present value of a perpetuity and a growing perpetuity
- Build a complete loan amortization schedule from scratch
- Explain how compounding frequency affects growth and calculate the EAR
- Apply the Rule of 72 for quick mental estimates
- Connect TVM to BBYM decisions: savings plans, loan evaluation, and Swanson Initiative endowment sizing
1. Why a Dollar Today Is Worth More
Three distinct reasons justify the time value of money:
| Reason | What It Means | BBYM Example |
|---|---|---|
| Investment Opportunity | A dollar today can be invested immediately and earn a return—growing into more than a dollar tomorrow | $5,000 at 7% today = $19,348 in 20 years via The Swanson Initiative |
| Inflation | Prices rise over time—future dollars buy less than today's dollars. Purchasing power erodes. | A $5,000 scholarship today must be larger in 10 years to cover the same tuition |
| Risk of Non-Payment | A future promise carries uncertainty—the payer may default. Cash in hand is certain. | A community loan repaid today has more certain value than one promised over 5 years |
Two Directions in TVM:
Compounding (forward): PV → multiply by (1+r)ⁿ → FV. "What will my savings be worth?"
Discounting (backward): FV → divide by (1+r)ⁿ → PV. "What is that future promise worth today?" Discounting is the exact inverse of compounding.
Compounding (forward): PV → multiply by (1+r)ⁿ → FV. "What will my savings be worth?"
Discounting (backward): FV → divide by (1+r)ⁿ → PV. "What is that future promise worth today?" Discounting is the exact inverse of compounding.
2. Future Value — Growing Money Forward
FV answers: "If I invest $X today at rate r for n periods, what will it grow to?" This is compounding—earning returns on both original principal and previously earned interest.
Future Value — Lump Sum
FV = PV × (1 + r)ⁿ
PV = present value · r = interest rate per period · n = number of periods
(1+r)ⁿ = Future Value Interest Factor (FVIF)
(1+r)ⁿ = Future Value Interest Factor (FVIF)
Swanson Initiative — $5,000 at 7%:
10 years: $5,000 × (1.07)¹⁰ = $5,000 × 1.9672 = $9,836
20 years: $5,000 × (1.07)²⁰ = $5,000 × 3.8697 = $19,348
30 years: $5,000 × (1.07)³⁰ = $5,000 × 7.6123 = $38,061
The extra 10 years (20→30) nearly doubles the outcome. The longer the horizon, the more explosive the compounding effect.
10 years: $5,000 × (1.07)¹⁰ = $5,000 × 1.9672 = $9,836
20 years: $5,000 × (1.07)²⁰ = $5,000 × 3.8697 = $19,348
30 years: $5,000 × (1.07)³⁰ = $5,000 × 7.6123 = $38,061
The extra 10 years (20→30) nearly doubles the outcome. The longer the horizon, the more explosive the compounding effect.
Rule of 72 — Mental Shortcut for Doubling Time:
Years to double ≈ 72 ÷ Interest Rate
At 6%: 72 ÷ 6 = 12 years | At 8%: 72 ÷ 8 = 9 years | At 12%: 72 ÷ 12 = 6 years
Use this to instantly sanity-check any FV claim. If someone says money doubles in 5 years at 6%, the Rule of 72 immediately shows that's wrong.
Years to double ≈ 72 ÷ Interest Rate
At 6%: 72 ÷ 6 = 12 years | At 8%: 72 ÷ 8 = 9 years | At 12%: 72 ÷ 12 = 6 years
Use this to instantly sanity-check any FV claim. If someone says money doubles in 5 years at 6%, the Rule of 72 immediately shows that's wrong.
3. Present Value — Discounting Future Money to Today
PV answers: "What is a future amount worth right now?" It is the foundation of all valuation—every bond, stock, and business is priced as the present value of its expected future cash flows.
Present Value — Lump Sum
PV = FV ÷ (1 + r)ⁿ
r = discount rate (required rate of return / opportunity cost of capital)
The PVIF = 1÷(1+r)ⁿ is always less than 1.0—future dollars are always worth less than today's
The PVIF = 1÷(1+r)ⁿ is always less than 1.0—future dollars are always worth less than today's
Evaluating a $10,000 Community Promise Due in 5 Years (r = 8%):
PV = $10,000 ÷ (1.08)⁵ = $10,000 ÷ 1.4693 = $6,806
That $10,000 future promise is worth only $6,806 today. BBYM should prefer $6,806 now unless it cannot earn 8% on the money—in which case waiting for $10,000 is better.
PV = $10,000 ÷ (1.08)⁵ = $10,000 ÷ 1.4693 = $6,806
That $10,000 future promise is worth only $6,806 today. BBYM should prefer $6,806 now unless it cannot earn 8% on the money—in which case waiting for $10,000 is better.
Why the Discount Rate Matters—Same $10,000 Due in 10 Years:
At 4%: $10,000 ÷ (1.04)¹⁰ = $6,756
At 8%: $10,000 ÷ (1.08)¹⁰ = $4,632
At 12%: $10,000 ÷ (1.12)¹⁰ = $3,220
Same cash flow, three very different present values. This is why when interest rates rise, bond and stock prices fall—the higher discount rate shrinks the PV of all future cash flows.
At 4%: $10,000 ÷ (1.04)¹⁰ = $6,756
At 8%: $10,000 ÷ (1.08)¹⁰ = $4,632
At 12%: $10,000 ÷ (1.12)¹⁰ = $3,220
Same cash flow, three very different present values. This is why when interest rates rise, bond and stock prices fall—the higher discount rate shrinks the PV of all future cash flows.
4. Annuities — Equal Payments Over Time
An annuity is a series of equal payments at regular intervals. Mortgages, car loans, student loans, pensions—all are annuities. Two types differ only in when the first payment occurs:
| Feature | Ordinary Annuity | Annuity Due |
|---|---|---|
| Payment timing | END of each period | BEGINNING of each period |
| Common uses | Loans, mortgages, bonds | Leases, insurance premiums, rent |
| Relative value | Lower FV and PV (baseline) | Higher FV and PV by factor of (1+r)—payments earn one extra period of interest |
| Conversion | — | FVA(due) = FVA(ord) × (1+r) | PVA(due) = PVA(ord) × (1+r) |
Retirement Savings—$200/month (end of month), 30 years, 7%:
r = 7%÷12 = 0.5833%/month · n = 360
FVA = $200 × [(1.005833)³⁶⁰ − 1] ÷ 0.005833 = $200 × 1,219.97 = $243,994
Total contributed: $200 × 360 = $72,000 | Interest earned: $171,994—more than 2× contributions. Compounding makes interest dwarf principal over long horizons.
r = 7%÷12 = 0.5833%/month · n = 360
FVA = $200 × [(1.005833)³⁶⁰ − 1] ÷ 0.005833 = $200 × 1,219.97 = $243,994
Total contributed: $200 × 360 = $72,000 | Interest earned: $171,994—more than 2× contributions. Compounding makes interest dwarf principal over long horizons.
5. Perpetuities — Equal Payments Forever
A perpetuity pays forever. Despite infinite payments, it has a finite PV because distant payments are so heavily discounted they contribute negligibly to the total.
Present Value of a Perpetuity
PV = PMT ÷ r
Growing perpetuity: PV = PMT ÷ (r − g) [requires r > g]
PMT = fixed payment · r = discount rate · g = constant annual growth rate
Swanson Initiative Endowment Sizing:
Distribute $30,000/year forever at 6% return:
Required endowment = $30,000 ÷ 0.06 = $500,000
If distributions grow 2%/year with inflation:
Required endowment = $30,000 ÷ (0.06 − 0.02) = $30,000 ÷ 0.04 = $750,000
This is exactly how university endowments and community foundations are sized. The perpetuity formula determines the capital required to sustain annual distributions indefinitely.
Distribute $30,000/year forever at 6% return:
Required endowment = $30,000 ÷ 0.06 = $500,000
If distributions grow 2%/year with inflation:
Required endowment = $30,000 ÷ (0.06 − 0.02) = $30,000 ÷ 0.04 = $750,000
This is exactly how university endowments and community foundations are sized. The perpetuity formula determines the capital required to sustain annual distributions indefinitely.
6. Compounding Frequency and the EAR
More frequent compounding = more growth. A 7% nominal rate compounded monthly produces more than 7% compounded annually because each month's interest immediately starts earning interest.
Effective Annual Rate (EAR)
EAR = (1 + r_nom ÷ m)^m − 1
r_nom = stated annual rate · m = compounding periods/year (12=monthly, 365=daily)
Example: 12% nominal, monthly: EAR = (1.01)¹² − 1 = 12.68%
Example: 12% nominal, monthly: EAR = (1.01)¹² − 1 = 12.68%
| Compounding | m | $1,000 @ 7% / 5 yrs | $1,000 @ 7% / 30 yrs |
|---|---|---|---|
| Annual | 1 | $1,403 | $7,612 |
| Quarterly | 4 | $1,415 | $7,918 |
| Monthly | 12 | $1,417 | $8,117 |
| Daily | 365 | $1,419 | $8,155 |
EAR for Fair Comparison: Bank A: 5.0% annual (EAR = 5.00%). Bank B: 4.9% monthly (EAR = (1+0.049/12)¹²−1 = 5.01%). Bank B pays more despite the lower stated rate. You cannot fairly compare without converting to EAR. Always ask for the effective annual rate on any loan or savings product.
7. Starting Early — The Most Important TVM Lesson
Early vs. Late Saver—Same $200/month at 7%:
Maya starts at 22 → 43 years → FVA ≈ $883,000 (contributed $103,200)
James starts at 32 → 33 years → FVA ≈ $320,000 (contributed $79,200)
Maya contributed $24,000 more but ends up with $563,000 more—nearly 3× as much. Those 10 extra early years of compounding outweigh all additional contributions from waiting. Start now. Always.
Maya starts at 22 → 43 years → FVA ≈ $883,000 (contributed $103,200)
James starts at 32 → 33 years → FVA ≈ $320,000 (contributed $79,200)
Maya contributed $24,000 more but ends up with $563,000 more—nearly 3× as much. Those 10 extra early years of compounding outweigh all additional contributions from waiting. Start now. Always.
Part 2 — All Formulas with Worked Examples
Every formula in one place—with a complete Birmingham-Bessemer worked example for each
Complete TVM Formula Reference
| Formula | Equation | Solves For |
|---|---|---|
| FV — Lump Sum | FV = PV × (1+r)ⁿ | How much a single investment grows to |
| PV — Lump Sum | PV = FV ÷ (1+r)ⁿ | What a future lump sum is worth today |
| FV — Ordinary Annuity | FVA = PMT × [(1+r)ⁿ−1] ÷ r | Final value of equal end-of-period payments |
| PV — Ordinary Annuity | PVA = PMT × [1−1/(1+r)ⁿ] ÷ r | Today's value of equal end-of-period payments (also: loan amount) |
| FV — Annuity Due | FVA(due) = FVA(ord) × (1+r) | Final value when payments are at start of period |
| PV — Annuity Due | PVA(due) = PVA(ord) × (1+r) | Today's value when payments are at start of period |
| PV — Perpetuity | PV = PMT ÷ r | Today's value of infinite equal payments |
| PV — Growing Perpetuity | PV = PMT ÷ (r−g) | Today's value of payments growing at rate g forever |
| Loan Payment (PMT) | PMT = PV × r ÷ [1−(1+r)^−ⁿ] | Equal periodic payment for a given loan |
| FV — Multiple Compounds | FV = PV × (1+r/m)^(m×n) | Growth with non-annual compounding |
| Effective Annual Rate | EAR = (1+r_nom/m)^m − 1 | True annual rate for any compounding frequency |
Worked Examples — Step by Step
Example 1 — FV Lump Sum (Assessment Q5)
Invest $2,000 today at 6% for 10 years. Find FV.
FV = $2,000 × (1.06)¹⁰ = $2,000 × 1.7908 = $3,582
The $2,000 grows by $1,582 through compounding alone over 10 years at 6%.
Example 2 — PV Lump Sum
Community grant of $15,000 promised in 8 years. Discount rate = 7%. Find PV.
PV = $15,000 ÷ (1.07)⁸ = $15,000 ÷ 1.7182 = $8,731
That $15,000 future promise is worth only $8,731 in today's dollars at 7%. Accept this deal only if BBYM cannot find a better use for $8,731 today.
Example 3 — FV Ordinary Annuity
Save $300/month (end of month) for 20 years at 6%/year (0.5%/month). Find FV.
FVA = $300 × [(1.005)²⁴⁰ − 1] ÷ 0.005 = $300 × [3.310 − 1] ÷ 0.005
= $300 × 462.04 = $138,612
Contributed: $300 × 240 = $72,000. Interest earned: $66,612—nearly equal to contributions after 20 years.
Example 4 — PV Ordinary Annuity (Loan Value)
CDFI loan: $500/month for 48 months at 8%/year (0.6667%/month). What loan amount?
PVA = $500 × [1 − 1/(1.006667)⁴⁸] ÷ 0.006667 = $500 × 40.96 = $20,480
Total paid: $500 × 48 = $24,000. Interest paid: $3,520 on a $20,480 loan over 4 years at 8%.
Example 5 — Loan Payment (PMT)
BBYM borrows $25,000 at 6%/year for 5 years (monthly). Find monthly PMT.
r = 0.5%/month · n = 60
PMT = $25,000 × 0.005 ÷ [1 − (1.005)^−⁶⁰] = $125 ÷ 0.2586 = $483.32/month
Total paid: $483.32 × 60 = $28,999. Total interest: $3,999 on a $25,000 loan at 6% over 5 years.
Example 6 — EAR
Credit card: 18% APR compounded monthly. What is the EAR?
EAR = (1 + 0.18/12)¹² − 1 = (1.015)¹² − 1 = 1.1956 − 1 = 19.56%
The card advertises 18% but actually costs 19.56%/year when monthly compounding is included.
Example 7 — Perpetuity
Swanson Initiative: distribute $30,000/year forever at 6% return. Required endowment?
PV = $30,000 ÷ 0.06 = $500,000
With 2% annual growth in distributions: PV = $30,000 ÷ (0.06−0.02) = $750,000
Part 3 — Loan Amortization
How to build an amortization schedule—a critical skill for evaluating any loan
What Is an Amortization Schedule?
An amortization schedule shows how each loan payment splits between interest and principal reduction, and tracks the declining balance over time. Every payment is the same amount—but the interest portion shrinks and the principal portion grows with each payment.
Why the Split Changes Each Period: Each payment covers that period's interest (Balance × r) first. Whatever remains reduces principal. As principal falls, next period's interest is smaller—so more of the fixed payment goes to principal. The last payments are almost entirely principal.
Complete Amortization Schedule — $10,000 Loan at 6%, 12 Months
Step 1 — Calculate monthly payment:
r = 6%/12 = 0.5% · n = 12 · PV = $10,000
PMT = $10,000 × 0.005 ÷ [1−(1.005)⁻¹²] = $50 ÷ 0.0582 = $860.66/month
Step 2 — For each row: Interest = Beg. Balance × 0.005 | Principal = PMT − Interest | End. Balance = Beg. Balance − Principal
r = 6%/12 = 0.5% · n = 12 · PV = $10,000
PMT = $10,000 × 0.005 ÷ [1−(1.005)⁻¹²] = $50 ÷ 0.0582 = $860.66/month
Step 2 — For each row: Interest = Beg. Balance × 0.005 | Principal = PMT − Interest | End. Balance = Beg. Balance − Principal
| Month | Beg. Balance | Payment | Interest (0.5%) | Principal | End. Balance |
|---|---|---|---|---|---|
| 1 | $10,000.00 | $860.66 | $50.00 | $810.66 | $9,189.34 |
| 2 | $9,189.34 | $860.66 | $45.95 | $814.71 | $8,374.63 |
| 3 | $8,374.63 | $860.66 | $41.87 | $818.79 | $7,555.84 |
| 4 | $7,555.84 | $860.66 | $37.78 | $822.88 | $6,732.96 |
| 5 | $6,732.96 | $860.66 | $33.66 | $827.00 | $5,905.96 |
| 6 | $5,905.96 | $860.66 | $29.53 | $831.13 | $5,074.83 |
| 7 | $5,074.83 | $860.66 | $25.37 | $835.29 | $4,239.54 |
| 8 | $4,239.54 | $860.66 | $21.20 | $839.46 | $3,400.08 |
| 9 | $3,400.08 | $860.66 | $17.00 | $843.66 | $2,556.42 |
| 10 | $2,556.42 | $860.66 | $12.78 | $847.88 | $1,708.54 |
| 11 | $1,708.54 | $860.66 | $8.54 | $852.12 | $856.42 |
| 12 | $856.42 | $860.66 | $4.28 | $856.38 | ~$0.04* |
| TOTALS | $10,327.92 | $327.96 | $9,999.96 |
*$0.04 rounding difference from using rounded payment amount.
Key observations:
Month 1 interest: $50.00 → Month 12 interest: $4.28 — shrinks every period
Month 1 principal: $810.66 → Month 12 principal: $856.38 — grows every period
Total interest on $10,000 for 12 months at 6% = $327.96
Extra principal payments in early months save the most—they eliminate all future interest that would have accrued on that amount.
Month 1 interest: $50.00 → Month 12 interest: $4.28 — shrinks every period
Month 1 principal: $810.66 → Month 12 principal: $856.38 — grows every period
Total interest on $10,000 for 12 months at 6% = $327.96
Extra principal payments in early months save the most—they eliminate all future interest that would have accrued on that amount.
30-Year vs. 15-Year Mortgage — The True Cost of a Longer Term
| Feature | 30-Year Mortgage | 15-Year Mortgage |
|---|---|---|
| Loan Amount | $150,000 | $150,000 |
| Annual Rate | 6.5% | 6.0% |
| Monthly Payment | $948 | $1,266 |
| Total Paid | $948 × 360 = $341,280 | $1,266 × 180 = $227,880 |
| Total Interest | $191,280 | $77,880 |
| Interest Savings | $113,400 saved by choosing the 15-year loan | |
The 30-year loan saves $318/month in payments but costs $113,400 more over its life. That $113,400 could have built family wealth, funded college, or been invested in a community enterprise. TVM analysis reveals the true cost of convenience—the monthly payment is not the cost of the loan.
Part 4 — Key Terms Defined
Master these 16 terms—they appear throughout the entire remaining curriculum
Time Value of Money (TVM)
The principle that a dollar received today is worth more than a dollar received in the future, because today's dollar can be invested to earn a return. TVM is the foundational concept underlying all financial valuation—bonds, stocks, loans, and capital budgeting decisions.
Future Value (FV)
The value of a present amount at a future date after earning compound interest. FV = PV × (1+r)ⁿ. Moving money forward in time—compounding. The further the horizon and the higher the rate, the larger the FV relative to PV.
Present Value (PV)
The current value of a future amount, discounted back at the required rate of return. PV = FV ÷ (1+r)ⁿ. Moving money backward in time—discounting. Every financial asset is ultimately valued as the PV of its expected future cash flows.
Compounding
Earning interest on both the original principal AND on previously earned interest—"interest on interest." The mechanism driving exponential growth over time. The longer the horizon and higher the rate, the more dramatic the compounding effect.
Discounting
The process of finding the present value of a future amount by dividing by (1+r)ⁿ. The exact inverse of compounding. The discount rate represents the opportunity cost—what could have been earned if the money were received today.
Discount Rate
The interest rate used to convert future cash flows into present values. Also called the required rate of return or opportunity cost of capital. Higher discount rates produce lower present values, reflecting higher risk or better available alternatives.
Annuity
A series of equal payments made at regular intervals over a specified number of periods. Most consumer loans (mortgages, car loans, student loans) are annuities. Key inputs: payment amount (PMT), rate per period (r), and number of periods (n).
Ordinary Annuity
An annuity in which payments occur at the END of each period. The most common type—used for most loans, bonds, and retirement accounts. Default assumption in all financial calculator annuity calculations unless stated otherwise.
Annuity Due
An annuity in which payments occur at the BEGINNING of each period. Common for leases, insurance premiums, and rent. Because payments arrive one period earlier, both FV and PV are higher than an equivalent ordinary annuity by exactly (1+r).
Perpetuity
An annuity that makes equal payments indefinitely—forever. Despite infinite payments, it has a finite present value: PV = PMT ÷ r. Used to value preferred stocks, endowments, and consol bonds. Growing perpetuity: PV = PMT ÷ (r−g).
Nominal Interest Rate
The stated or quoted annual interest rate, before adjusting for compounding frequency. Also called the Annual Percentage Rate (APR) in consumer lending. Understates the true cost when compounding occurs more than once per year.
Effective Annual Rate (EAR)
The actual annual rate of interest earned or paid, after accounting for compounding frequency. EAR = (1+r_nom/m)^m − 1. The only fair way to compare products with different compounding frequencies. Always higher than the nominal rate (except when m=1).
Amortized Loan
A loan repaid through equal periodic payments where each payment covers accrued interest first, then reduces principal. Early payments are mostly interest; later payments are mostly principal. Balance reaches zero at the final payment.
Amortization Schedule
A table showing the breakdown of each loan payment into interest and principal, and the remaining balance after each payment. Allows the borrower to see the true total cost of the loan and understand the dollar impact of making extra payments.
Rule of 72
A mental shortcut for estimating doubling time: Years to double ≈ 72 ÷ Interest Rate. At 8%, money doubles in roughly 9 years. Useful for quick sanity checks without a calculator—and a powerful way to demonstrate compounding to skeptics.
Opportunity Cost of Capital
The return foregone by choosing one investment over the next-best alternative. The discount rate in TVM calculations represents this cost—it is the minimum return required to justify an investment given what else could be done with the money.
Part 5 — Practice Questions
Show all work—these mirror the Unit 5 assessment format exactly
Attempt each calculation before revealing. TVM is learned by doing, not reading.
Conceptual Questions
Q1Explain in plain language why "a dollar today is worth more than a dollar tomorrow." Give three distinct reasons. Which reason is most directly captured in the TVM discount rate?▼
Three reasons:
1. Investment opportunity: A dollar today can be invested immediately and earn a return—growing into more than a dollar by tomorrow. This is the core TVM insight.
2. Inflation: Prices rise over time, so a dollar in the future buys less. Purchasing power erodes.
3. Risk of non-payment: A future promise carries uncertainty. A dollar in hand is certain; a promise is not.
The discount rate most directly captures the investment opportunity reason. It represents the return available on alternative investments of similar risk—the opportunity cost of waiting. Inflation and default risk are sometimes embedded in the discount rate as components, but the investment opportunity concept is what makes TVM a calculation rather than just a philosophy.
1. Investment opportunity: A dollar today can be invested immediately and earn a return—growing into more than a dollar by tomorrow. This is the core TVM insight.
2. Inflation: Prices rise over time, so a dollar in the future buys less. Purchasing power erodes.
3. Risk of non-payment: A future promise carries uncertainty. A dollar in hand is certain; a promise is not.
The discount rate most directly captures the investment opportunity reason. It represents the return available on alternative investments of similar risk—the opportunity cost of waiting. Inflation and default risk are sometimes embedded in the discount rate as components, but the investment opportunity concept is what makes TVM a calculation rather than just a philosophy.
Q2A BBYM student says: "I'll start saving for retirement at 35—I still have 30 years, that's plenty of time." Using TVM concepts, explain why starting at 25 instead of 35 makes such a dramatic difference.▼
Two compounding effects work together:
1. More compounding periods: 40 years vs. 30 is not a linear 33% difference—it is exponential. The growth from year 31 to year 40 is much larger in dollar terms than year 1 to year 10, because the base is far larger. The last decade of compounding can produce more wealth than the first three decades combined.
2. Early contributions compound the longest: Every dollar saved at 25 has 40 years to compound; every dollar at 35 has only 30. The first dollar saved is the most valuable. Waiting 10 years means your first contribution has 25% fewer compounding cycles—and loses all the explosive growth in those final years when the account balance is largest.
The practical result: starting at 25 with identical monthly contributions typically produces 2–3× more wealth at retirement than starting at 35.
1. More compounding periods: 40 years vs. 30 is not a linear 33% difference—it is exponential. The growth from year 31 to year 40 is much larger in dollar terms than year 1 to year 10, because the base is far larger. The last decade of compounding can produce more wealth than the first three decades combined.
2. Early contributions compound the longest: Every dollar saved at 25 has 40 years to compound; every dollar at 35 has only 30. The first dollar saved is the most valuable. Waiting 10 years means your first contribution has 25% fewer compounding cycles—and loses all the explosive growth in those final years when the account balance is largest.
The practical result: starting at 25 with identical monthly contributions typically produces 2–3× more wealth at retirement than starting at 35.
Q3What is the difference between an ordinary annuity and an annuity due? If you are receiving payments, which type do you prefer—and why?▼
An ordinary annuity pays at the END of each period. An annuity due pays at the BEGINNING—one period earlier.
If receiving payments, you prefer the annuity due. Each payment arrives one period sooner, so you can invest it immediately and earn one extra period of return on every payment. The FV and PV of an annuity due are always higher than an equivalent ordinary annuity by exactly (1+r).
If making payments (borrowing), you prefer the ordinary annuity—payments at the end of each period means you keep your money longer before paying, giving it more time to earn returns for you.
Key principle: earlier is better when receiving; later is better when paying. Every extra period money stays in your hands is valuable.
If receiving payments, you prefer the annuity due. Each payment arrives one period sooner, so you can invest it immediately and earn one extra period of return on every payment. The FV and PV of an annuity due are always higher than an equivalent ordinary annuity by exactly (1+r).
If making payments (borrowing), you prefer the ordinary annuity—payments at the end of each period means you keep your money longer before paying, giving it more time to earn returns for you.
Key principle: earlier is better when receiving; later is better when paying. Every extra period money stays in your hands is valuable.
Calculation Questions
Q4If you invest $2,000 today at 6% per year for 10 years, what is the future value? (This is the Unit 5 curriculum assessment question.)▼
FV = PV × (1+r)ⁿ
FV = $2,000 × (1.06)¹⁰ = $2,000 × 1.7908 = $3,582
The correct answer is $3,582. The $2,000 grows by $1,582 in interest over 10 years—a 79% increase purely through compounding.
FV = $2,000 × (1.06)¹⁰ = $2,000 × 1.7908 = $3,582
The correct answer is $3,582. The $2,000 grows by $1,582 in interest over 10 years—a 79% increase purely through compounding.
Q5The Swanson Initiative expects to receive a $50,000 donation in 6 years. If the required discount rate is 8%, what is the present value? What does this PV tell the board?▼
PV = FV ÷ (1+r)ⁿ
PV = $50,000 ÷ (1.08)⁶ = $50,000 ÷ 1.5869 = $31,511
What this tells the board: The $50,000 future promise is worth only $31,511 today at 8%. If the Initiative can invest $31,511 now at 8%, it grows to exactly $50,000 in 6 years—these options are financially equivalent. Any project requiring less than $31,511 upfront to generate $50,000 in 6 years creates value; more than $31,511 destroys value.
PV = $50,000 ÷ (1.08)⁶ = $50,000 ÷ 1.5869 = $31,511
What this tells the board: The $50,000 future promise is worth only $31,511 today at 8%. If the Initiative can invest $31,511 now at 8%, it grows to exactly $50,000 in 6 years—these options are financially equivalent. Any project requiring less than $31,511 upfront to generate $50,000 in 6 years creates value; more than $31,511 destroys value.
Q6A BBYM entrepreneur saves $150/month (end of month) for 5 years at 6% annual rate. (a) FV of savings. (b) Total contributed. (c) Total interest earned.▼
r = 6%/12 = 0.5% = 0.005 · n = 60
(a) FVA = $150 × [(1.005)⁶⁰ − 1] ÷ 0.005 = $150 × [1.3489−1] ÷ 0.005 = $150 × 69.77 = $10,466
(b) Total contributed = $150 × 60 = $9,000
(c) Interest earned = $10,466 − $9,000 = $1,466
At 5 years the interest is still modest relative to contributions. At 20–30 years, interest earned would far exceed contributions—illustrating why the long end of the compounding curve is so powerful.
(a) FVA = $150 × [(1.005)⁶⁰ − 1] ÷ 0.005 = $150 × [1.3489−1] ÷ 0.005 = $150 × 69.77 = $10,466
(b) Total contributed = $150 × 60 = $9,000
(c) Interest earned = $10,466 − $9,000 = $1,466
At 5 years the interest is still modest relative to contributions. At 20–30 years, interest earned would far exceed contributions—illustrating why the long end of the compounding curve is so powerful.
Q7A CDFI offers a $18,000 equipment loan at 7.2% annual (0.6%/month) for 36 months. (a) Monthly payment. (b) Total interest paid over the life of the loan.▼
(a) PMT = PV × r ÷ [1−(1+r)^−ⁿ]
PMT = $18,000 × 0.006 ÷ [1−(1.006)^−³⁶]
(1.006)³⁶ = 1.2408 → (1.006)^−³⁶ = 0.8060
PMT = $108 ÷ [1−0.8060] = $108 ÷ 0.1940 = $556.70/month
(b) Total paid = $556.70 × 36 = $20,041
Total interest = $20,041 − $18,000 = $2,041
On an $18,000 loan for 3 years at 7.2%, total interest is $2,041—about 11.3% of principal. This is the true cost of borrowing expressed in dollars, not percentages.
PMT = $18,000 × 0.006 ÷ [1−(1.006)^−³⁶]
(1.006)³⁶ = 1.2408 → (1.006)^−³⁶ = 0.8060
PMT = $108 ÷ [1−0.8060] = $108 ÷ 0.1940 = $556.70/month
(b) Total paid = $556.70 × 36 = $20,041
Total interest = $20,041 − $18,000 = $2,041
On an $18,000 loan for 3 years at 7.2%, total interest is $2,041—about 11.3% of principal. This is the true cost of borrowing expressed in dollars, not percentages.
Q8A credit card charges 24% APR, compounded monthly. (a) What is the EAR? (b) If you carry a $2,000 balance for one full year without payments, how much do you owe?▼
(a) EAR = (1 + 0.24/12)¹² − 1 = (1.02)¹² − 1 = 1.2682 − 1 = 26.82%
The card advertises 24% but the true annual cost is 26.82%.
(b) FV = $2,000 × (1.02)¹² = $2,000 × 1.2682 = $2,536
Doing nothing for one year turns $2,000 into $2,536—$536 in interest charges. With only minimum payments, the borrower might pay for years without meaningfully reducing principal.
The card advertises 24% but the true annual cost is 26.82%.
(b) FV = $2,000 × (1.02)¹² = $2,000 × 1.2682 = $2,536
Doing nothing for one year turns $2,000 into $2,536—$536 in interest charges. With only minimum payments, the borrower might pay for years without meaningfully reducing principal.
Q9The Swanson Initiative wants to establish a perpetual $4,000/year scholarship forever. Fund earns 5%. How large must the endowment be? What if the scholarship grows 2%/year with inflation?▼
PV = PMT ÷ r = $4,000 ÷ 0.05 = $80,000
At 5% return, $80,000 generates exactly $4,000/year ($80,000 × 5%). The principal is never touched, so the fund runs forever.
With 2% annual growth (growing perpetuity):
PV = $4,000 ÷ (0.05 − 0.02) = $4,000 ÷ 0.03 = $133,333
The inflation-adjusted version requires $53,333 more capital. Growing distributions require a larger principal. This is why endowment planning always accounts for inflation.
At 5% return, $80,000 generates exactly $4,000/year ($80,000 × 5%). The principal is never touched, so the fund runs forever.
With 2% annual growth (growing perpetuity):
PV = $4,000 ÷ (0.05 − 0.02) = $4,000 ÷ 0.03 = $133,333
The inflation-adjusted version requires $53,333 more capital. Growing distributions require a larger principal. This is why endowment planning always accounts for inflation.
Q10Using the Rule of 72: (a) At 9%, how many years to double? (b) At what rate does money double in 6 years? (c) Inflation is 4%—how long before purchasing power is cut in half?▼
(a) Years to double at 9%: 72 ÷ 9 = 8 years
(b) Rate to double in 6 years: 72 ÷ 6 = 12% per year
(c) Purchasing power halves when prices double—same math, applied to inflation: 72 ÷ 4 = 18 years
At 4% inflation, $100 of purchasing power today becomes worth only $50 in 18 years. This is why keeping large amounts of cash in non-interest-bearing accounts is a slow financial drain—inflation erodes real value even when the nominal balance stays the same.
(b) Rate to double in 6 years: 72 ÷ 6 = 12% per year
(c) Purchasing power halves when prices double—same math, applied to inflation: 72 ÷ 4 = 18 years
At 4% inflation, $100 of purchasing power today becomes worth only $50 in 18 years. This is why keeping large amounts of cash in non-interest-bearing accounts is a slow financial drain—inflation erodes real value even when the nominal balance stays the same.
Q11Refer to the amortization schedule in Part 3. In Month 1, the borrower pays $860.66. How much of that pays down principal vs. interest? Why does the interest portion shrink every month?▼
Month 1: Interest = $10,000 × 0.005 = $50.00. Principal = $860.66 − $50.00 = $810.66.
Why interest shrinks every month: Each payment reduces the outstanding principal balance. In Month 2, the balance is $9,189.34—so interest is $9,189.34 × 0.005 = $45.95 (less than Month 1's $50). In Month 3, the balance is lower still, so interest drops again. The payment amount stays fixed at $860.66, but because interest takes a smaller slice each month, the principal portion grows automatically.
By Month 12, interest is only $4.28 and nearly the entire $860.66 payment ($856.38) reduces principal. This is why making extra principal payments early in a loan saves dramatically more interest than making the same extra payments later.
Why interest shrinks every month: Each payment reduces the outstanding principal balance. In Month 2, the balance is $9,189.34—so interest is $9,189.34 × 0.005 = $45.95 (less than Month 1's $50). In Month 3, the balance is lower still, so interest drops again. The payment amount stays fixed at $860.66, but because interest takes a smaller slice each month, the principal portion grows automatically.
By Month 12, interest is only $4.28 and nearly the entire $860.66 payment ($856.38) reduces principal. This is why making extra principal payments early in a loan saves dramatically more interest than making the same extra payments later.
Q12Two banks offer home loans on $200,000: Bank A at 6% for 30 years; Bank B at 6% for 15 years. Calculate monthly payments for both and total interest paid. Which should a BBYM family choose, and why might they choose the other?▼
Bank A (30-year): r = 0.5%/month · n = 360
PMT = $200,000 × 0.005 ÷ [1−(1.005)^−³⁶⁰] = $1,000 ÷ 0.8342… = $1,199/month
Total paid: $1,199 × 360 = $431,640 · Total interest: $231,640
Bank B (15-year): r = 0.5%/month · n = 180
PMT = $200,000 × 0.005 ÷ [1−(1.005)^−¹⁸⁰] = $1,000 ÷ 0.5931… = $1,687/month
Total paid: $1,687 × 180 = $303,660 · Total interest: $103,660
Interest savings with 15-year loan: $127,980
Wealth-building choice: The 15-year loan saves $127,980 in interest—money that stays in the family. A financially stable BBYM family with reliable income should strongly prefer the 15-year loan.
Why someone might choose the 30-year: The $488/month lower payment provides cash flow flexibility during tight periods. For a young entrepreneur reinvesting heavily in a business that earns more than 6%, the freed-up cash flow might create more wealth than the interest saved—but this requires discipline and a reliably high return. For most families, the 15-year loan builds more wealth with less risk.
PMT = $200,000 × 0.005 ÷ [1−(1.005)^−³⁶⁰] = $1,000 ÷ 0.8342… = $1,199/month
Total paid: $1,199 × 360 = $431,640 · Total interest: $231,640
Bank B (15-year): r = 0.5%/month · n = 180
PMT = $200,000 × 0.005 ÷ [1−(1.005)^−¹⁸⁰] = $1,000 ÷ 0.5931… = $1,687/month
Total paid: $1,687 × 180 = $303,660 · Total interest: $103,660
Interest savings with 15-year loan: $127,980
Wealth-building choice: The 15-year loan saves $127,980 in interest—money that stays in the family. A financially stable BBYM family with reliable income should strongly prefer the 15-year loan.
Why someone might choose the 30-year: The $488/month lower payment provides cash flow flexibility during tight periods. For a young entrepreneur reinvesting heavily in a business that earns more than 6%, the freed-up cash flow might create more wealth than the interest saved—but this requires discipline and a reliably high return. For most families, the 15-year loan builds more wealth with less risk.
Part 6 — Quick Reference Summary
Read this the night before the assessment—the whole unit on one page
Unit 5 in 5 Essential Sentences
Sentence 1
A dollar today is worth more than a dollar in the future because it can be invested to earn a return—and TVM quantifies exactly how much more, using compounding (forward) and discounting (backward) as two sides of the same mathematical operation.
Sentence 2
FV = PV × (1+r)ⁿ grows money forward; PV = FV ÷ (1+r)ⁿ brings it back; the discount rate represents opportunity cost, so a higher rate always produces a lower present value.
Sentence 3
Annuities (equal payments over time) use FVA = PMT × [(1+r)ⁿ−1]÷r and PVA = PMT × [1−1/(1+r)ⁿ]÷r; annuity due payments occur at the start of each period and are worth more by (1+r); perpetuities simplify to PV = PMT÷r.
Sentence 4
In an amortized loan, each equal payment covers interest first then reduces principal—early payments are mostly interest, late payments mostly principal—and the EAR = (1+r/m)^m−1 converts any nominal rate into its true annual cost for fair comparison.
Sentence 5
Starting early is the most powerful personal finance application of TVM—10 extra years of compounding at the beginning of a savings plan can produce more wealth than decades of additional contributions made later.
Must-Know Facts for the Assessment
| Question / Formula | Answer |
|---|---|
| FV of a lump sum | FV = PV × (1+r)ⁿ |
| PV of a lump sum | PV = FV ÷ (1+r)ⁿ |
| FV of ordinary annuity | FVA = PMT × [(1+r)ⁿ−1] ÷ r |
| PV of ordinary annuity | PVA = PMT × [1−1/(1+r)ⁿ] ÷ r |
| Annuity due vs. ordinary | Multiply ordinary result by (1+r); payments at start of period |
| PV of perpetuity | PV = PMT ÷ r |
| Growing perpetuity | PV = PMT ÷ (r − g) [requires r > g] |
| Loan payment formula | PMT = PV × r ÷ [1−(1+r)^−ⁿ] |
| EAR formula | EAR = (1 + r_nom/m)^m − 1 |
| Rule of 72 | Years to double ≈ 72 ÷ interest rate |
| $2,000 at 6% for 10 years | $3,582 (the curriculum assessment answer) |
| Higher discount rate → PV | Lower PV. Higher r = more discounting = smaller present value |
| Amortization: early payments | Mostly interest. Later payments mostly principal. Total payment stays constant. |
| Swanson Initiative: $30K/yr perpetuity at 6% | $30,000 ÷ 0.06 = $500,000 endowment required |
| Why start saving early? | Compounding is exponential—early dollars earn returns on returns for decades; late dollars miss the explosive growth years |