TIME VALUE OF MONEY

In this session we shall cover

• Meaning of Time value of money• What is Compounding and Discounting?• Different patterns of Cash flows and adjusting

them for TVM• Multi period compounding/discounting• Use of TVM in finance

Time Value of Money

• The basic principle in finance is that“ A rupee today is worth more than a rupee

tomorrow”

Why?

“ A rupee today is worth more than a rupee tomorrow”

because

“ the former can be invested to start earning interest immediately ”

Example

• If I have a Re1 today with me and I can invest it at an interest rate = 10% p.a. for 5 years, then this Re1 shall grow to

Example

Rs 1.611 after 5 years.

How?

Example

• Using the formula

P(1+r)n

where ‘P’ is the principal amount today, ‘r’ is the rate of interest per annum and ‘n’ are the number of years

= 1 (1+0.1)5 = 1.611

Example

• This is to say that

‘A person would be indifferent between receiving Re 1 today and receiving Rs 1.611 after 5 years if the opportunity to earn a return is of 10% p.a.’

COMPOUNDING

Present Value

COMPOUNDING

Future Value = Present Value (1+r) n

COMPOUNDING

Future Value = Present Value (1+r) n

• The process of finding the future value (FV), when the present value (PV) is known for different combinations of ‘r’ and ‘n’ is called COMPOUNDING.

DISCOUNTING

Future value

DISCOUNTING

• Present value = Future value (1+r) n

DISCOUNTING

• Present value = Future value (1+r) n • The process of finding the present value (PV),

when the future value (FV) is known for different combinations of ‘r’ and ‘n’ is called DISCOUNTING.

Example

• Rs 50,000 is to be received after 15 years from now. What would be its present value if the interest rate is 9% p.a.?

Rs 50,000 is to be received after 15 years from now. What would be its present value if the interest rate is 9% p.a.?

• PV = FV / (1+r)n

= 50,000 / (1+0.09)15

= Rs 13,750• This means that Rs 13,750 received today or

Rs 50,000 received after 15 years shall have the same value if the opportunity to earn interest is 9% p.a.

CVF and PVF Tables (Single amounts)

• CVF Table: provides the FV of Re 1 today for various combinations of ‘r’ and ‘n’

FV = PV * CVF (r%, n)

• PVF Table: provides the PV of Re 1 in future for various combinations of ‘r’ and ‘n’

PV = FV * PVF (r%, n)

CVF and PVF Tables (Single amounts)

Insert tables here

CVF and PVF Tables (Single amounts)

• Insert tables here

Rs 50,000 is to be received after 15 years from now.What would be its present value if the interest rate is 9% p.a.?

• PV= FV * PVF (9%, 15) = 50,000 * 0.275 = Rs 13,750

Eg of FV of a Single amount

• Rs 10,000 is being deposited in a bank today for 5 years. What shall it grow to after 5 years if interest rate is 10% p.a.?

Rs 10,000 is being deposited in a bank today for 5 years. What shall it grow to after 5 years if interest rate is 10% p.a.?

• FV = PV * CVF (r%, n) = 10,000 * CVF (10%, 5) = 10,000 * 1.464

= Rs 14,640• It means that Rs 10,000 of today and Rs

14,640 after 5 years from now carry the same value if the interest rate is 10% p.a.

Different other forms of amounts

• Multiple cash flows• Annuity• Perpetuity• Annuity due• Growing annuity• Growing perpetuity

Multiple cash flows

• This refers to more than one amount occurring (cash inflow) or having to pay (cash outflow) at different points of time in future

Eg of Multiple cash flows

• An investment will pay Rs 100 after one year, Rs 300 in the second year, Rs 500 in the third year and Rs 1000 in the fourth year. If the interest rate is 10%, what is the present value of these multiple cash flows together?

Eg of Multiple cash flows

PV = 100/(1+0.1)1 + 300/(1+0.1)2 + 500/(1+0.1)3 + 1000/(1+0.1)4

OR

= 100*PVF(10%,1)

+ 300*PVF(10%,2)

+ 500*PVF(10%,3)

+ 1000*PVF(10%,4)

= Rs 1397.91

Annuity

• It refers to the same amount of cash flow at regular intervals of time for a specified period

• This cash flow occurs at the end of each year• If it occurs at the beginning of each year, we

call it ‘Annuity due’

Calculating PV of an Annuity

• The PVs of Re1 annuities for different combinations of ‘n’ and ‘r’ can be had from PVAF Table

• PV (any annuity amount)= Annuity amount * PVAF (r%, n)

Example of Annuity

• What would be the PV of Rs 900 to be paid at the end of each year for 3 years from now at interest rate of 10%?

Example of Annuity

• What would be the PV of Rs 900 to be paid at the end of each year for 3 years from now at interest rate of 10%?

• PV of Rs 900 of annuity is to be calculated

Example of Annuity

• What would be the PV of Rs 900 to be paid at the end of each year for 3 years from now at interest rate of 10%?

• PV of Rs 900 of annuity is to be calculated• PV (any annuity amount)

= Annuity amount * PVAF (r%, n)

Example of Annuity

• What would be the PV of Rs 900 to be paid at the end of each year for 3 years from now at interest rate of 10%?

• PV of Rs 900 of annuity is to be calculated• PV (any annuity amount)

= Annuity amount * PVAF (r%, n)• PV (Rs 900 annuity) = 900 * PVAF (10%, 3)

Example of Annuity

• What would be the PV of Rs 900 to be paid at the end of each year for 3 years from now at interest rate of 10%?

• PV of Rs 900 of annuity is to be calculated• PV (any annuity amount)

= Annuity amount * PVAF (r%, n)• PV (Rs 900 annuity) = 900 * PVAF (10%, 3)

= 900 * 2.487 = Rs 2,238

Example of Annuity

• What would be the PV of Rs 900 to be paid at the end of each year for 3 years from now at interest rate of 10%?

• PV (Rs 900 annuity) = 900 * PVAF (10%, 3) = 900 * 2.487 = Rs 2,238

Perpetuity

• It is the same amount of cash flow occurring at regular intervals of time for an infinite time in future

• This also occurs at the end of each year

Calculating present value of Perpetuity

• PV (Perpetuity) = Annual amount / r

Example of Perpetuity

• Find out the PV of an investment which is expected to give a return of Rs 2500 p.a. infinitely and the rate of interest is 12% p.a.

Example of Perpetuity

• Find out the PV of an investment which is expected to give a return of Rs 2500 p.a. infinitely and the rate of interest is 12% p.a.

• PV of (perpetuity=2500) = 2500/0.12 = Rs 20,833.33

Multi period Compounding / Discounting

• The compounding / discounting may take place more than once a year i.e. monthly, quarterly, semi annually, etc.

• In that case an adjustment would be required in ‘r’ and ‘n’ in the formula of PV and FV, all else remaining the same

Multi period Compounding / Discounting

• The compounding / discounting may take place more than once a year i.e. monthly, quarterly, semi annually, etc.

• In that case an adjustment would be required in ‘r’ and ‘n’ in the formula of PV and FV, all else remaining the same

• r% p.a. : (r/x) % per period of compounding / discounting

Multi period Compounding / Discounting

• The compounding / discounting may take place more than once a year i.e. monthly, quarterly, semi annually, etc.

• in that case an adjustment would be required in ‘r’ and ‘n’ in the formula of PV and FV, all else remaining the same

• r% p.a. : (r/x) % per period of compounding / discounting

• n : n x periods of compounding / discounting

Multi period Compounding / Discounting

• The compounding / discounting may take place more than once a year i.e. monthly, quarterly, semi annually, etc.

• in that case an adjustment would be required in ‘r’ and ‘n’ in the formula of PV and FV, all else remaining the same

• r% p.a. : (r/x) % per period of compounding / discounting• n : n x periods of compounding / discounting• Here ‘x’ are the number of times compounding /

discounting takes place in a year

Example of Multi period discounting

• Calculate PV of Rs 1000 to be received after 2 years at interest = 12% p.a. if discounted :– Annually– Semi annually– Quarterly– Monthly

Solution

• If discounted annually:r = 12%, n = 2

Then,PV = 1000 * PVF (12%, 2)

= 797

Solution

• If discounted semi-annually:r = 6%, n = 4

Then,PV = 1000 * PVF (6%,4)

= 792

Solution

• If discounted quarterly,r = 3%, n = 8

Then,PV = 1000 * PVF (3%,8)

= 789

Solution

• If discounted monthly,r = 1%, n = 24

Then,PV = 1000 * PVF (1%, 24)

= 780

How shall it be used in finance?

• Implicit rate of return• Determination of time period• Accept / Reject Capital budgeting decisions• Bond valuation• Stock valuation• Financial analysis of firm

…………….. and many more!!!