1

Chapter: 3

Time value of money

2

Time value of moneyThe phrase “time value of money” refers to the fact that a same amount of money in hand today is more worth than the same amount of money some time in future

because you can earn interest on present money by depositing it in a bank or you can use it for some important purposes.

3

Following sumbols are defined in this article r = Rate of interest or Discount rate n = Number of yearsPV0=Present value at time period 0FVn= Future value at the end of n years.FVAn=Future value of an (ordinary) Annuity for n years. FVADn=Future value of an Annuity due for n years. PVAn =Present value of an (ordinary) Annuity for n years.PVADn=Present value of an Annuity due for n years. R =Constant Yearly payment /periodic payment or Constant Yearly receipt /periodic receipt under annuity plan.m= Number of times compounded or discounted in a year ∞ = Infinite (Never finish)PVA∞=Present value of (ordinary) Annuity, which occur forever=Perpetuity

4

Interest rate The amount charged, expressed as a percentage of principal, by a lender to a borrower for the use of cash or other assets. Interest rates are typically noted on an annual basis, known as the annual percentage rate (APR). The assets borrowed could include, cash, consumer goods, large assets, such as a vehicle or building. Further interest rates are two types:

Simple Interest: It is paid or earned on only original/Principal money borrowed or lent.

Compound Interest: It is paid or earned on original/Principal money borrowed or lent & as well as on previous reinvested accumulated interest amount.

5

Future value or Compound value or Terminal value:When present sum of money is compounded at a given interest rate for certain years, then it is called Future value.Future value techniques typically measure cash flows at the end of a project’s life. Present value or discount value:When future sum of money is discounted at a given interest rate at time zero, then it is called present value. Present value technique typically measures cash flows at the start of a project’s life (i.e. at time zero).The present value is just like cash in hand today.

6

The following types of transactions occur in Time value of money:

1.Single cash flows

2.Annuities

3.Mixed or uneven cash streams

4.Perpetuity

7

1.Single cash flowsWhen payment/deposit or receipt occurs one time only. For example, you paid/deposited a certain sum of money in a bank one time only or receipt occurs one time only at a interest rate of “r” after “n” years Future value=Fvn=PV0 (1+r)n

Present value=PV0 = FVn

(1+r)n

8

2. Annuities When series of equal payments/deposits or receipts occur at regular interval of time over a specified years or periods.Further Annuities are divided into the following two categories:(a). Ordinary Annuity: When each payment or receipt occurs at the end of the year or period.(b) Annuity due: When each payment or receipt occurs at the beginning of the Year or period (a). Ordinary Annuity:[Formula]Future value of (ordinary) Annuities=FVAn= R (1+r)n –1 r Present value of (ordinary) Annuities=PVAn= R (1+r)n -1 r (1+r)n

(b). Annuity due:[Formula] Note: In case of Annuity due, we will just multiply the future and resent value respective above formula by (1+r) only also

9

3. Uneven or mixed cash flowsWhen the cash flows are uneven. Future value of Uneven cash flows[Formula] -> [If cash is deposited at year end]Future value=FVn= Y1Cash inflows(1+r)n-1 + Y2 Cash inflows(1+r)n-2 …… + Yn Cash inflows(1+r)n-n = Rs xx Present value of Uneven cash flows[Formula] -> [If cash is received at year end] Present value= Pvo = Y1 Cash inflows + y2 Cash inflows +……..

(1+r) (1+r)2 + Yn Cash inflows =Rs.xx (1+r)n

10

4.Perpetuity.Present value of Perpetuity:It is an ordinary Annuity, whose first receipt starts at the end of first period, but continues for indefinite time period. For examples, Interest on Perpetual Bonds, Dividend on Preferred stocks, etc.Present value of Perpetuity = PvA = R ÷ r

11

Other advance topics in Time value of Money1. Compounding more than once a yearUp to now, we have assumed that interest is paid annually; but practically Interest is often compounded more than once a year. Banks for example compound semi annually, quarterly, monthly oe even on daily basis.If interest rate is compounded “m” times a year, then in each above respective interest rate “r” will be divided by m & power “n” will be multiplied by m Thus those all above Formulas will become as under, if compounding more than once in a year:oIf single cash flow:Future value= FVn=PV0(1+r/m)mn

Present value= Pv0 = FVn

(1+r/m)mn

12

oSimilarly in case of Annuities:Future value of (ordinary) Annuities=FVAn= R (1+r/m)mn – 1 r/mPresent value of (ordinary) Annuities=PVAn= R (1+r/m)mn -1 r/m (1+r/m)mn

2.Effective Annual Interest Rate(EAR)If Annual percentage rate (APR) is compounded more than once in a year (i.e. Semi annually, Quarterly, Monthly, Weekly, etc.), then the actual rate will be higher than the APR, which is called Effective Annual Interest Rate.It is calculated by:Formula: Effective annual Interest rate (EAR) =(1+r/m)m - 1 Where as, m = number of times in a year.

13

3. Amortizing a loanIn simple terms, the process of reducing the balance of a loan by a yearly or periodic payment amount .It is repaid in equal periodic installments. Examples: Auto loan, mortgage loan, etc. Each time you make a payment on a loan you pay some interest along with a part of the principal. By making regular yearly or periodic payments, the principal gradually decreases, and finally it reaches zero. The payments can be made monthly, quarterly, semiannually, or yearly.Yearly payments of present loan can be found out by present value of annuity formula. If the loan is to be repaid at the end of the year, then we can determine yearly installment by the present value of annuity formula.]

PVAn=R r(1+r)n -1 r(1+r)

14

If the loan is to be repaid at the beginning of the year, then we can determine yearly installment by the present value of annuity formula.] PVAn =R (1+r)n -1 X (1+ r) r(1+r)n

Note: If installment is paid “m” times a year, then the above formula will be changed as discussed above To compute Interest payment and principle payment in each installment, we have to prepare a amortization schedule for this purpose. It is given as under: …………………………. Amortization schedule ………………………Years/Periods Installment Interest Principal Principal at yr end Payments payments payments

15

Application of Time Value of moneyTime value of money has numerous applications in financial and investment decisions, as discussed above. These are summarized as under:1. Finding out the future value of present sum deposits.2. Finding out the present value of future sum.3. Amortization of a loan4. Effective annual rate of interest.5. Finding out the Bonds/stock ‘s cost and value. 6. Capital budgeting techniques, etc.

…………….. ………………………………………….

16

☺The end ☺

17

Presented byS.Z.Jafar