Financial Series Applications

Introduction

Understanding financial series is vital for making informed financial decisions. Series assist in budgeting, planning for savings, and managing loans, offering insight into potential future financial outcomes based on historical patterns.

• Financial Series: A compilation of future financial transactions aimed at identifying patterns or trends to enhance planning and budgeting.

Real-World Example

• Consider a household managing their finances. They utilise series to forecast monthly income and expenses, assisting in planning for significant purchases like holidays and balancing savings objectives by analysing the trend of income and essential expenses.

Geometric Sequence

Definition and Properties

• Geometric Sequence: A geometric sequence is characterised by each term being derived by multiplying the preceding term by a constant, known as the common ratio.

Financial Applications

• Financial Growth: Examples include the growth of investments over time due to compound interest.
• Financial Decay: Instances where assets consistently lose value, often modelled using depreciation rates.

Formula and Its Application

The formula for the nth term in a geometric sequence is: $a_n = a_1 \times r^{(n-1)}$

• $a_1$: Denotes the initial term or starting value in financial scenarios.
• $r$: Common ratio.
  • $r > 1$ signifies financial growth.
  • $0 < r < 1$ signifies financial decay.
• $n$: Term position, indicating which term in the sequence.

Example Problems

Example 1: Investment Growth

• An initial investment of £1000 appreciates annually by 5%:
  • $a_1 = 1000$, $r = 1.05$ (growth factor)
  • Compute the value after three years ($a_3$):
    • Formula: $a_3 = 1000 \times 1.05^{2}$
    • Result: £1102.50

Example 2: Asset Depreciation

• A car valued at £15,000 depreciates by 10% annually:
  • $a_1 = 15000$, $r = 0.9$ (decay factor)
  • Determine the value after two years ($a_2$):
    • Formula: $a_2 = 15000 \times 0.9^{1}$
    • Result: £13,500

Compound Interest Problems

Definition and Explanation

Compound Interest: A method of accruing interest on the initial principal and on accumulated interest itself, enhancing fund growth.

Components:

• $A$: The future value of the investment.
• $P$: Principal amount—initial sum.
• $r$: Annual interest rate, expressed as a decimal.
• $n$: Compounding frequency per year.
• $t$: Time in years.

Example: Calculating Future Values

• £5,000 invested at an annual interest rate of 5%, compounded monthly for 10 years:
  • $P = 5000$, $r = 0.05$, $n = 12$, $t = 10$
  • Solution: $A = 5000 \left(1 + \frac{0.05}{12}\right)^{120}$ $A = 5000 \times 1.6436$ $A = £8,218$

Effective Annual Rate (EAR)

Definition and Calculation

• Definition: EAR is the measure of the annual interest rate incorporating compounding effects.

• Formula: $EAR = \left(1 + \frac{i}{n}\right)^n - 1$ Where $i$ is the nominal interest rate, and $n$ designates compounding periods per year.

  Example Calculation:

  • Nominal interest rate of 6% compounded monthly:
  • $i = 0.06$, $n = 12$
  • EAR: $EAR = \left(1 + \frac{0.06}{12}\right)^{12} - 1 \approx 0.0617$ or 6.17%

Reducing Balance Loans vs Flat-rate Loans

• Reducing Balance Loan: Interest is calculated on the remaining principal.
• Flat-rate Loan: Interest is charged on the original loan amount.

Calculating Balance After Repayments

• Interest Calculation: Compute for the current term based on the remaining balance.
• Balance Update: Subtract the principal payment from the outstanding balance.

Developing Repayment Strategies

• Strategies: Formulate comprehensive repayment strategies based on varying terms.

Visual Aid:

Definitions and Comparisons

Calculation Methods

To calculate the future value of an annuity: $FV = P\frac{(1 + r)^n - 1}{r}$

Enhance your understanding of financial products and loan strategies by consistently applying appropriate formulas and methodologies. Financial series and geometric progressions adeptly model complex real-world financial situations, facilitating informed decision-making.