Essential Time Value of Money Calculations and Financial Formulas

Posted on Dec 5, 2025 in Finance

1. Simple vs. Compound Interest Calculation (4 Years)

Scenario: 4 years at 3% on a principal of $1,330.

  • Simple Future Value (FVs): FVs = P(1 + rt) = 1330(1 + 0.03 · 4) = $1,489.60.

  • Compound Future Value (FVc): FVc = P(1 + i)n = 1330(1.03)4 = $1,496.93.

  • Extra Interest Earned: FVc – FVs = $7.33.

2. Comparing Simple and Compound Interest in Year 10

Scenario: Calculating interest earned in the 10th year (7% on $2,800 principal).

  • Simple Interest: Yearly interest = 2800 × 0.07 = $196.

  • Compound Interest in Year 10: Interest in Year 10 = Balance at end of Year 9 × 0.07. Assuming the balance at the end of Year 9 is $5,147.69, the interest is 5147.69 × 0.07 = $360.34.

  • Difference: 360.34 – 196 = $164.34.

3. Future Value of a Deferred Single Sum Investment

Scenario: Invest $5,350 in Year 5 for 5 years at 8.6%. Total time horizon is 10 years.

  • You invest $5,350 in Year 5 for 5 years at 8.6%.

  • Compute Future Value (FV) over 5 years: FV = 5350(1.086)5 = $8,081.70. (This is the amount 10 years from today).

  • TVM Keys: N=5, I/Y=8.6, PV=−5350, FV=? ⇒ $8,081.70.

4. Quarterly Compounding on a Lump Sum Deposit

Scenario: Deposit $3,900 for 6 years with quarterly compounding.

  • Deposit: $3,900. Quarterly interest rate (iq): 0.65% (0.0065). Total periods (N): 6 × 4 = 24 quarters.

  • Future Value (FV): FV = PV(1 + iq)N = 3900(1.0065)24 = $4,556.12.

  • TVM Keys: N=24, I/Y=0.65, PV=−3900, FV=? ⇒ $4,556.12.

5. Piece-wise Annual Compounding with Changing Rates

Scenario: Initial deposit of $7,250 subject to three different annual rates.

  1. Step 1 (First 5 years @ 4%): FV5 = 7250(1.04)5 = $8,820.73.

  2. Step 2 (Next 4 years @ 4.6%): FV9 = 8820.73(1.046)4 = $10,559.21.

  3. Step 3 (Next 8 years @ 5.3%): FV17 = 10559.21(1.053)8 = $15,960.94.

  4. (TVM calculations confirm these three forward rolls.)

6. Time to Reach a Target After a Delayed Payout

Scenario: Receive $11,200 in 4 years, then grow at 6.6% until reaching $19,300.

  • We need to solve for time (t) required for the growth phase: 19,300 = 11,200(1.066)t.

  • Time (t) after payout: t = ln(19300/11200) / ln(1.066) = 8.51 years.

  • Total Wait Time from Today: 4 + 8.51 = 12.51 years.

7. Present Value of Uneven Cash Flows (12% Discount Rate)

Cash flows: $4,700 (t=1), $9,700 (t=2), $15,900 (t=3). Discount rate = 12%.

  • Present Value (PV) Calculation: PV = 4700/1.12 + 9700/1.122 + 15900/1.123 = $23,246.52.

  • Calculator Method (TVM CF Register): CF0=0; C01=4700; C02=9700; C03=15900; I=12; NPV ⇒ $23,246.52.

8. Present Value of a Deferred Semiannual Annuity

Scenario: $3,700 payment every 6 months for 12 years. APR 4.7% compounded semiannually. First payment at 5.5 years.

  • Parameters: Total payments (N) = 12 years × 2 = 24 payments. Semiannual interest rate (i) = 0.047 / 2.

  • The first payment is at 5.5 years. The value of the annuity one period before the first payment is at 5 years (10 periods).

  • Step 1: PV at 5 Years (PV5): PV5 = 3,700 · [1 – (1 + i)-24] / i = $67,284.39.

  • Step 2: Discount to Today (PV0): PV0 = 67284.39(1 + i)-10 = $53,337.90.

  • TVM Keys Summary:

    1. N=24, I/Y=4.7/2, PMT=3700 ⇒ PV = $67,284.39.
    2. N=10, I/Y=4.7/2, PV=−67284.39 ⇒ PV0 = $53,337.90.

9. Future Value of an Ordinary Annuity (Annual Deposits)

Scenario: Deposit $1,200 at year-end for 10 years at 5.59%.

  • Future Value (FV) Calculation: FV = PMT · [(1 + i)n – 1] / i = 1200 · [(1.0559)10 – 1] / 0.0559 = $15,515.69.

  • TVM Keys: N=10, I/Y=5.59, PMT=−1200, FV=? ⇒ $15,515.69.

10. Level Payment Mortgage Calculation (Monthly Compounding)

Scenario: Borrow $240,000 over 25 years at 5.71% APR, compounded monthly.

  • Parameters: Principal (PV) = $240,000. Monthly interest rate (im) = 0.0571 / 12. Total periods (N) = 25 × 12 = 300 months.

  • Monthly Payment (PMT): PMT = [im · PV] / [1 – (1 + im)-N] = $1,504.06.

  • TVM Keys: N=300, I/Y=5.71/12, PV=240000, PMT=? ⇒ $1,504.06.