Compound Interest and Time Value of Money Guide
Discrete compounding means interest is applied at fixed intervals. A yearly APR can be split into annual, quarterly, monthly, weekly, or daily periods. The calculator converts annual inputs into the matching periodic rate, then reports future value, present value, interest, payment totals, a schedule, and calculation steps.
Formula Table
| Need | Formula | Use when |
|---|---|---|
| Future value | \(FV=PV(1+i)^n\) | Grow a present amount forward. |
| Present value | \(PV=\dfrac{FV}{(1+i)^n}\) | Discount a future amount to today. |
| Solve for rate | \(i=\left(\dfrac{FV}{PV}\right)^{1/n}-1\) | Find the periodic return for a lump sum. |
| Solve for time | \(n=\dfrac{\ln(FV/PV)}{\ln(1+i)}\) | Find periods needed for a lump sum target. |
| Ordinary annuity FV | \(FV=A\dfrac{(1+i)^n-1}{i}\) | Payments at the end of each period. |
| Ordinary annuity PV | \(PV=A\dfrac{1-(1+i)^{-n}}{i}\) | Value today of end-of-period payments. |
| Annuity due FV | \(FV=A\dfrac{(1+i)^n-1}{i}(1+i)\) | Payments at the start of each period. |
| Annuity due PV | \(PV=A\dfrac{1-(1+i)^{-n}}{i}(1+i)\) | Present value of start-of-period payments. |
| Payment from FV | \(A=\dfrac{FV\cdot i}{(1+i)^n-1}\) | Required deposit to reach a target future value. |
| Payment from PV | \(A=\dfrac{PV\cdot i}{1-(1+i)^{-n}}\) | Payment for a present-value loan or payout. |
| APR to EAR | \(EAR=(1+r/m)^m-1\) | Convert nominal APR to effective yearly return. |
| EAR to APR | \(r=m((1+EAR)^{1/m}-1)\) | Find equivalent nominal APR for a frequency. |
| Continuous compounding | \(FV=PV e^{rt}\) | Compare the continuous limit with discrete compounding. |
Worked Examples
Savings with monthly compounding: 10,000 at 6% APR compounded monthly for 10 years gives \(FV=10000(1+0.06/12)^{120}\approx 18193.97\). Interpretation: about 8,193.97 is interest before taxes or fees.
Loan discounting to present value: A 25,000 payment due in 5 years at 7% annual discount rate has \(PV=25000/(1.07)^5\approx 17824.66\). A buyer paying more than that is accepting a lower return.
Ordinary annuity deposits: Depositing 100 at the end of each month for 10 years at 6% APR compounded monthly gives \(FV=100((1.005)^{120}-1)/0.005\approx 16387.93\).
Annuity due rent payments: A 1,200 rent payment at the start of each month for 12 months discounted at 0.5% per month has \(PV=1200(1-(1.005)^{-12})/0.005(1.005)\approx 14005.87\).
APR to EAR: 12% APR compounded monthly gives \(EAR=(1+0.12/12)^{12}-1\approx 12.6825\%\), so the effective yearly rate is higher than 12%.
Required rate: To grow 5,000 to 7,500 in 6 years, \(i=(7500/5000)^{1/6}-1\approx 6.9915\%\) per year.
Common Mistakes
- Mixing annual and monthly units: 6% per year is not 6% per month. Use APR and frequency or convert to periodic rate first.
- Confusing PV and FV: PV is the value today; FV is the value at the end of the term.
- Ignoring payment timing: Annuity due payments are one period earlier than ordinary payments and produce a higher value when rates are positive.
- Expecting taxes or inflation: Results are nominal unless you adjust the rate or cash flows yourself.
- Reading APR as EAR: APR is nominal; EAR includes the compounding effect.
When to Use PV vs FV
Use future value when you want to know what an investment or savings plan may become. Use present value when you want to compare a future payment, bond cash flow, loan payoff, lease, or investment target to money today.