Session 08-01 - Compound Interest & Geometric Sequences
Section 08: Financial Mathematics
Entry Quiz - 10 Minutes
Quick Review from Section 07
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.
Welcome to Financial Mathematics!
New Section Overview
Section 08 covers essential financial topics:
- Session 08-01: Compound Interest & Geometric Sequences (today)
- Session 08-02: Annuities & Loan Amortization
- Session 08-03: Cost Analysis & Pricing Decisions
. . .
Cost analysis and pricing decisions are exam-critical topics!
Learning Objectives
What You’ll Master Today
- Understand geometric sequences and their formulas
- Calculate compound interest for different compounding periods
- Find effective annual rates for comparison
- Apply present value concepts to business decisions
- Use the Rule of 72 for quick estimates
Part A: Geometric Sequences
What is a Geometric Sequence?
A sequence where each term is multiplied by a constant ratio:
. . .
\[a_1, \, a_1 \cdot r, \, a_1 \cdot r^2, \, a_1 \cdot r^3, \, \ldots\]
. . .
ImportantKey Formulas
Explicit formula (nth term): \[a_n = a_1 \cdot r^{n-1}\]
Recursive formula: \[a_n = a_{n-1} \cdot r\]
Examples of Geometric Sequences
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)
. . .
- \(|r| > 1\): sequence grows (or alternates with growth)
- \(|r| < 1\): sequence decays toward zero
- \(r = 1\): constant sequence
Part B: Compound Interest
Simple vs. Compound Interest
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!
The Compound Interest Formula
Annual Compounding
\[A = P(1 + r)^t\]
. . .
where:
- \(A\) = Final amount (future value)
- \(P\) = Principal (initial investment)
- \(r\) = Annual interest rate (as decimal)
- \(t\) = Time in years
. . .
Example: Invest 1,000 at 5% for 10 years, \(A = 1000(1.05)^{10}\)
Multiple Compounding Periods
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 |
Compounding Comparison
1,000 invested at 6% for 50 years:
Continuous Compounding
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, \(A = 1000 \cdot e^{0.06 \times 5}\)
Break - 10 Minutes
Part C: Effective Annual Rate
Comparing Different Rates
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.
EAR Calculation Example
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%)!
EAR for Continuous Compounding
For continuous compounding:
. . .
\[r_{\text{eff}} = e^r - 1\]
. . .
Example: 6% continuous compounding \[r_{\text{eff}} = e^{0.06} - 1 \approx 0.0618 = 6.18\%\]
Part D: Present Value
The Present Value Concept
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
\[PV = \frac{FV}{(1 + r)^t} = FV \cdot (1 + r)^{-t}\]
Present Value Example
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: \[PV = \frac{10000}{(1 + 0.06/12)^{60}} = \frac{10000}{1.3489} = 7,413.72\]
Part E: Business Applications
Investment Growth Analysis
A company invests 50,000 in bonds paying 4.5% compounded quarterly.
- What is the value after 10 years?
- What is the effective annual rate?
- How long until the investment doubles?
Inflation Adjustment
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 \[r_{\text{real}} = \frac{1.06}{1.02} - 1 = 0.039 = 3.9\%\]
Guided Practice - 15 Minutes
Practice Problems
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?
Wrap-Up & Key Takeaways
Today’s Essential Concepts
- Geometric sequences have constant ratio: \(a_n = a_1 \cdot r^{n-1}\)
- Compound interest creates exponential growth
- More frequent compounding increases returns
- Effective annual rate allows fair comparison
- Present value discounts future amounts
- Rule of 72 estimates doubling time quickly
. . .
TipComing Next
Session 08-02: Annuities & Loan Amortization - regular payment streams!
Homework Assignment
Tasks 08-01
- Calculate geometric sequence terms and sums
- Compound interest with various compounding periods
- Compare investments using effective annual rates
- Present value calculations for financial planning
. . .
Practice with your calculator - these calculations appear frequently on exams!