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Number and Algebra - Annuities (HL)

Grade 11IB_AI

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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An annuity is a sequence of equal payments made at regular intervals (e.g., monthly, quarterly, or annually) over a specified period of time.

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The Future Value (FVFV) of an annuity is the total value of all payments plus the interest earned at the end of the term.

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The Present Value (PVPV) of an annuity is the current lump-sum value that is equivalent to a series of future periodic payments.

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In IB AI HL, calculations are typically performed using a Finance/TVM Solver on a GDC, where NN is the total number of payment periods, I%I\% is the annual interest rate, PVPV is the present value, PMTPMT is the periodic payment, FVFV is the future value, P/YP/Y is payments per year, and C/YC/Y is compounding periods per year.

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The formulas for the FVFV and PVPV of an annuity are derived from the sum of a geometric series: Sn=a(rn−1)r−1S_n = \frac{a(r^n - 1)}{r - 1}.

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For an ordinary annuity, payments are made at the end of each period. For an annuity due, payments are made at the beginning of each period.

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Amortization refers to the process of paying off a debt (like a mortgage) through a series of equal regular payments consisting of both interest and principal.

📐Formulae

FV=PMT((1+i)n−1)iFV = \frac{PMT \left( (1 + i)^n - 1 \right)}{i}

PV=PMT(1−(1+i)−n)iPV = \frac{PMT \left( 1 - (1 + i)^{-n} \right)}{i}

i=r100×ki = \frac{r}{100 \times k}

Sn=u1(rn−1)r−1S_n = \frac{u_1(r^n - 1)}{r - 1}

💡Examples

Problem 1:

An investor deposits 500 into a savings account at the end of every month for 5 years. The account pays an annual interest rate of 6%6\%, compounded monthly. Calculate the total amount in the account at the end of the 5 years.

Solution:

Using the TVM Solver on the GDC: N=5×12=60N = 5 \times 12 = 60 I%=6I\% = 6 PV=0PV = 0 PMT=−500PMT = -500 (outflow) P/Y=12P/Y = 12 C/Y=12C/Y = 12 Solve for FVFV: FV=500((1+0.0612)60−1)0.0612FV = \frac{500 \left( (1 + \frac{0.06}{12})^{60} - 1 \right)}{\frac{0.06}{12}} FV≈34885.02FV \approx 34885.02 The total amount is 34885.02.

Explanation:

We use the Future Value of an annuity formula because the payments are regular and we want to find the accumulated total. Note that PMTPMT is negative in GDC notation as it represents money leaving the investor's pocket.

Problem 2:

A student takes out a loan of 20000 to buy a car. The loan has an interest rate of 4.5%4.5\% per annum, compounded quarterly. The loan is to be repaid in equal quarterly installments over 4 years. Find the value of each quarterly payment.

Solution:

Using the TVM Solver: N=4×4=16N = 4 \times 4 = 16 I%=4.5I\% = 4.5 PV=20000PV = 20000 (inflow) FV=0FV = 0 (loan is paid off) P/Y=4P/Y = 4 C/Y=4C/Y = 4 Solve for PMTPMT: 20000=PMT(1−(1+0.0454)−16)0.045420000 = \frac{PMT \left( 1 - (1 + \frac{0.045}{4})^{-16} \right)}{\frac{0.045}{4}} PMT≈−1373.91PMT \approx -1373.91 Each quarterly payment is 1373.91.

Explanation:

This is a Present Value problem because the loan amount is the value of the payments today. We solve for PMTPMT to find the regular installment required to reduce the FVFV to zero.

Problem 3:

Show that the total interest paid on a loan of 10000 with monthly payments of 300 for 3 years is 800.

Solution:

Total amount paid over 3 years: 300×3610800\begin{array}{r} 300 \\ \times 36 \\ \hline 10800 \end{array} Total interest = Total amount paid - Principal 10800−10000800\begin{array}{r} 10800 \\ - 10000 \\ \hline 800 \end{array} Total interest is 800.

Explanation:

Total interest is calculated by subtracting the initial principal borrowed from the sum of all payments made over the duration of the loan.