Tasks 08-01 - Compound Interest & Geometric Sequences

Section 08: Financial Mathematics

Problem 1: Geometric Sequence Terms (x)

Find the specified term for each geometric sequence:

  1. \(a_1 = 3\), \(r = 2\): Find \(a_7\)
  2. \(a_1 = 100\), \(r = 0.5\): Find \(a_6\)
  3. \(a_1 = 5\), \(r = -3\): Find \(a_5\)
  4. \(2, 6, 18, 54, \ldots\): Find \(a_8\)

Problem 2: Geometric Series Sums (x)

Find the sum of each geometric series:

  1. \(1 + 3 + 9 + 27 + \ldots\) (first 8 terms)
  2. \(64 + 32 + 16 + 8 + \ldots\) (first 10 terms)
  3. \(\sum_{k=0}^{5} 4 \cdot 2^k\)
  4. \(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots\) (infinite sum)

Problem 3: Basic Compound Interest (x)

Calculate the future value for each investment:

  1. 5,000 invested at 4% annual interest for 10 years (annual compounding)
  2. 2,500 invested at 6% annual interest for 8 years (annual compounding)
  3. 10,000 invested at 3.5% annual interest for 15 years (annual compounding)

Problem 4: Multiple Compounding Periods (xx)

Calculate the future value with the given compounding frequency:

  1. 3,000 at 5% for 6 years, compounded semi-annually
  2. 8,000 at 4.8% for 4 years, compounded quarterly
  3. 1,500 at 6% for 3 years, compounded monthly
  4. 5,000 at 8% for 2 years, compounded daily

Problem 5: Effective Annual Rate (xx)

Find the effective annual rate (EAR) for each nominal rate:

  1. 6% compounded semi-annually
  2. 8% compounded quarterly
  3. 5% compounded monthly
  4. 4% compounded daily

Problem 6: Present Value (xx)

Find the present value (how much to invest today):

  1. To have 20,000 in 10 years at 5% annual interest
  2. To have 50,000 in 15 years at 4% annual interest
  3. To have 8,000 in 5 years at 6% compounded monthly
  4. To have 100,000 in 20 years at 7% compounded quarterly

Problem 7: Comparing Investments (xx)

Which investment offers a better return? Justify your answer.

  1. Bank A: 5.8% compounded monthly OR Bank B: 5.9% compounded annually

  2. Investment X: 7.2% compounded quarterly OR Investment Y: 7.0% compounded daily

  3. Option 1: 4.5% compounded semi-annually OR Option 2: 4.4% compounded monthly

Problem 8: Rule of 72 (x)

Use the Rule of 72 to estimate doubling time, then verify with exact calculation:

  1. 6% annual interest
  2. 9% annual interest
  3. 12% annual interest

Problem 9: Continuous Compounding (xx)

Calculate the future value with continuous compounding:

  1. 4,000 at 5% for 10 years
  2. 7,500 at 3.5% for 8 years
  3. 2,000 at 8% for 5 years

Then find the effective annual rate for each.

Problem 10: Business Application (xxx)

A company invests 250,000 Euro from profits into a bond fund.

  1. If the fund earns 5.5% compounded quarterly, what will it be worth in 7 years?
  2. What is the effective annual rate of return?
  3. How long until the investment doubles? (Use Rule of 72 and exact)
  4. If they need 400,000 Euro in 8 years, what annual rate (compounded annually) do they need?

Problem 11: Inflation and Real Returns (xxx)

An investment earns 7% nominal return while inflation is 2.5%.

  1. Calculate the approximate real return using the simple formula.
  2. Calculate the exact real return using the Fisher equation.
  3. If you invest 10,000 today, what is the real purchasing power after 5 years?

Problem 12: Mixed Problem (xxxx)

You have 15,000 Euro to invest. You’re comparing three options:

  • Option A: 4.8% compounded monthly for 10 years
  • Option B: 5.0% compounded annually for 10 years
  • Option C: 4.6% compounded continuously for 10 years
  1. Calculate the future value for each option.
  2. Calculate the effective annual rate for each option.
  3. Which option gives the best return?
  4. How much more does the best option earn compared to the worst?