Financial Mathematics – IB Math AI SL
1. Introduction
Financial Mathematics is a vital component of the IB Math Analysis & Approaches SL (AI SL) curriculum. It integrates algebraic techniques and real-world applications involving money, investments, interest rates, loans, and annuities. An understanding of how money grows over time (compound interest) and how regular payments or receipts are evaluated (annuities, amortization) is essential, both practically and within the IB exam framework.
In this guide, we'll explore key formulas and concepts such as compound interest, present and future values, annuities, perpetuities, and other relevant topics. We'll then practice them through progressively challenging problems—10 easy, 10 medium, and 10 hard—mirroring the style you might encounter in an IB exam.
2. Key Concepts & Formulas
Below is a concise outline of the critical formulas and ideas in Financial Mathematics for IB Math AI SL.
2.1 Simple Interest vs. Compound Interest
where \(A\) is the amount, \(P\) is the principal, \(r\) is the (annual) interest rate, and \(t\) is time in years.
Compound Interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]where \(n\) is the number of compounding periods per year, and \(t\) is the time in years.
2.2 Effective and Nominal Interest Rates
If an interest rate \(r_{\text{nominal}}\) is compounded \(n\) times a year, the effective interest rate \(r_{\text{eff}}\) is the single annual rate that would produce the same amount of growth:
2.3 Present Value and Future Value
The future value (FV) of an amount of money after \(t\) years with interest rate \(r\) (compounded once a year, for simplicity) is:
The present value (PV) is the current worth of a future sum of money or stream of cash flows given a specified rate of return. For one future payment:
2.4 Annuities
An annuity is a series of equal payments or receipts made at regular intervals. Two common types:
- Annuity-immediate (Ordinary Annuity): Payments are made at the end of each period.
- Annuity-due: Payments are made at the beginning of each period.
Future Value of an Ordinary Annuity (payment at end of period):
where \(R\) is the regular payment, \(i\) is the interest rate per period, and \(n\) is the total number of payments.
Present Value of an Ordinary Annuity:
For an annuity-due, each payment is compounded for one additional period, so you multiply the ordinary annuity formula by \((1 + i)\) to adjust accordingly.
2.5 Perpetuities
A perpetuity is a type of annuity that continues forever, with no end date. If a perpetuity pays \(R\) each period and the interest rate is \(i\):
2.6 Amortization
When repaying a loan through installments, each payment covers both interest and principal. The loan amortization process can be analyzed via the present value of an annuity:
Understanding the breakdown of each payment into interest vs. principal is often tested in Financial Mathematics questions.
3. Examples with Solutions
We now present 30 practice problems, divided into 10 easy, 10 medium, and 10 hard. Each example has a fully worked solution, reflecting typical IB Math AI SL exam-style questions on Financial Mathematics.
3.1 Easy Examples (1 to 10)
Example 1 Easy
Question: A principal of \$1000 is invested at a simple annual interest rate of 5% for 4 years. Find the total amount at the end of 4 years.
With simple interest, the formula is \( A = P(1 + rt) \). Here, \(P = 1000\), \(r = 0.05\), and \(t = 4\). Thus:
\[ A = 1000 \times \bigl(1 + 0.05 \times 4\bigr) = 1000 \times (1 + 0.20) = 1000 \times 1.20 = 1200. \]
Answer: \$1200
Example 2 Easy
Question: An initial deposit of \$500 grows at 6% annual interest (compounded once per year). Find the amount after 3 years.
For compound interest annually, \(\displaystyle A = P (1 + r)^t.\) Here, \(P = 500\), \(r = 0.06\), \(t = 3\).
\[ A = 500 \times (1.06)^3. \] Numerically, \((1.06)^3 \approx 1.191016.\) So, \(A \approx 500 \times 1.191016 = 595.508 \approx 595.51.\)
Answer: \$595.51
Example 3 Easy
Question: An investor deposits \$2000 into a savings account offering 4% effective annual interest. How much will the deposit be worth in 5 years?
Here, the interest rate of 4% is already effective annual rate. So we simply use \(A = P(1+r)^t\): \[ A = 2000 \times (1.04)^5. \] \((1.04)^5 \approx 1.2166529.\) Thus, \[ A \approx 2000 \times 1.2166529 = 2433.3058 \approx 2433.31. \]
Answer: \$2433.31
Example 4 Easy
Question: If \$10,000 is deposited at a nominal annual interest rate of 6% compounded quarterly, find the effective annual interest rate.
The nominal rate \( r_{\text{nominal}} = 0.06\). Compounded quarterly means \(n = 4\) compounding periods per year. The effective annual rate \( r_{\text{eff}} \) is found from: \[ 1 + r_{\text{eff}} = \left(1 + \frac{r_{\text{nominal}}}{n}\right)^n = \left(1 + \frac{0.06}{4}\right)^4 = (1 + 0.015)^4 = (1.015)^4. \] Numerically, \((1.015)^4 \approx 1.06136355.\) Hence, \(r_{\text{eff}} \approx 1.06136355 - 1 = 0.06136355 \approx 6.136\%\).
Answer: 6.136%
Example 5 Easy
Question: A payment of \$100 is due in 2 years. If the annual discount rate is 5%, what is the present value of this \$100?
Present value: \(\text{PV} = \frac{\text{FV}}{(1 + r)^t}\). \[ \text{PV} = \frac{100}{(1 + 0.05)^2} = \frac{100}{(1.05)^2} = \frac{100}{1.1025} \approx 90.7029 \approx 90.70. \]
Answer: \$90.70
Example 6 Easy
Question: A bank advertises a 3% nominal annual rate, compounded semi-annually. Find the amount of \$400 invested for 2 years under these conditions.
Nominal rate \(r = 0.03\), compounded semi-annually means \(n = 2\) periods per year. Time \(t = 2\) years. The total number of compounding periods is \(nt = 2 \times 2 = 4\). The interest rate per period is \(i = \frac{r}{n} = \frac{0.03}{2} = 0.015.\) Future Value: \(A = P (1 + i)^{nt} = 400 \times (1 + 0.015)^4.\) Numerically, \((1.015)^4 \approx 1.06136355.\) So, \(A \approx 400 \times 1.06136355 = 424.54542 \approx 424.55.\)
Answer: \$424.55
Example 7 Easy
Question: Define the term Effective Annual Rate (EAR) in a short sentence.
The Effective Annual Rate (EAR) is the actual annual rate of interest earned or paid on an investment or loan after considering the effect of compounding over a given period, typically one year.
Answer: It's the real annual rate of return taking compounding into account.
Example 8 Easy
Question: A project requires \$5,000 in 3 years. If the discount rate is 7% per year (compounded annually), find the present value you need to invest now.
Using the present value formula: \(\text{PV} = \frac{\text{FV}}{(1+r)^t}\). \[ \text{PV} = \frac{5000}{(1+0.07)^3} = \frac{5000}{(1.07)^3}. \] Numerically, \((1.07)^3 \approx 1.225043.\) So, \(\text{PV} \approx \frac{5000}{1.225043} \approx 4081.562 \approx 4081.56.\)
Answer: \$4081.56
Example 9 Easy
Question: A \$500 deposit grows to \$530 in one year. Find the effective annual interest rate.
The interest earned is \$530 - \$500 = \$30. The effective annual rate \(r_{\text{eff}} = \frac{\text{Interest Earned}}{\text{Principal}} = \frac{30}{500} = 0.06\). Alternatively, using \(A = P(1+r_{\text{eff}})^t\), we have \(530 = 500(1+r_{\text{eff}})^1\). So, \(1+r_{\text{eff}} = \frac{530}{500} = 1.06 \implies r_{\text{eff}} = 0.06\).
Answer: 6%
Example 10 Easy
Question: You have \$200 now and will deposit it in a simple interest account earning 4% per annum. How much interest will you earn after 5 years?
Simple interest formula: \( \text{Interest} = P \times r \times t.\)
Here, \( P = 200, r = 0.04, t = 5.\)
\[ \text{Interest} = 200 \times 0.04 \times 5 = 200 \times 0.20 = 40. \]
Answer: \$40
3.2 Medium Examples (11 to 20)
Example 11 Medium
Question: Find the present value of an ordinary annuity that pays \$1000 at the end of each year for 5 years, if the discount rate is 8% per year.
For an ordinary annuity with \(n=5\) payments, payment \(R = 1000\), and interest rate per period \(i=0.08\): \[ \text{PV} = R \times \frac{1 - (1 + i)^{-n}}{i} = 1000 \times \frac{1 - (1 + 0.08)^{-5}}{0.08}. \] Let's compute: \((1.08)^{-5} \approx 0.6805832.\) Then \(1 - 0.6805832 = 0.3194168.\) So, \(\frac{0.3194168}{0.08} \approx 3.99271.\) Thus, \(\text{PV} \approx 1000 \times 3.99271 = 3992.71.\)
Answer: \$3992.71
Example 12 Medium
Question: Suppose you receive \$500 at the beginning of each year for 4 years. The annual interest rate is 5%. Find the future value of this annuity-due at the end of 4 years.
For an annuity-due, payments are at the start of each period. The future value formula for an ordinary annuity is \(\text{FV}_{\text{ordinary}} = R \frac{(1+i)^n - 1}{i}\). For an annuity-due, we multiply this by \((1 + i)\). Here, \( R=500, i=0.05, n=4.\)
First, compute the ordinary annuity FV factor: \(\frac{(1.05)^4 - 1}{0.05}\). \((1.05)^4 \approx 1.21550625.\) Then \((1.21550625 - 1) = 0.21550625.\) \(\frac{0.21550625}{0.05} \approx 4.310125.\) So, \(\text{FV}_{\text{ordinary}} \approx 500 \times 4.310125 = 2155.0625.\) Now multiply by \((1+0.05)=1.05\) for annuity-due: \(\text{FV}_{\text{annuity-due}} = 2155.0625 \times 1.05 \approx 2262.8156 \approx 2262.82.\)
Answer: \$2262.82
Example 13 Medium
Question: You invest \$1,000 at an annual nominal rate of 9% compounded monthly. How much will you have after 2 years?
Nominal annual rate \(r=0.09\), compounded monthly means \(n=12\) periods per year. Time \(t=2\) years. The monthly interest rate is \(i = \frac{r}{n} = \frac{0.09}{12} = 0.0075.\) Total number of periods is \(nt = 12 \times 2 = 24\). Future Value: \(A = P (1 + i)^{nt} = 1000 \times (1 + 0.0075)^{24}.\) Numerically, \((1.0075)^{24} \approx 1.1964135.\) So, \(A \approx 1000 \times 1.1964135 = 1196.4135 \approx 1196.41.\)
Answer: \$1196.41
Example 14 Medium
Question: A loan of \$5000 is repaid in 3 equal annual payments at the end of each year. The interest rate is 6% per annum. Find the amount of each payment.
This is an ordinary annuity where the present value (loan amount) is \$5000. Let \(R\) be the annual payment. Interest rate per period \(i=0.06\), number of payments \(n=3\). Using the present value of an ordinary annuity formula: \(\text{PV} = R \frac{1 - (1 + i)^{-n}}{i}\). \[ 5000 = R \times \frac{1 - (1.06)^{-3}}{0.06}. \] First compute the factor: \((1.06)^{-3} \approx 0.83961928.\) Then \(1 - 0.83961928 = 0.16038072.\) So, \(\frac{0.16038072}{0.06} \approx 2.673012.\) Thus, \(5000 = R \times 2.673012 \implies R \approx \frac{5000}{2.673012} \approx 1870.543 \approx 1870.54.\)
Answer: \$1870.54
Example 15 Medium
Question: If a perpetuity pays \$200 at the end of each year indefinitely, and the annual interest rate is 5%, find its present value.
The present value of a perpetuity is given by \(\text{PV}_{\text{perpetuity}} = \frac{R}{i}\). Here \(R = 200\) and \(i = 0.05\). \[ \text{PV}_{\text{perpetuity}} = \frac{200}{0.05} = 4000. \]
Answer: \$4000
Example 16 Medium
Question: A 2-year certificate of deposit (CD) offers a 2.5% nominal annual interest rate compounded quarterly. What is the effective annual rate (EAR)?
Nominal annual rate \(r_{\text{nominal}} = 0.025\), compounded quarterly means \(n=4\). The effective annual rate \(r_{\text{eff}}\) is found using: \[ 1 + r_{\text{eff}} = \left(1 + \frac{r_{\text{nominal}}}{n}\right)^n = \left(1 + \frac{0.025}{4}\right)^4 = (1 + 0.00625)^4. \] \((1.00625)^4 \approx 1.0252352.\) So, \(r_{\text{eff}} \approx 1.0252352 - 1 = 0.0252352 \approx 2.524\%\).
Answer: 2.524%
Example 17 Medium
Question: You plan to save for college by depositing \$250 at the end of each month into an account paying 6% annual interest, compounded monthly. How much will you have after 5 years?
This is an ordinary annuity with monthly payments. Regular payment \(R = 250\). Annual interest rate is 6%, so monthly interest rate \(i = \frac{0.06}{12} = 0.005\). Number of deposits over 5 years is \(n = 5 \times 12 = 60\). The future value formula for an ordinary annuity: \(\text{FV} = R \frac{(1 + i)^n - 1}{i}\). \[ \text{FV} = 250 \times \frac{(1 + 0.005)^{60} - 1}{0.005}. \] Compute step by step: \((1.005)^{60} \approx 1.34885015.\) Then \(1.34885015 - 1 = 0.34885015.\) \(\frac{0.34885015}{0.005} \approx 69.77003.\) So, \(\text{FV} \approx 250 \times 69.77003 = 17442.5075 \approx 17442.51.\)
Answer: \$17,442.51
Example 18 Medium
Question: A \$1500 investment grows to \$1800 in 3 years. Assuming annual compounding, find the annual interest rate.
Using the compound interest formula \(A = P(1+r)^t\): \[ 1800 = 1500 \times (1+r)^3. \] Divide by 1500: \(\frac{1800}{1500} = (1+r)^3 \implies 1.2 = (1+r)^3.\) Take the cube root of both sides: \(1+r = \sqrt[3]{1.2}\). \(\sqrt[3]{1.2} \approx 1.0626585.\) So, \(1+r \approx 1.0626585 \implies r \approx 0.0626585 \approx 6.266\%\).
Answer: 6.266%
Example 19 Medium
Question: A 5-year loan of \$10,000 is to be repaid by equal annual payments at the end of each year, with an interest rate of 7% per annum. Find the amount of each annual payment.
This is an ordinary annuity where the present value (loan amount) is \$10,000. Let \(R\) be the annual payment. Interest rate per period \(i=0.07\), number of payments \(n=5\). \[ 10000 = R \times \frac{1 - (1.07)^{-5}}{0.07}. \] Compute the factor: \((1.07)^{-5} \approx 0.71298618.\) Then \(1 - 0.71298618 = 0.28701382.\) So, \(\frac{0.28701382}{0.07} \approx 4.1001974.\) Thus, \(10000 = R \times 4.1001974 \implies R \approx \frac{10000}{4.1001974} \approx 2438.906 \approx 2438.91.\)
Answer: \$2438.91
Example 20 Medium
Question: A perpetuity pays \$500 at the end of each year, with an annual interest rate of 10%. After receiving the first 10 payments, the holder sells this perpetuity. Calculate the selling price if the interest rate is still 10% at the time of sale.
A perpetuity that pays \$R per period at an interest rate \(i\) per period has a present value of \(\frac{R}{i}\). The value of the perpetuity is calculated based on all future payments from the point of valuation. After 10 payments have been received, the perpetuity still promises to pay \$500 at the end of each year forever, starting from the end of the 11th year. At the moment right after the 10th payment is received (which is the end of the 10th year), the remaining stream of payments is identical in structure to the original perpetuity but starting one period later from that point. Therefore, the present value of the remaining payments, valued at the end of the 10th year (which is the "present" for the buyer), is still \(\frac{R}{i}\). \[ \text{Selling Price} = \text{PV}_{\text{perpetuity}} = \frac{500}{0.10} = 5000. \]
Answer: \$5000
3.3 Hard Examples (21 to 30)
Example 21 Hard
Question: A person needs to accumulate \$10,000 in 4 years. They plan to make equal deposits at the end of each year into an account paying 8% interest compounded annually. Find the amount of each annual deposit \(X\).
This is finding the regular payment \(R\) for an ordinary annuity where the future value (FV) is known. \(\text{FV} = 10000\), number of payments \(n=4\), interest rate per period \(i=0.08\). \[ \text{FV} = R \times \frac{(1 + i)^n - 1}{i} \implies 10000 = X \times \frac{(1.08)^4 - 1}{0.08}. \] Compute the factor: \((1.08)^4 \approx 1.36048896.\) Then \(1.36048896 - 1 = 0.36048896.\) So, \(\frac{0.36048896}{0.08} \approx 4.506112.\) Thus, \(10000 = X \times 4.506112 \implies X \approx \frac{10000}{4.506112} \approx 2219.207 \approx 2219.21.\)
Answer: \$2219.21
Example 22 Hard
Question: A \$50,000 loan is amortized over 10 years at 6% annual interest, with equal payments made at the end of each year. For the first payment, calculate the amount of interest paid and the amount of principal repaid.
- Find the annual payment (R): The loan is the present value of an ordinary annuity. \(\text{PV} = 50000, n=10, i=0.06\). \[ 50000 = R \times \frac{1 - (1.06)^{-10}}{0.06}. \] \((1.06)^{-10} \approx 0.55839477.\) \(1 - 0.55839477 = 0.44160523.\) Factor \(\approx \frac{0.44160523}{0.06} \approx 7.3600871.\) So, \(R \approx \frac{50000}{7.3600871} \approx 6793.395 \approx 6793.40.\)
- Interest in the first payment: The interest is calculated on the outstanding principal at the beginning of the period. Interest for 1st year = Principal \(\times\) interest rate = \(50000 \times 0.06 = 3000.\)
- Principal repayment in the first payment: Principal Repaid = Total Payment - Interest Paid = \(6793.40 - 3000 = 3793.40.\)
Answer: Interest = \$3000, Principal Repaid = \$3793.40
Example 23 Hard
Question: A company issues a bond that pays a \$100 annual coupon at the end of each year forever (a perpetuity). However, the company has the option to call (buy back) the bond for \$1100 at the end of 5 years, immediately after the 5th coupon is paid. If the market discount rate is 8% per annum, what is the fair price of this callable bond today?
A callable bond's price is the minimum of its price if not called (value as a perpetuity) and its price if called at the earliest call date, both discounted to today. The company will call the bond if the market value of the perpetuity (at year 5) is greater than the call price (\$1100).
Scenario 1: Bond is called at year 5. The investor receives 5 annual coupons of \$100 and a call price of \$1100 at year 5. The present value of this stream is: PV = (PV of 5-year annuity of \$100) + (PV of \$1100 in 5 years) \[ \text{PV} = 100 \times \frac{1 - (1.08)^{-5}}{0.08} + \frac{1100}{(1.08)^5}. \] \((1.08)^{-5} \approx 0.6805832.\) Annuity factor \(\approx \frac{1 - 0.6805832}{0.08} \approx \frac{0.3194168}{0.08} \approx 3.99271.\) PV of coupons \(\approx 100 \times 3.99271 = 399.27.\) PV of call price \(\approx 1100 \times 0.6805832 = 748.64.\) Total PV if called \(\approx 399.27 + 748.64 = 1147.91.\)
Scenario 2: Bond is not called (becomes a perpetuity). This scenario is only relevant if the company would *not* want to call it. At year 5, after the 5th coupon is paid, the remaining bond is a perpetuity paying \$100 per year. Its value at year 5 would be \(\frac{100}{0.08} = \$1250\). Since \$1250 (value if not called) > \$1100 (call price), the company *will* call the bond to save money.
Therefore, the fair price of the bond is based on the assumption it will be called.
Answer: \$1147.91
Example 24 Hard
Question: Bank A offers a nominal annual rate of 10% compounded monthly. Bank B offers a nominal annual rate of 10.1% compounded quarterly. Which bank offers a better effective annual rate (EAR), and by how much?
Bank A: \(r_{\text{nominal}} = 0.10\), \(n=12\). \[ \text{EAR}_A = \left(1 + \frac{0.10}{12}\right)^{12} - 1 = (1 + 0.0083333...)^{12} - 1. \] \((1.0083333...)^{12} \approx 1.104713067.\) \(\text{EAR}_A \approx 0.104713067 \approx 10.4713\%.\)
Bank B: \(r_{\text{nominal}} = 0.101\), \(n=4\). \[ \text{EAR}_B = \left(1 + \frac{0.101}{4}\right)^{4} - 1 = (1 + 0.02525)^{4} - 1. \] \((1.02525)^{4} \approx 1.1049096.\) \(\text{EAR}_B \approx 0.1049096 \approx 10.4910\%.\)
Comparing EARs: Bank B (10.4910%) offers a slightly better rate than Bank A (10.4713%). Difference = \(10.4910\% - 10.4713\% = 0.0197\%\).
Answer: Bank B is better. Its EAR is approximately 0.0197 percentage points higher than Bank A's EAR.
Example 25 Hard
Question: You need to have \$20,000 in an account in exactly 3 years. You make two deposits: \$X now and \$Y in 1 year (at the end of year 1). The account pays 6% annual interest, compounded annually. If the total amount you deposit (\(X+Y\)) is \$15,000, find the values of X and Y.
Let X be the deposit now (at t=0) and Y be the deposit at the end of year 1 (t=1). The target amount is \$20,000 at the end of year 3 (t=3). Interest rate is 6% per annum.
Future value of X at t=3: \(X(1.06)^3\).
Future value of Y at t=3 (deposited at t=1, so it grows for 2 years): \(Y(1.06)^2\).
The sum of these future values must equal \$20,000: \[ X(1.06)^3 + Y(1.06)^2 = 20000 \quad (*) \]
We are also given that \(X + Y = 15000\). So, \(Y = 15000 - X\). Substitute this into equation (*): \[ X(1.06)^3 + (15000 - X)(1.06)^2 = 20000 \]
Calculate powers of 1.06:
\((1.06)^2 = 1.1236\)
\((1.06)^3 = 1.1236 \times 1.06 = 1.191016\)
Substitute these values: \[ 1.191016 X + (15000 - X)(1.1236) = 20000 \] \[ 1.191016 X + 15000 \times 1.1236 - 1.1236 X = 20000 \] \[ 1.191016 X + 16854 - 1.1236 X = 20000 \]
Combine terms with X: \[ (1.191016 - 1.1236) X = 20000 - 16854 \] \[ 0.067416 X = 3146 \]
Solve for X: \[ X = \frac{3146}{0.067416} \approx 46663.999... \] This result for X is much larger than the total deposit of \$15,000, which indicates an issue with the problem statement or my interpretation of deposit timing. Let's re-read: "deposit \$Y in 1 year". This means at t=1. So it grows for \(3-1=2\) years. This seems correct.
Re-checking the algebra: \(1.191016X + 16854 - 1.1236X = 20000\) \( (1.191016 - 1.1236)X = 20000 - 16854 \) \( 0.067416 X = 3146 \) If this algebra is correct, then the conditions \(X+Y=15000\) and the target of \$20,000 are inconsistent with a 6% interest rate. The growth factor difference \( (1.06)^3 - (1.06)^2 = (1.06)^2 (1.06-1) = (1.06)^2 \times 0.06 = 1.1236 \times 0.06 = 0.067416\). This is correct.
The equation is essentially \(X \cdot (1.06)^3 + (15000-X) \cdot (1.06)^2 = 20000\). This can be rewritten as \(X((1.06)^3 - (1.06)^2) = 20000 - 15000(1.06)^2\). \(X \cdot 0.067416 = 20000 - 15000 \times 1.1236 = 20000 - 16854 = 3146\). This calculation is consistent.
Conclusion: The problem as stated (with X+Y=15000) leads to X being approx \$46,664, which is impossible if X and Y are positive and sum to 15000. It implies that if X were \$46,664, then Y would be \(15000 - 46664 = -31664\), meaning you'd withdraw money. This problem might be designed to show that not all financial goals are achievable under given constraints, or there's a typo in the numbers. Assuming the question expects a mathematical derivation based on the setup:
If X \(\approx 46663.99\), then Y = \(15000 - 46663.99 = -31663.99\). This means you deposit \$46,663.99 now, and withdraw \$31,663.99 in one year. Let's check if this works: FV of X: \(46663.99 \times (1.06)^3 \approx 46663.99 \times 1.191016 \approx 55578.21\) FV of Y: \(-31663.99 \times (1.06)^2 \approx -31663.99 \times 1.1236 \approx -35578.21\) Total FV = \(55578.21 - 35578.21 = 20000\). The math works out, but the scenario is unusual.
Answer: Assuming the setup is intentional and negative deposits (withdrawals) are allowed for Y: \(X \approx \$46,663.99\) and \(Y \approx -\$31,663.99\). If X and Y must be positive, there is no solution under the given constraints.
Example 26 Hard
Question: You deposit \$5,000 now (t=0). Then, starting at the end of year 1 (t=1) and continuing for 5 years (i.e., payments at t=1, 2, 3, 4, 5), you deposit \$2,000 annually. The annual interest rate is 7%, compounded annually. Find the total accumulated amount at the end of year 5 (t=5).
This problem has two parts: the future value of the initial lump sum, and the future value of the annuity.
Part 1: Future Value of the initial \$5,000 deposit. This deposit is made at t=0 and grows for 5 years. \[ \text{FV}_1 = 5000 \times (1 + 0.07)^5 = 5000 \times (1.07)^5. \] \((1.07)^5 \approx 1.4025517.\) \[ \text{FV}_1 \approx 5000 \times 1.4025517 = 7012.7585 \approx 7012.76. \]
Part 2: Future Value of the annual \$2,000 deposits. These are 5 payments of \$2,000 made at the end of years 1, 2, 3, 4, and 5. This is an ordinary annuity. \(R = 2000, i = 0.07, n = 5\). \[ \text{FV}_2 = R \times \frac{(1 + i)^n - 1}{i} = 2000 \times \frac{(1.07)^5 - 1}{0.07}. \] We already have \((1.07)^5 \approx 1.4025517.\) So, \((1.07)^5 - 1 \approx 0.4025517.\) The factor is \(\frac{0.4025517}{0.07} \approx 5.7507385.\) \[ \text{FV}_2 \approx 2000 \times 5.7507385 = 11501.477 \approx 11501.48. \]
Total Accumulated Amount: Total FV = \(\text{FV}_1 + \text{FV}_2 \approx 7012.76 + 11501.48 = 18514.24.\)
Answer: \$18,514.24
Example 27 Hard
Question: A loan of \$40,000 is to be amortized by equal annual payments at the end of each year for 7 years. The annual interest rate is 6%. Calculate the outstanding loan balance immediately after the 4th payment has been made.
- Find the annual payment (R): \(\text{PV} = 40000, n=7, i=0.06\). \[ 40000 = R \times \frac{1 - (1.06)^{-7}}{0.06}. \] \((1.06)^{-7} \approx 0.6650571.\) Factor \(\approx \frac{1 - 0.6650571}{0.06} = \frac{0.3349429}{0.06} \approx 5.582381.\) So, \(R \approx \frac{40000}{5.582381} \approx 7165.575 \approx 7165.58.\)
- Outstanding balance after the 4th payment: This is the present value of the remaining \(7 - 4 = 3\) payments, valued at t=4. \[ \text{Balance}_4 = R \times \frac{1 - (1.06)^{-3}}{0.06}. \] \((1.06)^{-3} \approx 0.83961928.\) Factor \(\approx \frac{1 - 0.83961928}{0.06} = \frac{0.16038072}{0.06} \approx 2.673012.\) \[ \text{Balance}_4 \approx 7165.58 \times 2.673012 = 19156.108 \approx 19156.11. \]
Answer: \$19,156.11
Example 28 Hard
Question: A bond with a face value of \$6000 is redeemable at \$6500 in 3 years. It pays semi-annual coupons based on a nominal annual coupon rate of 10% of its face value. If the market yield rate for similar bonds is 8% nominal, compounded semi-annually, find the current fair price of this bond.
Bond details: Face value for coupon calculation = \$6000. Annual coupon rate = 10%, so annual coupon = \(0.10 \times 6000 = \$600\). Coupons are paid semi-annually, so each coupon payment \(C = \frac{600}{2} = \$300\). Redemption value (FV) = \$6500 at the end of 3 years. Time to maturity = 3 years, so \(3 \times 2 = 6\) semi-annual periods.
Market yield rate: Nominal annual yield = 8%, compounded semi-annually. So, the semi-annual discount rate \(i = \frac{0.08}{2} = 0.04\).
The price of the bond is the present value of all future cash flows (coupons + redemption value) discounted at the market yield rate.
PV of coupons (ordinary annuity): \(R=300, n=6, i=0.04\).
\[ \text{PV}_{\text{coupons}} = 300 \times \frac{1 - (1.04)^{-6}}{0.04}. \]
\((1.04)^{-6} \approx 0.7903145.\)
Annuity factor \(\approx \frac{1 - 0.7903145}{0.04} = \frac{0.2096855}{0.04} \approx 5.242137.\)
\(\text{PV}_{\text{coupons}} \approx 300 \times 5.242137 = 1572.6411 \approx 1572.64.\)
PV of redemption value: \(\text{PV}_{\text{redemption}} = \frac{6500}{(1.04)^6} \approx 6500 \times 0.7903145 = 5137.04425 \approx 5137.04.\)
Total fair price = \(\text{PV}_{\text{coupons}} + \text{PV}_{\text{redemption}} \approx 1572.64 + 5137.04 = 6709.68.\)
Answer: \$6709.68
Example 29 Hard
Question: A 10-year annuity-due has annual payments of \$2000 starting today (t=0). The annual interest rate is 5%. Immediately after the 4th payment is made, the annuitant wants to sell the rights to the remaining payments. What is the fair surrender value at that time?
- Annuity-due payments are at the beginning of each period. 1st payment at t=0. 2nd payment at t=1. 3rd payment at t=2. 4th payment at t=3.
- "Immediately after the 4th payment is made" means we are at t=3.
- There are a total of 10 payments. Payments made are at t=0, 1, 2, 3. Remaining payments are at t=4, 5, 6, 7, 8, 9. This is a stream of \(10 - 4 = 6\) payments.
- The first of these remaining payments occurs at t=4. We need to find the present value of these 6 remaining payments, valued at t=3. The payment at t=4 is one period away from t=3. The payment at t=5 is two periods away from t=3, and so on. The payment at t=9 is six periods away from t=3. This stream of payments (first one at t=4, valued at t=3) is an ordinary annuity with \(R=2000, n=6, i=0.05\).
- Surrender Value = PV of remaining payments at t=3: \[ \text{Value} = 2000 \times \frac{1 - (1.05)^{-6}}{0.05}. \] \((1.05)^{-6} \approx 0.746215396.\) Factor \(\approx \frac{1 - 0.746215396}{0.05} = \frac{0.253784604}{0.05} \approx 5.075692.\) \[ \text{Value} \approx 2000 \times 5.075692 = 10151.384 \approx 10151.38. \]
Answer: \$10,151.38
Example 30 Hard
Question: A stock index is currently at 2000 points. It is expected to have a total return of 10% per annum, compounded continuously. However, it also pays out a continuous dividend yield equivalent to 2% per annum. What is the expected level of the stock index itself (price return only) after 3 years?
The total return of an asset includes both price appreciation and any income (like dividends). If the total return is 10% continuously, and 2% is paid out as continuous dividends, then the rate of price appreciation of the index itself is the total return rate minus the dividend yield rate.
Net continuous growth rate of the index price = (Total continuous return rate) - (Continuous dividend yield rate)
Net rate = \(0.10 - 0.02 = 0.08\) or 8% per annum, compounded continuously.
The formula for continuous compounding is \(A = P e^{rt}\). Here, \(P = 2000\), net rate \(r = 0.08\), time \(t = 3\) years. \[ \text{Index Level} = 2000 \times e^{(0.08) \times 3} = 2000 \times e^{0.24}. \] \( e^{0.24} \approx 1.27124915.\) So, Index Level \(\approx 2000 \times 1.27124915 = 2542.4983 \approx 2542.50.\)
Answer: 2542.50 points
4. Conclusion
Financial Mathematics in IB Math AI SL encompasses essential life skills, from understanding interest rates and effective vs. nominal rates to more advanced topics like annuities, perpetuities, and bond pricing. Whether dealing with personal finance, corporate investments, or exam-style problems, the overarching ideas remain consistent: calculating present and future values under various compounding assumptions, analyzing payment structures, and evaluating the time value of money.
The 30 examples provided here (10 easy, 10 medium, 10 hard) aim to systematically build confidence and problem-solving capabilities. In actual IB exams, clarity on concepts—like the difference between annuity-immediate and annuity-due, or nominal vs. effective rates—can be the key to success. Always sketch a timeline, verify whether payments occur at the beginning or end of a period, and carefully interpret the interest rate’s compounding. With thorough practice and a solid conceptual foundation, you will excel in Financial Mathematics questions and apply these skills beyond the classroom as well.
