Study Notes on Corporate Finance & Financial Strategies - Time Value of Money

Definition: The principle that £1 today does not have the same value as £1 tomorrow.
Example Question: Would you prefer receiving £100 today or £100 in one year?
Investment Scenario: If £100 is received today and invested at an annual interest rate of 6%:
- Future amount after 1 year is calculated as:
  
  £100 imes (1 + 0.06) = £106
- This future value is only possible if funds are received today.

Future Value Calculation: To find future value after successive periods, you can continue applying the interest rate.
- Example: Leaving £100 in the bank account for two years:
- After the first year:
  
  £106 imes (1 + 0.06) = £112.36
- After the second year:
  
  £112.36 imes (1 + 0.06) = £119.10
- General Formula: Future Value of £100 invested for t years:
  
  FV = £100 imes (1 + r)^t

Definition: Interest earned on interest; involves reinvestment of earned interest.
Comparison Example: Simple vs Compound interest at 6%:
- Simple Interest:
- Year 1: Interest Earned = £6, Total Value = £106
- Year 2: Interest Earned = £6, Total Value = £112
- Year 3: Interest Earned = £6, Total Value = £118
- Compound Interest:
- Year 1: £6 Earned, Total Value = £106
- Year 2 Interest = £6.36, Total Value = £112.36
- Year 3 Interest = £6.74, Total Value = £119.10

Scenario: Deposit €100 today in an account paying 8%, then deposit another €100 in one year. Total after 2 years?
- Calculating FV:
- €100 Initial Deposit:
  - After 1 Year: = €108
  - After 2 Years: = €116.64
- Future Value Calculation for €2,000 deposits over 5 years at 10%:
  - Current balance = €0
  - Future Value growth over 5 years:
  - At end of Year 1:
    
    €2,200
  - At end of Year 2:
    
    €4,620
  - At end of Year 3:
    
    €7,282
  - At end of Year 4:
    
    €10,210.20
- Total Future Value after 5 years = €12,210.20

Definition: The value today of future cash flows.
Calculation Method: Discounting future values back to the present using the discount rate (r).
- Example: Present value of £300 to be received in one year at a 2% interest rate:
  
  PV = £300 / (1 + 0.02) = £294
  
  Alternatively, expressed as:
  
  PV = FV imes (1 + r)^{-t}

Definition: The present value of £1 received in year t discounted at r.
Formula:

DF = rac{1}{(1 + r)^t}
Allows calculation of the present value of future cash flows by multiplying with the appropriate discount factor.

Definition: A perpetuity is a stream of equal cash flows occurring at regular intervals that continues indefinitely.
Present Value of Perpetuity Formula: PV = rac{C}{r}
- Where C is the reoccurring cash flow starting one period from now and r is the interest rate.

Definition: An annuity is a series of equal cash flows paid at regular intervals for a finite term.
Present Value of Annuity Formula: PV = C imes iggl[ rac{1}{r} - rac{1}{r(1 + r)^t}iggr]
- Where C is the cash flow, r is the interest rate, and t is the number of periods.

Future Value of Annuity Formula: FV = C imes iggl[ rac{(1 + r)^t - 1}{r}iggr]
- C is the cash flow, r is the interest rate, and t is the number of periods.

Effective Interest Rate: Rate at which money grows, considering the effects of compounding.
APR (Annual Percentage Rate): Often used in financial contexts, expressed simply without accounting for compounding.
- For a monthly rate:
- APR = ext{monthly rate} imes 12

Inflation: The pace at which prices for goods and services rise, typically measured via the Consumer Price Index (CPI).
Nominal Interest Rate: Interest rate before adjusting for inflation.
Real Interest Rate: Adjusted for inflation, reflecting the true increase in purchasing power.
- Formula for the Real Interest Rate:
  
  r{real} = r{nominal} - ext{inflation rate}

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