Introduction to Financial Maths

• Financial maths is a branch of mathematics that deals with financial problems. particularly monetary value of things.

Monetary value

• Refers to the value of a product or service measured in terms of money. E.g., The bread costs approximately R20, meaning its monetary value is R20.
• Monetary value is subject to change over time, which may be due to one of the following:
  • Interest - Is the money charged when a debt is incurred through borrowing money, taking a car, phone, house, etc. on credit.
  • Inflation - Inflation is the rate of increase in prices over a given period of time. for example, monetary value or price of bread in 2005 was R5, today in 2024 it is R20.
  • Deflation - Deflation is the rate of decrease in prices over a given period of time. for example, monetary value or price of petrol was R23.01 in May 2023, and 22.79 in December 2023.
  • Depreciation - Things lose value over time due to wear and tear or becoming outdated. for example, you buy a phone for R5000, after a year you want to sell it. The fair price should be below R5000 due to wear and tear.
  • Appreciation - Things gain value over time due to inflation, market demand, or improvement to the asset. House prices usually increase there's not enough houses to house everybody, hence there is always a high demand for houses.
• Interest, Inflation, Deflation, Depreciation, and Appreciation can be represented as capital 'I'.

Rate of Change

• Percentage with which value changes over time, represented by small letter 'i' or r.
• Represents all monetetary changes, Interest rate, Inflation rate, Deflation rate, Depreciation rate, or Appreciation rate
• Expressed as a percentage. E.g., Interest rate of 7% or Inflation rate of 3%.

• Two types of changes: Growth and Decay

  • Simple growth and Decay: Constant growth or decrease
  • Compound growth and Decay: Constant rate calculated on a different amount each year

Simple Change in monetary value

Equation: I = Pit, where:

• I is the total amount of monetary value change
• P is the initial monetary value of asset, or money borrowed or lended.
• i is the rate of change in percentage, from the principal each passing time, which is usually per year for simple rate of change. Note: rate of change can also be represented as r.
• t is the amount of time the rate of change counted apon. Note: duration can also be represented as n.
  
• You deposit R2000 at Capitec bank for a 5% interest rate annual.
  

  How much interest would you have earned after 5 years?

  
  first divide the given interest rate by 100.
  i = 5/100
  i = 0.05
  
  I = Pit
  I = R2000 * 0.05 * 5
  I = R500
  Therefore, you would have earned R500 interest in your R2000 deposit.
  

• Now, how much in total will be available in your account after those 5 years?

  The total is usually represented by capital letter 'A'. Therefore, A is the sum of Principal and Interest earned.
  A = P + I
  A = R2000 + R500
  A = R2500
• You need to calculate the final amount available after a given duration without calculating interest earned first
  A = P + I
  I = Pit, hence you can replace I with Pit
  A = P + Pit
  Factorize, A = P (1 + it)

Simple Interest

• The interest earned is equal to the principal times the rate times the time. Meaning, the principal never changes, the interest is always counted using the initial investment or principle. Hence, also interest earned in a given timeline remains the same.
• You deposit R2000 at Capitec bank for a 5% interest rate annual.
  
  
  first divide the given interest rate by 100.
  i = 5/100
  i = 0.05
  

  first year interest

  I = Pit
  I = R2000 * 0.05 * 1
  I = R100
  

  second year interest

  I = Pit
  I = R2000 * 0.05 * 1
  I = R100
  

  third year interest

  I = Pit
  I = R2000 * 0.05 * 1
  I = R100
  

Compound Interest

• In contrast to Simple Interest, Compound interest The interest earned each timeline (yearly, Quarterly, Monthly, Weekly, daily, hourly) is added to the principal for the next timeline's calculation.
• You deposit R2000 at Capitec bank compounded at 5% annual.
  
  
  first divide the given interest rate by 100.
  i = 5/100
  i = 0.05
  

  first year interest

  I = Pit
  I = R2000 * 0.05 * 1
  I = R100
  

  second year interest

  The new principal is R2100, because the previous timeline earned interest get added to the initial principal (R2000 + R100 = R2100). I = Pit
  I = R2100 * 0.05 * 1
  I = R105
  

  third year interest

  The new principal is R2250, because R2100 + R105 = R2250. I = Pit
  I = R2250 * 0.05 * 1
  I = R110,25
  
• Therefore in the period of three years, you will earn R300 with a total of R2300 in your account if your R2000 deposit is 5% Simple or earn R312,25 interest and total amount of R2315,25 in your account if compounded.

Compound Interest Formula

• The accumulated amount (A) is calculated using the formula:
• A = P(1 + r/m)^mt
  Where:
  • A = accumulated amount
  • P = principal invested
  • r = interest rate
  • m = number of conversions per year
  • t = number of years
  Example: A = $2,000(1 + 0.05/1)^(1*3)

• You deposit R2000 at Capitec bank compounded at 5% annual.
  

  Find the accumulated amount in 3 years

  
  first divide the given interest rate by 100.
  i = 5/100
  i = 0.05
  A = P(1 + r/m)^mt
  A = R2000(1 + 0.05/1)⁽¹⁾⁽³⁾
  A = R2000(1 + 0.05)³
  A = R2000(1.05)³
  A = R2000(1.1025)
  A = R2315,25

Effective Rate of Interest

• It calculates the rate required for the same earnings with different compounding frequencies.

Calculation for Monthly Compounding

• r_eff = (1 + r_nom / m)^m - 1 Where:
  r_eff = is the effective rate of interest.
  r_nom = is the nominal rate of interest.
  m = is the number of conversions per year.

Example: For R1000 invested at 5% with monthly compounding:

• m = 12 because there are 12 months in a year
• r_eff = (1 + r_nom / m)^m - 1
  r_eff = (1 + 0.05 / 12)¹² - 1
  r_eff = 0.05116 * 100
  r_eff = 5.116%
  

Calculation daily Compounding

• m = 365, because there are 365 days in a year.
• r_eff = (1 + r_nom / m)^m - 1
  r_eff = (1 + 0.05 / 365)³⁶⁵ - 1
  r_eff = 0.05127 * 100
  r_eff = 5.127%
  

Comparison of Monthly and Daily Compounding

• The effective rate of interest for daily compounding is slightly higher at 5.127% compared to 5.116% for monthly compounding.
• This means the equivalent rate needed to earn the same amount of money is higher for daily compounding.

Future Value and Present Value

• Future value (FV) represents the value of money in the future, while present value (PV) represents the value of money today.
• Find the future value of $10,000 in 20 years at an annual interest rate of 6%

Future value/Present value Formula

• It is almost exactly the same as the formula for compound interest, the only difference is that A is FV and P is PV.
• FV = PV * (1 + r)ⁿ
• m = number of conversions per year
• t = number of years

• The future value of $10,000 in 20 years at 6% interest is $32,071.35.
• Present value is the value of an amount of money at the current time
• Present value of $100,000 in 10 years at an annual interest rate of 6% is calculated using the formula: PV = FV / (1 + r)ⁿ.
• The present value of $100,000 in 10 years at 6% interest is $55,839.48.
• Illustrates the time value of money, where money is worth more now than in the future.
• Purchasing power decreases as time progresses.
• The same amount of money is worth more now than in the future.

Depreciation and Reducing Balance Method

• Depreciation is the loss of the value of an asset over a period of time.

Example Calculation

• Anthony bought a delivery vehicle for $80,000 with an estimated useful life of three years at a 30% rate.
• Year 1: Depreciation = $80,000 - 30% = $24,000.
• Year 2: Depreciation = $80,000 - ($24,000 + 30% of $80,000) = $16,800.
• Year 3: Depreciation = $80,000 - ($24,000 + $16,800) - 30% = $11,760.
• Overall Depreciation in those three years is therefore, the sum of all Depreciation.
  Overall Depreciation = Year 1: Depreciation + Year 2: Depreciation + Year 2: Depreciation
  Overall Depreciation = Year 1: $24000 + $16800 + 11760
  Overall Depreciation = $52560

Book Value

• Net book value represents the present value of the asset in financial statements.
• And is calculated by substracting the overall depreciation by original price of the asset.
• E.g., Net book value of Anthony's vehicle after 3 years = $80000 - $52560
  Net book value = $27440
• Or you can use the fornula, A = P(1 - r)ⁿ
• A = $80000(1 - 0.3)³
• A = $27440

Conclusion

• The reducing balance method results in higher depreciation in the first year, gradually reducing over time.