Lecture 6: Time value of money

Question 1

What is time value of money

Answer 1

• Present value (PV): the value of the asset now
• For example, in the case we mentioned earlier, the £100 you deposited in the bank today is the present value)
• Future value (FV): the value of the asset on a specific date in the future
• Similarly, the £110 that you will receive after one year is the future value
  Assuming the interest rate is 10% per annum, and you deposited £100 in the bank today (at present).
  
  • How much are you going to have in you bank
    account in a year?
    Balance in a year = 100 + 10010% = 110
    where 100 is your ‘principal’ and 10010% is the interest payment
    Time value of the money! £110 receiving in a year is only worth £100 today. In other words, you deposited £100 in the bank today, and after one year, you will receive £110.
    The money you have now is worth more than the identical sum in the future due to its potential earning capacity. This core principle of finance holds that provided money can earn interest, any amount of money is worth more the sooner it is received.
    One of the incentives to save (money) is the possibility of gaining a higher level of future consumption by sacrificing some present consumption.
    The time value of money draws from the idea that rational investors prefer to receive money today rather than the same amount of money in the future because of money’s potential to grow in value over a given period of time. For example, money deposited into a savings account earns a certain interest rate and is therefore said to be compounding in value.

Question 2

Opportunity cost

Answer 2

• Opportunity cost: the sacrifice of the return available on the best forgone alternative. Opportunity costs represent the potential benefits an individual, investor, or business misses out on when choosing one alternative over another.
• Time value of money is the compensation to the opportunity cost of investors.
• i.e., if you choose to deposit your money in a commercial bank and may receive interests paid by the bank rather than invest your money in stock markets, the potential capital gains and dividends are the ‘opportunity costs’ because doing so you will only receive the interests (associated with your deposits), and miss out the potential capital gains and dividends from investing in stock markets.
• Will you lend me £1,000, if I tell you that I will repay you the same amount in 10 years?
• I should repay you the principal plus addition interest payment; otherwise, you may not lend me £1,000.

Question 3

Discounted cash flow

Answer 3

What is discounted cash flow?
- Future cash flows are converted into the common denominator of time zero by adjusting for the time value of money.
‘Time zero’ denotes ‘now’ or ‘at present’
- In a simple word, we know the future value (FV) of the cash flows, and looking for the present value (PV) of them.
Formula: FV = PV * (1+R)
where R is the interest rate (or required rate of return, discount rate… ), PV is the present value of the cash flow and FV is the future value of the cash flow.
Rearrange above equation, we have:
PV = FV/(1+R)
Please do not be worried about that – you would see some simple examples later.

Example 1: assuming a sum of £10 is now deposited in a bank account that pays 12 per cent per annum interest rate. How much will be in the account at the end of year 1? (What is the FV of £10 in a year
FV = PV * (1+R)
= £10*(1+12%)
= £11.2
Based on this formula, FV = PV * (1+R), we have already got the present value (PV) which is £10, and the interest rate (R) is 12%. We next insert these numbers into the formula, and then we could get the future value (FV).

Question 4

Annuities

Answer 4

What is an annuity?
- An annuity is a financial instrument which offers the holder a series of identical payments over a period of years.
What is the PV of an annuity?
- PV of an annuity is the sum of PVs of all future payments.
PV and market price
Note: we only consider the annuities that offer the equally payment at the end of each year in this module.

Question 5

Perpetuities

Answer 5

What is a perpetuity?
- A perpetuity is a financial instrument which offers the holder a regular sum of identical payments received at intervals forever.
What is the PV of a perpetuity?
- PV of a perpetuity is the sum of PVs of all future payments.
PV and market price
Note: we only consider the perpetuities that offer the equally payment at the end of each year in this module.
General formula of PV of a perpetuity is:
PVP = A/R
where A is the amount of annual payment and R is the discount rate (i.e., interest rate). Therefore, for our example,
A is £10 and R is 12%. The present value of this perpetuity is
PVP = £10/12%
≈ £ 83.3