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Financial Mathematics SL+HL

IB Mathematics: Applications & Interpretation · Topics 1.4, 1.7

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Compound Interest & Depreciation

SL 1.4 — Financial applications of geometric sequences

Formula: $FV = PV \times \left(1 + \frac{r}{100k}\right)^{kn}$
where $r$ = annual rate (%), $k$ = compounding periods per year, $n$ = years

► In IB, always use the TVM solver on your GDC instead of this formula. The formula is given for understanding — the GDC is faster and less error-prone.

TVM Solver Setup — Compound Interest

N	Total number of compounding periods ($k \times n$)
I%	Annual interest rate (always per annum, never divide yourself)
PV	Present value — negative if you're paying/investing
PMT	Payment per period (0 for simple compound interest)
FV	Future value — positive if you receive it
P/Y	= C/Y always in IB (compounding periods per year)

✗ Sign convention: Money OUT is negative, money IN is positive. If you invest 5000, PV = −5000. If you receive the future value, FV is positive.

Worked Example — Compound Interest

10000 dollars invested at 4.8% compounded monthly for 6 years.

            TI-84: APPS → Finance → TVM Solver

            N=72, I%=4.8, PV=−10000, PMT=0, FV=?, P/Y=12, C/Y=12 → Solve for FV

            FV = 13330.90

            Nspire: Menu → Finance → TVM Solver

            N=72, I(%)=4.8, PV=−10000, PMT=0, FV=?, PpY=12, CpY=12 → Solve

            Casio: MENU → Financial → TVM → Compound Interest

            n=72, I%=4.8, PV=−10000, PMT=0, FV=?, P/Y=12, C/Y=12 → Solve

► Depreciation: Same setup but the value decreases. Annual depreciation of 15% → I% = −15, or model as $FV = PV(1 - 0.15)^n$.

Loans & Amortization

SL 1.7 — Amortization and annuities using technology

A loan is repaid with equal monthly payments. The TVM solver finds the payment (PMT) or the number of payments (N).

Key setup: PV = loan amount (positive — you receive it), FV = 0 (loan fully repaid), PMT = negative (you pay it out)

Worked Example — Car Loan

Loan of 25000 dollars at 6.9% nominal annual rate, compounded monthly, repaid over 5 years.

            N=60, I%=6.9, PV=25000, PMT=?, FV=0, P/Y=12, C/Y=12

            PMT = −493.87 (monthly payment of 493.87 dollars)

Total paid: 60 × 493.87 = 29632.20 dollars
Total interest: 29632.20 − 25000 = 4632.20 dollars

✗ N is NOT years when P/Y = 12. If monthly payments for 5 years: N = 60 (not 5).

Savings Plans & Annuities

Regular deposits into a growing fund

Regular payments INTO an account. PV = 0 (start with nothing), PMT = negative (you pay in), FV = positive (what you accumulate).

Worked Example — Monthly Savings

Save 200 dollars per month at 5.4% compounded monthly for 10 years.

            N=120, I%=5.4, PV=0, PMT=−200, FV=?, P/Y=12, C/Y=12

            FV = 31,541.16 dollars

Total deposited: 120 × 200 = 24000. Interest earned: 7541.16 dollars.

Outstanding Balance & Early Repayment

Finding balance after partial repayment

Method: To find the outstanding balance after $k$ payments, use the TVM solver with N = $k$ (payments made so far), solve for FV. The FV is the remaining balance.

Worked Example — Outstanding Balance

From the car loan above (25000 at 6.9%, 60 monthly payments of 493.87). Find balance after 24 payments.

            N=24, I%=6.9, PV=25000, PMT=−493.87, FV=?, P/Y=12, C/Y=12

            FV = −16,118.42 (outstanding balance is 16,118.42)

► Increasing payments: If someone increases their payment, find the outstanding balance first, then use that as the new PV with the new PMT to find the new N.

Things to Watch Out For

✗ "Per annum" means per year. I% is always the annual rate. Never divide I% yourself — the calculator handles it using P/Y.

✗ P/Y = C/Y in IB. Always set both to the same value (compounding frequency per year).

✗ Rounding the final payment. If N = 68.3 months, the borrower makes 68 full payments plus a smaller final payment. Find the final payment by solving with N = 68 for FV, then that FV plus one month's interest = final payment.

✗ "How much interest was paid?" = Total payments − original loan. Don't forget to include the final smaller payment if N is not a whole number.

► Real value with inflation: If inflation is 3% per year, the real value of money in $n$ years is $\frac{\text{nominal value}}{(1.03)^n}$. This is an HL extension topic.

► Effective annual rate: $(1 + \frac{r}{100k})^k - 1$ gives the equivalent annual rate. Useful for comparing different compounding frequencies.

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