Lecture 6 - Time Value of Money Flashcards
(26 cards)
Let’s warm up
- Assuming the interest rate is 10% per annum,
and you saved £100 in the bank today. - How much are you going to have in you bank
account in a year?
balance in a year = 100 + 100*10% = 110 - Time value of the money! £110 receiving in a
year is only worth £100 today.
- What is time value of money?
- Present value (PV): the value of the asset now
- Future value (FV): the value of the asset on a
specific date in the future. - Time value of money: A pound to be received
in the future is worth less than a pound received
today. The present value of a sum of money
depends on the date of it receipt.
The same example:
assuming the interest rateis 10% per annum, and you saved £100 in the
bank today. How much are you going to have
in you bank account in a year?
balance in a year = £100 + £100*10% = 110
- Present value (PV): £100
- Future value (FV): £110
- Why does money have time value?
- Opportunity cost: the sacrifice of the return
available on the best forgone alternative. - Time value of money is the compensation to
the opportunity cost of investors’. - Will you lend me £1,000, if I tell you that I will
repay you the same amount in 10 years?
Time value of money
- e.g. saving money in the bank, we receive
interest. The interest rate is the time value of
money to against inflation, compensating your
loss of liquidity and potential return from
alternative investment (opportunity cost). - You will generously lend me £1,000 if I repay you
the £1,000 plus interest in 10 years time. The
interest is the time value of this £1,000 to
compensate your opportunity cost over this 10
years.
- 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. - In a simple word, we know the future value
(FV) of the cash flows, and looking for the
present value (PV) of them. (How much are
they worth now?)
Discounted cash flow
- Single cash flow in single period
FV = PV * (1+R)
where R is the interest rate (required rate of
return, discount rate… ), PV is the present valueof the cash flow and FV is the future value of thecash flow. Rearrange above equation, we have
PV = FV/(1+R)
Example 1
assuming a sum of £10 is deposited in a bank account that pays 12 percent 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
- Example 2:
Given the interest rate of 12 per cent
per annum. If you anticipate the receipt of £11.2in the bank account in one year, how much do
you need to deposit now? (what is the PV of£11.2 you are to receive in one year?)
PV = FV/(1+R)
= £11.2/(1+12%)
= £10
In this case, we say we discount the FV for one year.
Single cash flow in multiple periods
FV = PV * (1+R)T
where R is the interest rate (required rate of
return, discount rate…), T is the number of
period, 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)T
Example 1:
assuming a sum of £10 is deposited in
a bank account that pays 10 per cent per annum interest rate. How much will be in the account at
the end of year 2? (What is the FV of £10 in two years)
FV = PV * (1+R)T
= £10 *(1+10%)2
= £12.1
In this case we say the interest rate is compounding
for 2 years.
- Illustration of the example
Year FV
1 £10*(1+10%) = £11
2 £11(1+10%)
= £10(1+10%)*(1+10%)
= £10 *(1+10%)2
= £12.1
Example 2:
Given the interest rate of 10 per cent
per annum. If you anticipate the receipt of £12.1 in the bank account in two years, how much do you need to deposit now? (what is the PV of
£12.1 you are to receive in two years?)
PV = FV/(1+R)T
= £12.1/(1+10%)2
= £10
In this case, we discount the FV for 2 years.
- The general formula for both Single cash flow in
single period and Single cash flow in multiple
periods are
FV = PV * (1+R)T
and
PV = FV/(1+R)T
where R is the interest rate (required rate of return,
discount rate…), T is the number of period, PV is the
present value of the cash flow and FV is the future
value of the cash flow.
Discounted cash flow – Using the
tables
- Tired of using your calculator? We can use FV and PV table to find the answer!!!
- For FV and PV of single cash flow, we use Table A-1 and A-3 provided, respectively.
- Table A-1: The future value of £1 in T years.
- Table A-3: The present value of £1 to be received in T years.
- Various interest rates and time periods are
given on the table.
Discounted cash flow – Using the
tables
- Find the FV value of £10 twenty years from
now, given the interest rate is 15%.(Table A-1)
£10*16.367 = £163.67 - Find the PV value of £10 to be received in
fifteen years, given the interest rate is
12%.(Table A-3)
£10*0.1827 = £1.827
- Multiple cash flows in multiple periods
- Annuities
- Perpetuities
Note: we only consider ordinary annuity and
ordinary perpetuity in this module.
- 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.
- Example:
A five-year annuity offers the annual
payment of £10 per year, the payment starts at the end
of year one, and given the interest rate is 12% per annum, what is the PV of this annuity.
- Illustration (breakdown to each year)
Year FV PV
1 10 10/(1+12%)
2 10 10/(1+12%)2
3 10 10/(1+12%)3
4 10 10/(1+12%)4
5 10 10/(1+12%)5
illustration continued
So the PV value of this annuity is:
PV=10/(1+12%)+10/(1+12%)’2+10/(1+12%)’3+10/(1+12%)’4+10/(1+12%)’5 = £36.05
R is the discount rate (E.g interest rate), T is the number of years that this annuity will last, and A is the amount of annual payment. In our example, R=12%, T=5 and A=£10. Then PVA=£36.05
WE can also use PV of annuity table to find the PV of an annuity
Table A-4 is the PV of an ordinary annuity offering £1 per annual payment. Various interest rates and time periods are given.
In our example, we can find the annuity factor
A’5,12% = 3.605
Note the annuity above stands for the payment of £1 per annum, in our example is £10 per annum, so the PV of the annuity in our example is
10*A’5,12% = 36.05
Exercise
Try to find the value of A’15,10% =
A’50,3% =
Using Table A-4, we get:
A’15,10% = 7.6061
A’50,3% = 25.730
- 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.
Example
A perpetuity offers the annual payment of £10 per year, the payment starts at the end of year one to infinity, and given the interest rate is 12% per annum, what is the PV of this perpetuity
Year FV PV
1 10 10/(1+12%)
2 10 10/(1+12%)2
3 10 10/(1+12%)3
4 10 10/(1+12%)4
5 10 10/(1+12%)5
: (infinity)
