Financial applications

The future value $FV$ is the value of a present amount after repeated compound growth. If the periodic decimal interest rate is $i$ for $n$ periods, then $FV=PV(1+i)^n$. This is the nth-term formula of a geometric sequence.

Each arrow is one compounding period. Four arrows give $FV=1000(1.05)^4=1215.51$ to the nearest cent.

When interest is compounded annually at annual rate $r$ for $t$ years, use $i=r$ and $n=t$. For any other frequency, first convert the annual rate and time to matching periodic values.

The formula can be rearranged to find an unknown starting amount. To find a rate, isolate the power and take an nth root. To find time, use logarithms or a GDC solver because the unknown appears in an exponent.

A calculated time such as $11.90$ years may need contextual interpretation. If interest is credited only once per whole year and the question asks when a target is first reached, test years 11 and 12 and choose the first valid whole period.

A nominal annual rate may be compounded more than once per year. If the nominal decimal rate is $r$ and there are $m$ equal compounding periods each year, then the periodic rate is $i=\frac{r}{m}$. Over $t$ years, the number of periods is $n=mt$.

Common values are $m=12$ for monthly, $m=4$ for quarterly, $m=2$ for semi-annual and $m=365$ for daily compounding when the question specifies 365 days. Do not infer a convention that the problem does not state.

The effective annual rate is the actual percentage increase over one year after within-year compounding. For nominal rate $r$ compounded $m$ times, $EAR=(1+\frac{r}{m})^m-1$. More frequent compounding makes the effective rate slightly larger than the nominal rate when the nominal rate is positive.

When comparing financial products, rates must be converted to a common time basis. Comparing a nominal monthly-compounded rate directly with an effective annual rate is not a fair comparison.

The time value of money means the same nominal amount at two different dates does not have the same financial value. Money available now can earn interest, so a future cash amount normally has a smaller present value when the discount rate is positive.

Rearranging the compound-interest formula gives $PV=\frac{FV}{(1+i)^n}$. This process is called discounting. Compounding moves a value forward in time; discounting moves it backward.

For a future target, present value answers: ‘How much must be invested now?’ For a future payment, it answers: ‘What is that future cash flow worth today at the chosen rate?’ The chosen discount rate is an assumption and should match the period and context.

Appreciation is repeated percentage growth in an asset's value. A rate of $a$ per period uses multiplier $1+a$, so $V_n=V_0(1+a)^n$.

Reducing-balance depreciation is repeated percentage decay. A depreciation rate $d$ per period uses multiplier $1-d$, so $V_n=V_0(1-d)^n$. The same percentage is applied to a smaller balance each period, so the money lost each period is not constant.

Straight-line depreciation subtracts a fixed money amount each period and is arithmetic, not geometric. Read the wording carefully: ‘loses 12% of its value each year’ is reducing-balance depreciation; ‘loses $3000 each year’ is straight-line.

A model can be rearranged to find an unknown appreciation or depreciation rate. Check that the result makes contextual sense: a depreciation rate should normally lie between 0% and 100%, and asset values from this model remain non-negative.

A lump-sum investment uses one compound-interest term. Regular equal deposits form a geometric series because each deposit earns interest for a different number of periods.

For an ordinary annuity, deposits of $R$ are made at the end of each period. Immediately after the nth deposit, the future value is $FV=\frac{R[(1+i)^n-1]}{i}$. The final deposit earns no interest before that valuation time. Use a positive integer payment count $n$ and a constant periodic rate $i>-1$. For quotient formulas assume $i\ne0$; at zero interest, $FV=nR$ and $R=\frac{FV}{n}$.

For an annuity due, the $n$ deposits are made at times $0,1,\ldots,n-1$. Compare both streams at the same terminal date $t=n$: each due deposit then earns one extra period relative to ordinary deposits at $1,\ldots,n$, so multiply the ordinary future value by $1+i$. This is not the value immediately after the last due deposit at $t=n-1$.

The savings formula can be rearranged to find a regular deposit needed for a target. Deposits, compounding and the periodic rate must use the same interval. If deposits are monthly but interest is compounded annually, a more careful timeline or appropriate calculator setup is required.

An amortizing loan is repaid through equal periodic payments. Each payment first covers the interest charged on the outstanding balance; the remainder reduces the principal. Early payments often contain more interest because the balance is larger.

If $PV$ is borrowed now and $R$ is paid at the end of each period for $n$ periods at periodic rate $i$, then the loan principal equals the present value of the repayments: $PV=\frac{R[1-(1+i)^{-n}]}{i}$. Use a positive integer $n$, equal end-of-period repayments and constant $i>-1$. The quotient formulas require $i\ne0$. At zero interest, $PV=nR$, $R=\frac{PV}{n}$, and $B_k=PV-kR$.

Rearranging gives the payment formula $R=\frac{PVi}{[1-(1+i)^{-n}]}$. The outstanding balance after $k$ payments can be found by growing the original loan for $k$ periods and subtracting the accumulated value of the payments made. Use a positive integer $n$, equal end-of-period repayments and constant $i>-1$. The quotient formulas require $i\ne0$. At zero interest, $PV=nR$, $R=\frac{PV}{n}$, and $B_k=PV-kR$.

Total repayment is $Rn$ when every payment is equal. Total interest is $Rn-PV$, although an actual lender may adjust the final payment because regular payments are rounded to cents and may include fees not represented by the model.

Inflation is a general rise in prices. If prices rise by decimal rate $f$ per year, a current cost $C_0$ is modeled by $C_t=C_0(1+f)^t$.

The real value of a future money amount $M$, measured in today's purchasing power, is $\frac{M}{(1+f)^t}$. This is discounting by the inflation factor and is an explicit AI SL syllabus application.

To compare an investment with inflation, place both rates on a compatible effective annual basis. The exact real growth factor is $\frac{(1+r)}{(1+f)}$, so $r_{\mathrm{real}}=\frac{(1+r)}{(1+f)}-1$. Subtracting rates is only an approximation.

A positive money balance does not guarantee increased purchasing power. If the effective investment return is lower than inflation, the real value falls.

Compare rates on the same effective time basis and compare cash amounts at the same date. Otherwise the comparison is not like-for-like.

For investments, consider effective return, fees, taxes, risk, liquidity and inflation. For loans, consider payment size, total interest, fees, variable-rate risk, early-repayment conditions and affordability.

A smaller loan payment does not prove a loan is cheaper: a longer term may create much more total interest. Calculate total repayments and interpret the trade-off.

A good conclusion states the mathematical advantage and at least one assumption. For example, “Option A has the larger five-year future value, assuming its rate remains fixed and fees are equal.”

Step 1: mark every cash flow and date. Step 2: select simple interest, one compound amount, depreciation, regular savings or a loan model. Step 3: match rate and period. Step 4: count periods and decide BEGIN or END timing.

Step 5: choose a formula, spreadsheet recurrence or GDC finance solver. Step 6: calculate with full precision. Step 7: check direction and approximate size. Step 8: interpret with currency, date, units and assumptions.

Financial models commonly assume a fixed rate, regular timing, no missed payments and no omitted fees or taxes. Inflation, market returns, depreciation and variable loan rates can change over time, so long extrapolations are less reliable.

A spreadsheet is useful for an amortization schedule or changing rates. Each row can show opening balance, periodic interest, payment and closing balance, making the recurrence visible.

Continuous compounding and the constant $e$ are enrichment in the current guide and are not examined here. Do not use a continuous model unless a problem explicitly defines one.

Borrowing and lending also involve affordability, transparent fees, unequal access to credit and cultural or religious views of interest. Mathematics describes consequences but does not make ethical decisions by itself.

Many GDCs provide a TVM or financial solver. Typical fields are $N$ (number of periods), $I\%$ (nominal annual percentage rate), $PV$, $PMT$, $FV$, $\frac{P}{Y}$ (payments per year) and $\frac{C}{Y}$ (compounding periods per year). Labels differ by calculator, so know your approved model's menu.

Use a consistent cash-flow sign convention. Money received and money paid must have opposite signs. For a loan, the amount borrowed may be positive and repayments negative. For savings, deposits may be negative and the future value positive. If all cash flows have the same sign, the calculator may report an error or a meaningless result.

Set $\frac{P}{Y}$ and $\frac{C}{Y}$ deliberately. A monthly loan normally has both set to 12; annual lump-sum growth normally uses 1. Check whether the calculator expects $I\%$ as a percentage such as 5.4 or as a decimal—TVM solvers usually use percentage form.

For hand calculations and GDC methods alike, show the model, identify values and interpret the output. Calculator screens do not replace mathematical communication. Restore changed financial settings after the problem so they do not affect later work.

Attempt each question before reading its solution. Decide first whether the situation uses simple interest, one compound amount, reducing-balance change, regular deposits or loan repayments.

For every answer, check rate-period consistency, calculator precision, money rounding, units and whether a whole-number period must be tested.

Choose a formula only after identifying the cash flows, period and payment timing. Use this sheet to check a completed solution rather than to skip the model.