Time Value of Money (Treasury Examples)

You're offered $1,000 today or $1,000 in five years. Every rational investor takes the immediate payment (because money available now can be invested to grow, while delayed money incurs opportunity cost). At a 4% Treasury yield (current 10-year rate), that $1,000 today compounds to $1,217 in five years—a $217 advantage over waiting.¹ The practical antidote: understand present value and future value mechanics (the foundation of all investment decisions, from bond pricing to retirement planning).

The Two Pillars of TVM (Opportunity Cost and Inflation)

Time value of money (TVM) rests on two principles that make today's dollar worth more than tomorrow's:

Opportunity cost → Money received today can be invested to earn returns (interest, dividends, capital gains). Delaying receipt means forfeiting those returns. At 5% annual return, $10,000 today becomes $16,289 in 10 years. Waiting 10 years to receive $10,000 costs you $6,289 in foregone growth.²

Inflation risk → Future purchasing power is uncertain (inflation erodes the real value of fixed nominal payments). $1,000 received in 20 years at 3% inflation buys only $554 worth of today's goods.³ Receiving money sooner locks in current purchasing power.

The point is: TVM isn't a theoretical construct (useful for finance textbooks). It's the reason bond prices fluctuate with interest rates, the mechanism that determines mortgage affordability, and the math that dictates whether your retirement savings suffice. Every financial decision involves choosing between cash flows at different points in time.

Future Value Formula (What $1 Today Becomes)

Future value (FV) answers: "If I invest this amount today, what will it grow to?" The formula: FV = PV × (1 + r)^n, where PV = present value (today's amount), r = annual return rate, n = number of years.

Example using current 10-year Treasury yield (4.12%):⁴

You invest $5,000 today at 4.12% for 10 years.

FV = $5,000 × (1.0412)^10

FV = $5,000 × 1.498

FV = $7,492

Your $5,000 compounds to $7,492 over a decade (a $2,492 gain). That's the opportunity cost of delaying receipt—if someone offers you $5,000 in 10 years, it's worth only $5,000 then but costs you $2,492 in foregone growth.

Why this matters: the formula reveals how small rate differences compound into massive wealth gaps over time. At 4% for 30 years, $10,000 → $32,434. At 7% for 30 years (equity-like returns), $10,000 → $76,123.⁵ That 3 percentage point spread (7% vs 4%) creates a $43,689 difference on every $10,000 invested.

Compounding Frequency (Annual vs Monthly)

The basic formula assumes annual compounding (interest paid once per year). Most investments compound more frequently (bonds pay semi-annually, savings accounts compound monthly or daily). The adjusted formula: FV = PV × (1 + r/m)^(m×n), where m = compounding periods per year.

Example: $10,000 at 5% for 10 years.

Annual compounding (m = 1): FV = $10,000 × (1.05)^10 = $16,289

Monthly compounding (m = 12): FV = $10,000 × (1 + 0.05/12)^(12×10) = $10,000 × (1.004167)^120 = $16,470

Daily compounding (m = 365): FV = $10,000 × (1 + 0.05/365)^(365×10) = $16,487⁶

The durable lesson: more frequent compounding delivers marginally higher returns (the difference between monthly and daily is trivial). The big driver is the base rate (5% vs 7% matters far more than annual vs monthly). Don't optimize compounding frequency (negligible impact). Optimize asset allocation (massive impact).

Present Value Formula (What Future $1 Is Worth Today)

Present value (PV) answers: "What would I pay today for a payment I'll receive in the future?" The formula: PV = FV / (1 + r)^n (the inverse of future value).

Example using Fed funds rate (3.50% midpoint):⁷

You'll receive $20,000 in 15 years. What's it worth today (discounted at 3.50%)?

PV = $20,000 / (1.035)^15

PV = $20,000 / 1.675

PV = $11,940

That future $20,000 payment is worth $11,940 in today's dollars (if you can invest at 3.50% annually). The $8,060 difference represents opportunity cost (you could turn $11,940 into $20,000 by investing it for 15 years at 3.50%).

The test: which is the better deal: $12,000 today or $22,000 in 15 years? Discount the future payment: $22,000 / (1.035)^15 = $13,134 present value. The future payment wins ($13,134 > $12,000). At what discount rate would you be indifferent? Solve for r: $12,000 = $22,000 / (1 + r)^15 → r = 4.0%. If you can earn more than 4% annually elsewhere, take the $12,000 today. If not, wait for the $22,000.

The Discount Rate Choice (What Rate to Use)

Choosing the discount rate determines present value calculations (and whether an investment looks attractive). Three common approaches:

Risk-free rate (Treasury yields): Use for guaranteed cash flows (government bonds, FDIC-insured CDs). Current 10-year Treasury: 4.12%. This is the floor (if an investment can't beat Treasuries, why take additional risk?).

Opportunity cost rate (your best alternative investment): Use for comparing options. If your stock portfolio returns 9%, discount future payments at 9% (you need to beat what you'd otherwise earn).

Required return (risk-adjusted): Use for risky cash flows (corporate bonds, dividend stocks, real estate). Higher risk → higher discount rate. Investment-grade corporate bonds: 5-6%. High-yield bonds: 7-9%. Stocks: 9-11% (reflecting equity risk premium over Treasuries).⁸

Why this matters: changing the discount rate flips investment decisions. A project delivering $100,000 in 10 years is worth $67,556 at 4% discount rate (attractive if it costs $60,000 today). The same project is worth $42,241 at 9% discount rate (unattractive if it costs $60,000). Higher discount rates penalize distant cash flows (which is rational—future payments carry more uncertainty and opportunity cost).

Treasury Bond Pricing (TVM in Action)

Treasury bonds demonstrate TVM mechanics directly (prices move inversely with interest rates because future cash flows get discounted at prevailing rates).

A 10-year Treasury bond with 3% coupon pays $30 annually per $1,000 face value (plus $1,000 at maturity). When issued, prevailing yields are 3%, so the bond trades at par ($1,000).

Scenario 1: Rates rise to 5%

New bonds pay $50/year. Your 3% bond becomes less attractive (who wants $30 when new bonds pay $50?). To compete, your bond's price drops until its yield matches 5%.

Present value calculation (discounting all future cash flows at 5%):

PV of coupons = $30/(1.05) + $30/(1.05)^2 + ... + $30/(1.05)^10 = $231.65

PV of principal = $1,000/(1.05)^10 = $613.91

Total PV = $845.56 (the bond's new market price)

You'd lose $154.44 per bond (15.4%) if you sold immediately (the opportunity cost of holding a 3% bond in a 5% rate environment).⁹

Scenario 2: Rates fall to 2%

New bonds pay only $20/year. Your 3% bond delivers more income (attractive relative to alternatives). Price rises until yield drops to 2%.

PV of coupons = $30/(1.02) + ... + $30/(1.02)^10 = $268.96

PV of principal = $1,000/(1.02)^10 = $820.35

Total PV = $1,089.31

You'd gain $89.31 per bond (8.9%) if you sold (the premium investors pay for above-market coupon payments).

The point is: bond prices fluctuate because TVM dictates that future cash flows are worth more when discount rates (yields) fall and less when they rise. You can't "avoid" TVM risk by buying "safe" bonds (unless you hold to maturity and never mark-to-market).

Retirement Planning Applications (The 4% Rule Math)

The 4% rule (you can withdraw 4% of a retirement portfolio annually for 30 years without running out of money) is a TVM calculation in disguise.¹⁰

Start with the question: "How much do I need to retire?" If you want $60,000/year income, you need $1,500,000 ($60,000 / 0.04). The math:

Assume 6% portfolio return (60/40 stock/bond mix), 3% inflation, 30-year retirement. Real return: 6% - 3% = 3%. You withdraw 4% initially ($60,000), adjust for inflation annually.

PV of withdrawal stream (simplified): $60,000 × [1 - (1.03)^-30] / 0.03 ≈ $1,476,000 (the portfolio value needed to fund 30 years of inflation-adjusted withdrawals at 3% real return).

Why this matters: small changes in assumptions create massive swings in required savings. At 4% real return (more optimistic), you need $1,383,000. At 2% real return (conservative), you need $1,632,000 (an 18% difference based on 1% real return variation).¹¹

The test: if you're 40 with $200,000 saved and want to retire at 65 with $1,500,000, can you get there? FV formula: $200,000 × (1 + r)^25 = $1,500,000 → r = 8.1%. You need 8.1% annual returns (achievable with 80-90% stock allocation historically, but not guaranteed). If you can contribute $1,000/month, required return drops to 5.8% (more conservative allocation works).

Mortgage Affordability (TVM Dictates Payment Size)

Mortgage payments are TVM in reverse (calculating the monthly payment that amortizes a present value loan over n periods at rate r). Formula: Payment = PV × [r(1+r)^n] / [(1+r)^n - 1].

Example: $400,000 mortgage at 6.5% for 30 years (360 months, r = 6.5%/12 = 0.542% per month).

Payment = $400,000 × [0.00542(1.00542)^360] / [(1.00542)^360 - 1]

Payment = $400,000 × [0.00542 × 6.64] / [5.64]

Payment = $2,528/month¹²

Over 30 years, you pay $910,080 total ($2,528 × 360 months). Of that, $400,000 is principal, $510,080 is interest (the time value of borrowing money for three decades).

The practical point: rate differences reshape affordability. At 5% (instead of 6.5%), the same $400,000 loan costs $2,147/month—saving $381/month or $137,160 over 30 years. That's why refinancing during rate drops is valuable (you're reducing the TVM cost of the loan).

Why 15-Year Mortgages Cost Less (Compounding Works Both Ways)

A 15-year mortgage typically carries rates 0.5-0.75% lower than 30-year (lenders take less duration risk). But the real savings come from half the compounding time.

$400,000 at 5.75% for 15 years: Payment = $3,315/month, total paid = $596,700 (interest = $196,700).

$400,000 at 6.5% for 30 years: Payment = $2,528/month, total paid = $910,080 (interest = $510,080).

The 15-year mortgage saves $313,380 in total interest (61% less interest than the 30-year option). The tradeoff: monthly payment is $787 higher (requiring more cash flow).¹³

The durable lesson: compounding cuts both ways. It accelerates wealth when you're the investor (earning returns on capital). It accelerates costs when you're the borrower (paying interest on debt). Minimize time for debt, maximize time for investments.

Detection Signals (You're Ignoring TVM If...)

You're not applying TVM thinking to financial decisions if:

You compare dollar amounts without adjusting for time. "I'll have a $2,000,000 house in 30 years" sounds impressive. Adjusted for 3% inflation, that's $823,800 in today's purchasing power (less impressive). Always convert future values to present dollars.

You choose investments based on nominal returns alone (ignoring time horizon). A 5-year CD at 4.5% looks better than stocks (volatile, uncertain returns). But over 30 years, stocks at 9-10% compound to 16x your principal, while 4.5% delivers 3.7x.¹⁴ TVM makes long time horizons favor equities overwhelmingly.

You delay retirement contributions "until you earn more". Waiting from age 25 to 30 costs you 5 years of compounding on every dollar. $5,000 contributed at 25 and left untouched until 65 (40 years at 8%) becomes $108,622. The same $5,000 contributed at 30 becomes $73,926 (a $34,696 penalty for waiting 5 years).¹⁵

You pay off low-interest debt aggressively while avoiding equity investing. Paying down a 3.5% student loan saves you 3.5% annually (locked in, risk-free). Investing in diversified stocks historically returns 9-10% (variable, risky). The TVM-optimal choice: minimum payments on sub-5% debt, maximize equity investing. The spread (10% - 3.5% = 6.5%) compounds in your favor over decades.

Annuities and Perpetuities (TVM for Streams of Cash Flows)

Single cash flows (one payment at a point in time) use basic FV/PV formulas. Annuities (repeated equal payments for n periods) and perpetuities (payments forever) require adjusted formulas.

Ordinary annuity PV (equal payments at end of each period for n years): PV = PMT × [1 - (1+r)^-n] / r

Example: You'll receive $10,000/year for 20 years (starting in 1 year). Discount rate: 5%. What's the present value?

PV = $10,000 × [1 - (1.05)^-20] / 0.05

PV = $10,000 × 12.462

PV = $124,620

That stream of 20 payments totaling $200,000 nominal is worth $124,620 today (opportunity cost of tying up capital for 20 years at 5% discounts future payments).¹⁶

Perpetuity PV (payment forever): PV = PMT / r

Example: A preferred stock pays $100/year forever. Discount rate: 6%. What's it worth?

PV = $100 / 0.06 = $1,667

The point is: perpetuities sound exotic (payments that never end). They're common in finance (preferred stock dividends, some real estate leases, endowment spending). The formula is elegantly simple because distant cash flows approach zero present value (a payment 100 years from now at 6% discount rate: $100/(1.06)^100 = $0.29—essentially worthless).

Next Step (Calculate Your Portfolio's Required Return)

Run a TVM reverse calculation on your retirement plan:

Step 1: Define target. How much do you need at retirement? Use 4% rule: Annual income needed ÷ 0.04 = Portfolio target. Example: $80,000/year → $2,000,000 target.

Step 2: Inventory current savings. Sum all retirement accounts (401k, IRA, taxable). Example: $150,000 current.

Step 3: Calculate years to retirement. Example: Age 35, retire at 65 = 30 years.

Step 4: Solve for required return (assuming no additional contributions).

FV = PV × (1 + r)^n

$2,000,000 = $150,000 × (1 + r)^30

(1 + r)^30 = 13.33

r = 9.1% annually

You need 9.1% returns for 30 years (achievable historically with 90%+ stock allocation, but risky—depends on equity risk premium holding). If that feels aggressive, add contributions.

Step 5: Add monthly contributions to reduce required return.

Using financial calculator or spreadsheet (PMT function): To reach $2,000,000 in 30 years from $150,000 starting point at 7% return → monthly contribution = $1,680. That's more achievable (7% is conservative equity return, contributions reduce dependence on high returns).

The practical antidote: TVM math exposes whether your plan is feasible (given realistic return assumptions). If required returns exceed 10-11% (or required contributions exceed your capacity), you adjust targets (retire later, spend less, save more). The formula doesn't lie—running the numbers forces honesty about tradeoffs.

The durable lesson: time value of money isn't an academic abstraction. It's the mechanism explaining why starting early dominates contributing more later, why bond prices fall when rates rise, why low mortgage rates save hundreds of thousands, and why retirement planning requires compound growth. Every dollar's value depends on when you receive it and what you can earn elsewhere. Internalize that principle, and financial decision-making becomes systematic calculation instead of guesswork.

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Footnotes

1. Author calculation. FV = $1,000 × (1.04)^5 = $1,217. Opportunity cost of waiting 5 years: $217 foregone growth. ↩

2. Author calculation. FV = $10,000 × (1.05)^10 = $16,289. Opportunity cost: $16,289 - $10,000 = $6,289 foregone if payment delayed 10 years. ↩

3. Author calculation. Purchasing power of $1,000 in 20 years at 3% inflation: $1,000 / (1.03)^20 = $554. ↩

4. Trading Economics, United States 10-Year Treasury Yield, accessed December 29, 2025, https://tradingeconomics.com/united-states/government-bond-yield. Current yield: 4.12%. ↩

5. Author calculations. $10,000 at 4% for 30 years: $10,000 × (1.04)^30 = $32,434. At 7% for 30 years: $10,000 × (1.07)^30 = $76,123. Difference: $43,689. ↩

6. Author calculations. Annual: (1.05)^10 = 1.6289. Monthly: (1 + 0.05/12)^120 = 1.6470. Daily: (1 + 0.05/365)^3650 = 1.6487. Compounding frequency impact: $16,487 - $16,289 = $198 (1.2% gain from daily vs annual). ↩

7. Federal Reserve, FOMC Statement (December 18, 2025), accessed December 29, 2025, https://www.federalreserve.gov/newsevents/pressreleases/monetary20251210a.htm. Federal funds rate: 3.50-3.75% (midpoint 3.625%, rounded to 3.50% for examples). ↩

8. Equity risk premium estimates: 3-6% over Treasuries. Current 10-year Treasury: 4.12%. Expected equity return: 4.12% + 4% premium = 8.12%, rounded to 9% for conservative planning. Corporate bond spreads: investment-grade +1-2% over Treasuries (5-6% yield), high-yield +3-5% (7-9% yield). ↩

9. Bond pricing calculation uses standard PV formula for coupon bonds. PV of annuity (coupons) + PV of lump sum (principal at maturity). Discount rate changes from 3% (issue) to 5% (new rate environment). Price drops from $1,000 par to $845.56 (15.4% loss). ↩

10. Bengen, William P. "Determining Withdrawal Rates Using Historical Data." Journal of Financial Planning (1994). The 4% rule assumes 60/40 portfolio, 30-year horizon, inflation-adjusted withdrawals. Historical success rate: 95%+ (portfolio survives 30 years without depletion). ↩

11. Author calculations. PV of annuity formula adjusted for inflation-adjusted withdrawals. At 3% real return: $60,000 × [1 - (1.03)^-30] / 0.03 = $1,476,000. At 4% real: $1,383,000. At 2% real: $1,632,000. Sensitivity: 1% real return change = 18% portfolio requirement change. ↩

12. Mortgage payment formula: M = P × [r(1+r)^n] / [(1+r)^n - 1], where M = monthly payment, P = principal ($400,000), r = monthly rate (6.5%/12), n = months (360). Calculation: $400,000 × 0.00633 = $2,528/month. Total paid: $2,528 × 360 = $910,080. ↩

13. Author calculations. 15-year at 5.75%: Payment = $3,315, total = $596,700, interest = $196,700. 30-year at 6.5%: Payment = $2,528, total = $910,080, interest = $510,080. Interest savings: $313,380 (61% reduction). ↩

14. Author calculations. 5-year CD at 4.5% over 30 years: continuously rolled at same rate: $10,000 × (1.045)^30 = $37,453 (3.7x). Stocks at 9.5%: $10,000 × (1.095)^30 = $148,575 (14.9x). Time horizon transforms compounding outcomes. ↩

15. Author calculations. $5,000 at 8% for 40 years (age 25-65): FV = $5,000 × (1.08)^40 = $108,622. For 35 years (age 30-65): FV = $5,000 × (1.08)^35 = $73,926. Delay cost: $34,696 (32% less terminal wealth for 5-year delay). ↩

16. Annuity PV formula: PV = PMT × [1 - (1+r)^-n] / r. Example: $10,000 annual payment, 20 years, 5% discount: PV = $10,000 × [1 - 0.377] / 0.05 = $10,000 × 12.462 = $124,620. Perpetuity PV: PMT / r = $100 / 0.06 = $1,667. ↩