Maths-Chapter5

Types of Financial Contracts

• Perpetuity: A constant cash flow that lasts forever.

• Growing Perpetuity: A cash flow that grows at a constant rate forever.

• Annuity: A constant cash flow that lasts for a fixed number of periods.

• Growing Annuity: A cash flow that grows at a constant rate for a fixed number of periods.

Review of Geometric Series

• Formula: $a + aq + aq^2 + … + aq^{n-1} = a\frac{1 - q^n}{1 - q}$

• Note: If q < 1 and $n$ approaches infinity, then $q^n \to 0$, leading to the series converging to $\frac{a}{1 - q}$.

Present Value of Perpetuities

• Formula: $PV = \frac{C}{r}$ where $C$ is the cash flow and $r$ is the interest rate.

  • Example: Value of an investment paying £10 each year at a 10% interest rate:

  • Calculation: $PV = \frac{10}{0.1} = £100$

Growing Perpetuity

• Formula: $PV = \frac{C}{r - g}$ where $g$ is the growth rate.

  • Example: Expected cash flow of $10 growing at 3% with a discount rate of 10%:

  • Calculation: PV = \frac{10}{0.1 - 0.03} = \frac{10}{0.07} \approx $142.86

Present Value of Annuities

• Formula: $PV = C \left( \frac{1 - (1 + r)^{-T}}{r} \right)$ where $T$ is the number of periods.

  • Example 1: 10-year mortgage with payments of €1000 at 0.25% monthly interest calculation.

  • Example 2: For a £5000 loan paid off in four annual payments at 10% interest:

    • Each payment (solution from balance table).

Growing Annuity

• Formula: $PV = \frac{C}{r - g} \left(1 - (1 + g)^T(1+r)^{-T} \right)$

  • Example: Cash flow of $10 growing at 3% lasting for 10 years with a discount rate of 10%.

Savings for Future Goals

• Exercise 1: Calculate annual savings required from age 36 to 70 at 5% interest to fund future spending goals.

• Exercise 2: Compare costs of education paths (medicine vs. law) with a 6% discount rate.

Stated vs. Effective Interest Rates

• Distinction between quoted (stated) rates and actual (effective) rates.

• Stated rate is annual without compounding, whereas effective considers compounding frequency.

• Effective Annual Rate (EAR): $EAR = (1 + \frac{r}{m})^{m} - 1$

  • m = number of compounding periods per year.

• Example shows calculations for monthly compounded interest.

Continuous Compounding

• As compounding frequency increases, it approaches the continuous compounding formula:

  • Future Value: $FV = C_0 e^{rT}$

  • Example: For an investment of $100 at a continuously compounded rate of 10% for 3 years, calculate effective annual return and future value.

Methodology for using the formulas

• Perpetuity

  • Use the perpetuity formula when you have a constant cash flow for an infinite duration, such as property investments or bonds that pay a fixed amount.

  • Sample question: What is the present value of a constant cash flow of £X?

• Increasing perpetuity

  • This formula applies when the cash flow increases at a constant rate to infinity. Use it to value the cash flows of companies that grow their profits every year.

  • Sample question: What would be the present value of an increasing cash flow of $X at a growth rate of Y%?

• Annuity

  • Use the annuity formula when you have constant payments over a fixed period, such as personal loans and mortgages.

  • Sample question: What is the total amount I will pay on a loan of £X with monthly payments of £Y for Z years?

• Annuity increasing

  • This case applies to cash flows that increase over a defined period. Use it for insurance contracts or increasing pension payments.

  • Sample question: What is the present value of an increasing annuity of $X over Y years at a growth rate of Z%?

• Effective interest rate

  • Apply the formula when you need to compare interest rates with different capitalisation frequencies.

  • Sample question: What is the effective interest rate for a nominal rate of X% compounded Y times a year?

• Continuous compounding

  • Use this formula for long-term investments where the interest rate is compounded continuously, as in the case of investment funds.

  • Sample question: What would be the value of a continuously compounded investment of $X after T years with a rate of r%?