Amortization and annuities

An ordinary annuity makes each payment at the end of a period. Its first payment is at $t=1$ and its last is at $t=n$. Most loan repayments and many savings questions use this convention unless the problem says payments begin immediately.

An annuity due makes each payment at the beginning of a period. Its first payment is at $t=0$ and its last is at $t=n-1$. Rent, leases and “deposit today, then monthly” plans often use this pattern.

With the same payment and number of deposits, an annuity due earns one additional period of interest.

Because every annuity-due payment is invested for one extra period, its value at a common future date is the corresponding ordinary-annuity value multiplied by $1+i$. This relationship is valid only when the payment, rate and number of payments are otherwise identical.

A timeline is the safest way to decide the calculator setting. Mark the valuation date, first payment, last payment and any period with no cash flow before selecting END or BEGIN.

Financial models work one period at a time. If a nominal annual rate $r$ is compounded $m$ times per year, the periodic rate is $i=\frac{r}{m}$. If the term is $t$ years, the period count is $n=mt$. Use $r$ as a decimal in formulas.

A quoted rate of $6\%$ per year compounded monthly gives $i=\frac{0.06}{12}=0.005$, or $0.5\%$ per month. A five-year monthly loan has $n=5(12)=60$ payments.

Payment frequency and compounding frequency are often equal, but not always. When they differ, use the effective rate for one payment interval or the GDC's P/Y and C/Y settings correctly. Do not divide an annual rate by a guessed number without checking the contract convention.

The number of periods is a count, so it must usually be a whole number. A computed value such as 47.3 payments requires contextual interpretation: the target is reached during a period, or the next scheduled full payment may be needed.

The future value of a savings annuity is the account balance at a chosen future date after all modeled deposits and interest. End-of-period deposits form a geometric series because earlier deposits earn interest for more periods.

For an ordinary annuity with payment $R$, periodic rate $i\ne0$ and $n$ payments, $FV=R\frac{(1+i)^n-1}{i}$. For $i=0$, there is no interest and $FV=nR$.

For beginning-of-period deposits, multiply the ordinary result by $1+i$. To find a regular deposit needed for a target, rearrange or use a financial solver with the target entered as $FV$.

The future value includes deposits and interest. Total contributions are nR, so interest earned equals future value minus contributions when there are no fees or withdrawals.

The present value of an annuity is the single amount today that is financially equivalent to its future payments at a stated interest rate. It discounts every payment back to the valuation date.

For $n$ end-of-period payments of $R$, $PV=R\frac{1-(1+i)^{-n}}{i}$. This model applies to loan repayments, fixed withdrawals, pensions, leases and any level stream valued today.

An annuity-due present value is multiplied by $1+i$ because every payment occurs one period earlier. If the first withdrawal is today, select BEGIN mode rather than END.

Present value is not simply nR. Future payments are worth less today when the discount rate is positive. The choice of rate is a model assumption and can substantially change the answer.

An amortizing loan is repaid through regular payments that cover both interest and principal, reducing the balance to a specified final value—usually $0$—after $n$ payments. Examples include mortgages, car loans and personal loans.

Interest is calculated on the outstanding balance each period. Early in a long loan, the balance is large, so a greater part of each payment is interest. As the balance falls, interest decreases and more of the fixed payment repays principal.

At the loan date, the amount borrowed is the present value of the repayment annuity: $PV=R\frac{1-(1+i)^{-n}}{i}$. A financial solver normally finds the payment directly.

The simple model excludes arrangement fees, insurance, penalties, variable rates, irregular payments and balloon amounts unless they are entered as separate cash flows. A lender's quoted annual percentage measure may also include charges not captured by the nominal rate alone.

For a fully amortizing ordinary loan with present value $PV$, periodic rate $i$ and $n$ payments, the equal payment is $R=PV\frac{i}{1-(1+i)^{-n}}$. In AI HL, a TVM solver or spreadsheet is the expected main method; the formula explains and checks the technology.

If each repayment is $R$, total scheduled repayments are $nR$. Ignoring fees, total interest is $nR-PV$. A longer term usually reduces the periodic payment but increases the total interest because the debt remains outstanding for longer.

Currency payments are normally rounded to the nearest cent. Calculators use unrounded values internally, whereas lenders may round each payment and adjust the final one by a few cents. Keep full precision for mathematical balances, then explain real payment rounding.

The outstanding balance after $k$ payments is the debt immediately after payment $k$, unless the question states another time. Timing matters because the balance immediately before a payment includes one more period of interest.

One method grows the original loan for $k$ periods and subtracts the accumulated value of the first $k$ payments: $B_k=PV(1+i)^k-R\frac{(1+i)^k-1}{i}$.

A second method values the remaining $n-k$ payments at time $k$: $B_k=R\frac{1-(1+i)^{-(n-k)}}{i}$. The two methods agree when the same timing and unrounded payment are used.

An early-settlement quote may differ from the mathematical balance because of payment rounding, daily interest, fees or penalties. Treat the model result as conditional on its stated assumptions.

An amortization table follows a loan period by period. Typical columns are payment number, opening balance, interest, payment, principal repaid and closing balance. Each row's closing balance becomes the next row's opening balance.

If $B_{j-1}$ is the opening balance for period $j$, then interest is $I_j=iB_{j-1}$. Principal repaid is $P_j=R-I_j$, and the new balance is $B_j=B_{j-1}-P_j$. Equivalently, $B_j=B_{j-1}(1+i)-R$.

A spreadsheet should keep unrounded internal values while formatting currency cells to two decimal places. If every row is mathematically rounded to cents, the final balance may need a small adjustment; that is normal in real schedules.

Tables reveal structure that one TVM answer cannot: how interest changes, how much principal has been repaid, the effect of an extra payment and the balance at any date.

Regular-contribution models support education funds, emergency savings, retirement accounts and sinking funds for future purchases. The mathematical output should be interpreted alongside affordability, inflation, investment risk, fees and possible rate changes.

A target problem may ask for a deposit, number of periods or rate. Technology is especially useful when the unknown appears in an exponent or when the required number of payments must be found by iteration.

Comparisons must use the same valuation date and cash-flow timing. A plan with a larger final value may simply contain larger or earlier deposits. Compare contributions, interest earned, fees and assumptions, not only the headline balance.

Investment returns are often uncertain, unlike a fixed loan rate. Treat a constant-return annuity calculation as a scenario, not a guaranteed forecast, unless the product genuinely guarantees the rate.

A TVM solver commonly uses N, I%, PV, PMT and FV. N is the total number of payment periods. I% is usually a quoted annual percentage when P/Y and C/Y are configured, but calculator conventions differ. PV, PMT and FV are cash flows at different dates.

Set P/Y to payments per year and C/Y to compounding periods per year when those fields exist. Select END for ordinary annuities and BEGIN for annuities due. Clear previous TVM values before starting a new question.

Use a consistent sign convention: cash received is one sign and cash paid is the opposite. For a borrower, PV can be positive and PMT negative. For a savings plan, PMT can be negative and FV positive. If every cash flow has the same sign, many solvers cannot find a meaningful solution.

Spreadsheet functions often use PMT$rate,nper,pv,fv,type$, FV$rate,nper,pmt,pv,type$ and PV$rate,nper,pmt,fv,type$, where type = 0 means end and type = 1 means beginning. Enter the periodic rate, not an annual percentage, unless the function documentation says otherwise.

Attempt each question by drawing or imagining the cash-flow timeline first. Then state the rate per period, total periods, payment mode and TVM signs before using technology.

An annuity has equal cash flows at equal intervals. Convert to one period with $i=\frac{r}{m}$ and $n=mt$, then decide whether payments occur at the end or beginning.

Use financial technology as the main method: clear old values, set N, I%, P/Y, C/Y and END/BEGIN, enter opposite signs for inflows and outflows, solve, and interpret the date and sign of the result.

For loans, each payment splits into interest $iB$ and principal $R-iB$. Outstanding balance can be found from the original loan and completed payments or as the present value of remaining payments.

Finish with currency accuracy, total cost or target interpretation, and a limitation such as fees, changing rates, inflation, missed payments or investment uncertainty.