Section 08: Financial Mathematics
Test your understanding of Probability
In a survey, 60% of customers are satisfied, 40% are repeat buyers, and 30% are both. Find \(P(\text{Satisfied OR Repeat})\).
A medical test has sensitivity 95% and specificity 90%. If disease prevalence is 2%, find the PPV.
A coin is flipped 4 times. Find \(P(\text{exactly 2 heads})\).
Create a contingency table: 100 people surveyed, 55 own cars, 40 own bikes, 20 own both.
Section 08 covers essential financial topics:
Cost analysis and pricing decisions are exam-critical topics!
A sequence where each term is multiplied by a constant ratio:
\[a_1, \, a_1 \cdot r, \, a_1 \cdot r^2, \, a_1 \cdot r^3, \, \ldots\]
Key Formulas
Explicit formula (nth term): \[a_n = a_1 \cdot r^{n-1}\]
Recursive formula: \[a_n = a_{n-1} \cdot r\]
Common ratio determines behavior:
\(r = 2\): \(3, 6, 12, 24, 48, \ldots\) (growth)
\(r = \frac{1}{2}\): \(16, 8, 4, 2, 1, \ldots\) (decay)
\(r = -2\): \(1, -2, 4, -8, 16, \ldots\) (alternating)
Two ways to grow money:
Simple Interest: Interest earned only on principal \[A = P(1 + rt)\]
Compound Interest: Interest earned on principal AND accumulated interest \[A = P(1 + r)^t\]
Compound interest creates exponential growth - this is where geometric sequences appear!
Annual Compounding
\[A = P(1 + r)^t\]
where:
Example: Invest 1,000 at 5% for 10 years, \(A = 1000(1.05)^{10}\)
What if interest compounds more frequently?
General Compound Interest Formula
\[A = P\left(1 + \frac{r}{n}\right)^{nt}\]
where \(n\) = number of compounding periods per year
| Compounding | n |
|---|---|
| Annual | 1 |
| Semi-annual | 2 |
| Quarterly | 4 |
| Monthly | 12 |
| Daily | 365 |
1,000 invested at 6% for 50 years:
As \(n \to \infty\), we get continuous compounding:
Continuous Compounding Formula
\[A = Pe^{rt}\]
where \(e \approx 2.71828\)
Derivation: \(\lim_{n \to \infty} \left(1 + \frac{r}{n}\right)^n = e^r\)
Example: 1,000 at 6% continuous for 5 years, \(A = 1000 \cdot e^{0.06 \times 5}\)
Problem: Bank A offers 6% compounded monthly. Bank B offers 6.1% compounded annually. Which is better?
We need a common basis for comparison!
Effective Annual Rate (EAR)
\[r_{\text{eff}} = \left(1 + \frac{r}{n}\right)^n - 1\]
This gives the equivalent annual rate for any compounding frequency.
Bank A: 6% compounded monthly \[r_{\text{eff}} = \left(1 + \frac{0.06}{12}\right)^{12} - 1 = (1.005)^{12} - 1 = 0.0617 = 6.17\%\]
Bank B: 6.1% compounded annually \[r_{\text{eff}} = 6.1\%\]
Bank A is slightly better (6.17% > 6.10%)!
For continuous compounding:
\[r_{\text{eff}} = e^r - 1\]
Example: 6% continuous compounding \[r_{\text{eff}} = e^{0.06} - 1 \approx 0.0618 = 6.18\%\]
Question: How much invest today to have 10,000 in 5 years at 6%?
We need to discount future values back to today.
Present Value Formula
\[PV = \frac{FV}{(1 + r)^t} = FV \cdot (1 + r)^{-t}\]
Goal: 10,000 in 5 years at 6%
\[PV = \frac{10000}{(1.06)^5} = \frac{10000}{1.3382} = 7,472.58\]
You need to invest 7,472.58 today to have 10,000 in 5 years!
With monthly compounding: \[PV = \frac{10000}{(1 + 0.06/12)^{60}} = \frac{10000}{1.3489} = 7,413.72\]
A company invests 50,000 in bonds paying 4.5% compounded quarterly.
Real Interest Rate (Fisher Equation)
\[r_{\text{real}} \approx r_{\text{nominal}} - r_{\text{inflation}}\]
Exact formula: \[1 + r_{\text{real}} = \frac{1 + r_{\text{nominal}}}{1 + r_{\text{inflation}}}\]
Example: 6% nominal return, 2% inflation \[r_{\text{real}} = \frac{1.06}{1.02} - 1 = 0.039 = 3.9\%\]
Work in pairs
Problem 1: Calculate the future value of 2,500 invested at 5.5% compounded monthly for 8 years.
Problem 2: Bank A offers 4.8% compounded daily. Bank B offers 4.9% compounded annually. Which bank offers a better return?
Problem 3: You need 25,000 in 6 years. How much must you invest today at 7% compounded quarterly?
Coming Next
Session 08-02: Annuities & Loan Amortization - regular payment streams!
Practice with your calculator - these calculations appear frequently on exams!
Session 08-01 - Compound Interest & Geometric Sequences | Dr. Nikolai Heinrichs & Dr. Tobias Vlćek | Home