Financial Mathematics SL+HL

IB Mathematics: Applications & Interpretation · Topics 1.4, 1.7

JMaths

www.jmaths.xyz

1

Compound Interest & Depreciation

SL 1.4 — Financial applications of geometric sequences

Formula (given in booklet): $ 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

\$10 000 invested at 4.8% compounded monthly for 6 years. Find the future value.

N = 72 (= 6 × 12), I% = 4.8, PV = −10000, PMT = 0, FV = ?, P/Y = C/Y = 12

FV = \$13 329.91

TI-84 Plus CE

[APPS] → Finance → TVM Solver
N = 72 I% = 4.8 PV = −10000 PMT = 0
FV = ? P/Y = 12 C/Y = 12 → cursor on FV, [ALPHA][ENTER]

TI-Nspire CX II

[Menu] → Finance → Finance Solver
N = 72 I(%) = 4.8 PV = −10000 Pmt = 0
FV = (blank) PpY = 12 CpY = 12 → tab to FV, [Enter]

Casio fx-CG50

[MENU] → Financial → Compound Interest
n = 72 I% = 4.8 PV = −10000 PMT = 0
FV = 0 P/Y = 12 C/Y = 12 → highlight FV, [SOLVE] (F6)

2

Annual Depreciation

SL 1.4.2 — Value decreasing by a fixed percentage

Method: Depreciation of $ d\% $ per year keeps a fraction $ (1 - d/100) $. Model as $ FV = PV(1 - d/100)^n $, or on the TVM solver set I% = $ -d $.

Worked Example — Car Depreciation

A car worth \$18 000 depreciates 12% per year. Find its value after 5 years.

By formula: $ 18000 \times (1 - 0.12)^5 = 18000 \times 0.88^5 $

TVM: N = 5, I% = −12, PV = −18000, PMT = 0, FV = ?, P/Y = C/Y = 1

FV = \$9498.16

TI-84 Plus CE — Depreciation

[APPS] → Finance → TVM Solver
N = 5 I% = −12 PV = −18000 PMT = 0
FV = ? P/Y = 1 C/Y = 1 → solve FV

TI-Nspire CX II — Depreciation

[Menu] → Finance → Finance Solver
N = 5 I(%) = −12 PV = −18000 Pmt = 0
FV = (blank) PpY = 1 CpY = 1 → solve FV

Casio fx-CG50 — Depreciation

[MENU] → Financial → Compound Interest
n = 5 I% = −12 PV = −18000 PMT = 0
FV = 0 P/Y = 1 C/Y = 1 → highlight FV, [SOLVE]

► Simple vs compound: Simple interest is arithmetic — the same amount each year ($ I = PV \times r/100 $ per year, total $ = PV(1 + rn/100) $). Compound interest is geometric — interest earns interest. The TVM solver models compound; don't use it for simple-interest questions.

Financial Mathematics SL+HL

IB Mathematics: Applications & Interpretation · Topics 1.4, 1.7

JMaths

www.jmaths.xyz

3

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 \$25 000 at 6.9% nominal annual rate, compounded monthly, repaid over 5 years. Find the monthly payment.

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

Total paid = 60 × 493.85 = \$29 631.08

Total interest = 29631.08 − 25000 = \$4631.08

PMT = −\$493.85 per month (negative = paid out)

TI-84 Plus CE — Loan

[APPS] → Finance → TVM Solver
N = 60 I% = 6.9 PV = 25000 PMT = ?
FV = 0 P/Y = 12 C/Y = 12 → cursor on PMT, [ALPHA][ENTER]

TI-Nspire CX II — Loan

[Menu] → Finance → Finance Solver
N = 60 I(%) = 6.9 PV = 25000 Pmt = (blank)
FV = 0 PpY = 12 CpY = 12 → solve Pmt

Casio fx-CG50 — Loan

[MENU] → Financial → Compound Interest
n = 60 I% = 6.9 PV = 25000 PMT = ?
FV = 0 P/Y = 12 C/Y = 12 → highlight PMT, [SOLVE]

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

4

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 per month at 5.4% compounded monthly for 10 years. Find the accumulated value.

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

Total deposited = 120 × 200 = \$24 000

Interest earned = 31730.20 − 24000 = \$7730.20

FV = \$31 730.20

TI-84 Plus CE — Savings

[APPS] → Finance → TVM Solver
N = 120 I% = 5.4 PV = 0 PMT = −200
FV = ? P/Y = 12 C/Y = 12 → solve FV

TI-Nspire CX II — Savings

[Menu] → Finance → Finance Solver
N = 120 I(%) = 5.4 PV = 0 Pmt = −200
FV = (blank) PpY = 12 CpY = 12 → solve FV

Casio fx-CG50 — Savings

[MENU] → Financial → Compound Interest
n = 120 I% = 5.4 PV = 0 PMT = −200
FV = 0 P/Y = 12 C/Y = 12 → highlight FV, [SOLVE]

Financial Mathematics SL+HL

IB Mathematics: Applications & Interpretation · Topics 1.4, 1.7

JMaths

www.jmaths.xyz

5

Outstanding Balance & Early Repayment

Finding the balance after partial repayment

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

Worked Example — Outstanding Balance

From the car loan above (\$25 000 at 6.9%, 60 monthly payments of \$493.85). Find the balance after 24 payments.

N = 24, I% = 6.9, PV = 25000, PMT = −493.85, FV = ?, P/Y = C/Y = 12

FV = −\$16 017.33 (outstanding balance is \$16 017.33)

TI-84 Plus CE — Balance

[APPS] → Finance → TVM Solver
N = 24 I% = 6.9 PV = 25000 PMT = −493.85
FV = ? P/Y = 12 C/Y = 12 → solve FV

TI-Nspire CX II — Balance

[Menu] → Finance → Finance Solver
N = 24 I(%) = 6.9 PV = 25000 Pmt = −493.85
FV = (blank) PpY = 12 CpY = 12 → solve FV

Casio fx-CG50 — Balance

[MENU] → Financial → Compound Interest
n = 24 I% = 6.9 PV = 25000 PMT = −493.85
FV = 0 P/Y = 12 C/Y = 12 → highlight FV, [SOLVE]

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

6

Inflation, Real Value & Compounding Frequency

SL 1.7 — Financial modelling with technology

Real (inflation-adjusted) value: If inflation is $ i\% $ per year, the real value of a future sum in $ n $ years is $ \dfrac{\text{nominal value}}{(1 + i/100)^n} $.

Worked Example — Real Value

A sum grows at 5% nominal for 4 years while inflation runs at 2% per year. Find the real growth.

Nominal factor over 4 years: $ 1.05^4 = 1.21551 $

Real value = $ \dfrac{1.05^4}{1.02^4} = \dfrac{1.21551}{1.08243} = 1.12294 $

Real annual growth $ \approx \dfrac{1.05}{1.02} - 1 = 0.0294 $

Real growth ≈ 12.29% over 4 years, ≈ 2.94% per year

Effective annual rate (EAR): $ \text{EAR} = \left[\left(1 + \dfrac{r}{100k}\right)^k - 1\right] \times 100\% $ — converts a nominal rate $ r\% $ compounded $ k $ times a year into one equivalent annual %.

Worked Example — Compounding Frequency

Compare 6% nominal on \$1000 over 1 year at different compounding frequencies.

Frequency	k	FV	EAR
Annually	1	\$1060.00	6.00%
Monthly	12	\$1061.68	6.17%
Daily	365	\$1061.83	6.18%

More frequent compounding gives a higher FV and EAR — keep P/Y = C/Y to match.

7

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 it 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. Include the final smaller payment if N is not a whole number.

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)