TIME VALUE OF MONEY:- Ved Prakash panda

Which would you prefer – Rs.1,000,00 today 1,000,00 today or Rs.1,000,00 after 10 years1,000,00 after 10 years?

Obviously, Rs.1,000,00 today1,000,00 today.

You already recognize that there is

TIME VALUE TO MONEYTIME VALUE TO MONEY!!

Why TIME?Why is TIMETIME such an important element in your decision?

TIMETIME allows you the opportunity to earn the

INTERESTINTEREST.

What is The Time Value of Money?• Money value today is worth more than received

tomorrow• This is because a rupee received today can be invested to

earn the interest• The amount of interest earned depends on the rate of return

that can be earned on the investment• Time value of money quantifies the value of a rupee

through time

Required Rate of Return• The time preference for money is generally expressed by an interest rate. This rate will be positive even in the absence of any risk. It may be therefore called the risk-free rate.

• An investor requires compensation for assuming risk, which is called risk premium.

• The investor’s required rate of return is:Risk-free rate + Risk premium

Time Value Adjustment• Two most common methods of adjusting cash flows

for time value of money: • Compounding — the process of calculating future

values of cash flows and • Discounting — the process of calculating present

values of cash flows.

Uses of Time Value of Money• Time Value of Money, or TVM, is a concept that is used in all aspects of finance including:• Bond valuation• Stock valuation• Accept/reject decisions for project management• Financial analysis of firms• And many others!

SYMBOLS• where

• i = rate of return or interest rate• n = time periods• A = Annuity• PV = present value• PVA = present value of an annuity• FV = future value• FVA = future value of an annuity• FVIF = Future value interest factor• PVIF = Present value interest factor• FVIFA = Future value interest factor of an annuity• PVIFA = Present value interest factor of an annuity

Future value of a lump sum

FV = PV X (1+i)n = PV (FVIFi,n)

EXAMPLE• A person deposits a sum of rs. 30,000 at the interest of

8%, compounded annually for 5 years. Find the maturity value after 5 years.

Solution• PV= 30,000• i = 8%• n = 5 years

FV = PV X (1+i)n = PV (FVIFi,n)

= 30,000( 1 + 0.08)5 = 44,079.84 The maturity value of rs30,000 invested now at 8%

compounded yearly is equal to rs 44,079.84 after 5 years.

•

Present value of a lump sum

PV = FV / (1+i)n

= FV (PVIFi,n)

Example• A person wishes to have a future sum of Rs. 5,00,000 for

his son’s education at U.K. after 10 years from now. What is the single payment that he should deposit now so that he will get the desired amount after 10 years? The bank gives 7% interest rate, compounded annually.

Solution PV = FV / (1+i)n

= FV (PVIFi,n) = 5,00,000 / (1+ 0.07)10

= 2,54,174.67

The person has to invest rs 2,54,174.67, now so that he will get a sum ofrs.5,00,000 after 10 years at 7% interest rate, compounded annually.

Future value of an annuityFVA = A X {[(1+i)n - 1]/i}

= A ( FVIFAi,n)

Example• A person who is now 30 years old is planning for his

retired life. He plans to invest an equql sum of rs. 10,000 at the end of every year for the next 30 years starting from the end of the next year. The bank gives 8% interest rate, compounded annually. Find the maturity value of his account when he is 60 years old.

Solution• A= 10,000• n= 30 years• i= 8%• F = ?• FVA = A X {[(1+i)n - 1]/i}• = 10,000X {[(1+0.08)30 - 1]/ 0.08}• = Rs. 11,32,831.50• The future sum of the annual equal payments after 30

years is equal to rs. 11,32,831.50.

Annuity for the future value

• Sinking Fund

• A= FVA / {[(1+i)n - 1]/i}

Example• A firm has to replace a machine after 10 years at an

outlay of Rs. 4,00,000. It plans to deposit an equal amount at the end of every year for the next 10 years at an interest rate of 6%, compounded annually. Find the equivalent amount that must be deposited at the end of every year for the next 10 years.

Solution• F=Rs.4,00,000• n= 10 years• i= 6%• A= ?• A= FVA / {[(1+i)n - 1]/i}• = 4,00,000/ {[(1+ 0.06)10 - 1]/ 0.06}• = Rs. 30,347.16• The annual equal amount which must be deposited

for 10 years is Rs.30,347.16.

Present value of an annuity

PVA = A x {[(1+i)n - 1]/ [i (1+i)n]} = A ( PVIFAi,n)

Example• A company wants to set up a reserve which will help the

company to have an annual equivalent amount of rs. 20,00,000 for the next 20 years towards its employees welfare measures. The reserve is assumed to grow out the rate of 10% annually. Find the single payment that must be made now as the reserve amount.

• PVA = A x {[(1+i)n - 1]/ [i (1+i)n]} • = 20,00,000 x {[(1+0.10)20 - 1]/ [ 0.10(1+0.10)20]}• = Rs. 170,27,128.00

• The amount of reserve which must be set up now is equal to Rs. 170,27,128.00.

Annuity for the present value

Loan Amortization or capital recoveryA = PVA / {[(1+i)n - 1]/ [i (1+i)n]}

Example• The State Bank of India gives a loan to a company to

purchase a machine worth RS. 5,00,000 at an interest rate of 12% compounded annually. This amount should be repaid in 10 yearly equal installments. Find the installments amount that the company has to pay to the bank.

Solution• P = Rs. 5,00,000• i= 12%• n= 10 years• A= ?

A = PVA / {[(1+i)n - 1]/ [i (1+i)n]}

= 5,00,000 / {[(1+ 0.12)10 - 1]/ [0.12 (1+0.12)10]}

= 88,492.05

The annual equivalent installment to be paid by the company to the bank is Rs.88,492.05.

The “Rule-of-72 and 69” How long does it take to double Rs.5,000 at a compound

rate of 12% per year (approx.)? Approx. Years to Double = 72 / i or,

= 0.35 + 69 / i

• Approx. Years to Double = 72 72 / i• 7272 / 12 = 6 Years6 Years

• If you deposit Rs.50,000 today in a financial institute at the rate of 8 per cent in how many (roughly) years will this double using rule 72 and rule 69.

• Solution. (a) Years to Double = 72 / i = 72 / 8 = 9 years

•  or, 0.35 + 69 / i = 0.35 + 69 / 8 = 8.975 years

Types of AnnuitiesAn Annuity represents a series of equal payments (or receipts) occurring over a specified number of equal periods.•Ordinary Annuity: Payments or receipts occur at the end of each period.•Annuity Due: Payments or receipts occur at the beginning of each period.

Examples of Annuities• Student Loan Payments• Car Loan Payments• Insurance Premiums• Mortgage Payments• Retirement Savings

Future value of an annuity dueFuture value of an annuity due = A X {[(1+i)n - 1]/i} X ( 1 + i )

= A ( FVIFAi,n) X ( 1 + i )•

Present value of an annuity due• Present value of an annuity due = A x {[(1+i)n - 1]/ [i (1+i)n]} X ( 1 + i )

= A ( PVIFAi,n) X ( 1 + i )

Multi period compounding• The interest rate is usually specified on an annual basis,

in a loan agreement or deposits , and is known as the nominal interest rate.

• If compounding is done more than once in a year , the actual annualized rate of interest would be higher than the nominal interest rate and it is called the effective interest rate.

Effective interest rate• EIR = { 1 +( i/m)} nm - 1

• Where • i = annual nominal rate of interest• n = the number of years• m = the number of compounding per year

Example• You have invested Rs 100.00 in a bank, interest rate

being 10% in a year. The bank will compound interest semi-annually( i.e twice a year ) . What is the effective interest rate.

• EIR = { 1 +( 0.10/ 2)} 2 - 1• = 1.1025 – 1 = 0.1025 = 10.25%

Compounded value of a sum in case of multi- period compounding • This concept can be used for multi- period compounding

or discounting.• FV = PV { 1 +( i/m)} nm

• Where• FV = Future value• PV = Present value • i = annual nominal rate of interest• n = the number of years• m = the number of compounding

Future value of an annuity in case of multi- period compounding • FVA = A X {[(1+ (i/m) )nm - 1] / (i/m) }

Example• What is the compound value of rs 1000 , interest rate

being 12% per annum if compounded annually, semi-annually, quarterly and monthly for 2 years.

Solution• 1. Annual compounding = 1,000 x ( 1 + 0.12) 2 • = 1,000 x 1.254 = 1254• 2. Half- yearly compounding = 1,000 x { 1 + (0.12/2)} 2x2 • = 1,000 x 1.262 = 1262• 3. Quarterly compounding = 1,000 x { 1 + (0.12/4)} 2x4 • = 1,000 x 1.267 = 1267• 4. Monthly compounding = 1,000 x { 1 + (0.12/12)} 2x12 • = 1,000 x 1.270 = 1270

Question• A person deposits a sum of rs. 1,00,000 at the interest of

8%, compounded semi- annually for 10 years. Find the maturity value after 10 years.

• FV= 1,00,000 x 2.191 = 2,19,100

Question• A person wants a future sum of Rs. 8,00,000 for his son’s

education after 7 years from now. What is the single payment that he should deposit now so that he will get the desired amount after 7 years? The bank gives 10% interest rate, compounded semi- annually.

• PV = 8,00,000 x 0.505= 4,04,000