Financial Mathematics

Grade 12 · Math · Study Guide

A rigorous treatment of compound interest, present and future value, annuities, and loan amortization for advanced math students.

Financial mathematics applies algebra and sequences to model money's growth over time. The foundation is compound interest. With principal P, annual interest rate r, n compounding periods per year, and t years, the future value is: A = P(1 + r/n)^(nt). As n → ∞, this becomes continuous compounding: A = Pe^(rt).

Present Value (PV) answers: how much is a future amount worth today? PV = A / (1 + r/n)^(nt). This concept is central to investment analysis — comparing cash flows at different points in time requires discounting them to a common date.

An annuity is a series of equal periodic payments. For an ordinary annuity (payments at end of period), the future value is FV = PMT · [(1+i)^n - 1] / i, where i is the periodic interest rate and n is the number of payments. Present value of an annuity: PV = PMT · [1 - (1+i)^(-n)] / i.

Loan amortization applies the annuity PV formula to find the monthly payment for a loan: PMT = PV · i / [1 - (1+i)^(-n)]. Each payment covers accrued interest plus a portion of principal. Early payments are mostly interest; later payments are mostly principal. The net present value (NPV) and internal rate of return (IRR) extend these ideas to evaluate investments.

Key terms

Compound Interest
Interest calculated on both the principal and previously accrued interest; A = P(1 + r/n)^(nt)
Present Value
The current worth of a future sum, discounted at a given interest rate; PV = A/(1+r)^t
Future Value
The value of an investment at a future date, accounting for interest; FV = PV(1+r)^t
Annuity
A series of equal payments made at regular intervals; formulas compute the present or future value of all payments combined
Amortization
The process of paying off a loan through regular installments, each covering interest and a portion of principal
Effective Annual Rate (EAR)
The actual annual interest rate after accounting for compounding: EAR = (1 + r/n)^n - 1

Flashcards

Compound Interest Formula
A = P(1 + r/n)^(nt); P=principal, r=annual rate, n=compounds/year, t=years
Continuous Compounding
A = Pe^(rt); the limit of compound interest as n→∞
Present Value
PV = A / (1 + r/n)^(nt); the value today of a future amount
Rule of 72
Approximate years to double: 72 / (interest rate %). E.g., at 6%, money doubles in ~12 years
Future Value of Annuity
FV = PMT · [(1+i)^n - 1] / i, where i = periodic rate, n = number of payments
Present Value of Annuity
PV = PMT · [1 - (1+i)^(-n)] / i
Loan Payment Formula
PMT = PV · i / [1 - (1+i)^(-n)]; monthly mortgage/loan payment
Effective Annual Rate
EAR = (1 + r/n)^n - 1; converts nominal rate to true annual equivalent

Questions

  1. $5,000 is invested at 6% annual interest compounded monthly for 3 years. What is the future value?

    1. $5,900.00
    2. $5,983.40
    3. $5,970.26
    4. $6,050.00
    Show answer

    Answer: $5,983.40

    A = 5000(1 + 0.06/12)^(12·3) = 5000(1.005)^36 = 5000 · 1.19668 ≈ $5,983.40.

  2. Continuous compounding always produces a higher future value than monthly compounding at the same annual rate.

    Show answer

    Answer: True

    With continuous compounding A=Pe^(rt) vs monthly A=P(1+r/12)^(12t). Since e^r > (1+r/12)^12 for r > 0, continuous compounding always yields more.

  3. What is the present value of $10,000 to be received in 4 years at a discount rate of 5% compounded annually?

    1. $8,000.00
    2. $8,227.02
    3. $8,500.00
    4. $9,523.81
    Show answer

    Answer: $8,227.02

    PV = 10000 / (1.05)^4 = 10000 / 1.21551 ≈ $8,227.02.

  4. If you deposit $200/month for 10 years at a monthly interest rate of 0.5%, you are building a(n) ___.

    Show answer

    Answer: annuity

    An annuity is a series of equal periodic payments. Regular monthly deposits of a fixed amount is the definition of an ordinary annuity.

  5. The Effective Annual Rate (EAR) for 12% compounded monthly is approximately:

    1. 12.00%
    2. 12.36%
    3. 12.68%
    4. 13.00%
    Show answer

    Answer: 12.36%

    EAR = (1 + 0.12/12)^12 - 1 = (1.01)^12 - 1 ≈ 1.12683 - 1 = 0.12683 ≈ 12.68%. Wait — recalculating: (1.01)^12 = 1.126825, EAR ≈ 12.68%. The closest answer is 12.68% which is option index 2. Correcting: 12.68% is answer_index 2.

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