Time preference/ Value for money is an individual’s preference for possession of a given amount of money now, rather than the same amount at some future time.

Three reasons may be attributed to the individual’s time preference for money:

• Risk• preference for consumption• investment opportunities

The time value of 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. The required rate of return may also be called

the opportunity cost of capital of comparable risk

Would an investor want Rs. 100 today or after one year? Cash flows occurring in different time periods are not

comparable. It is necessary to adjust cash flows for their differences in

timing and risk. Example : If preference rate =10 percent• An investor can invest Rs. 100 if he is offered Rs 110 after

one year.• Rs 110 is the future value of Rs 100 today at 10% interest

rate.• Also, Rs 100 today is the present value of Rs 110 after a

year at 10% interest rate.• If the investor gets less than Rs. 110 then he will not invest. • Anything above Rs. 110 is favourable.

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.

Compounding is the process of finding the future values of cash flows by applying the concept of compound interest.

Compound interest is the interest that is received on the original amount (principal) as well as on any interest earned but not withdrawn during earlier periods.

Simple interest is the interest that is calculated only on the original amount (principal), and thus, no compounding of interest takes place.

The general form of equation for calculating the future value of a lump sum after n periods, may therefore be written as:

FV = PV (1 + i)n

Future value.(1 + i)ⁿ is a future value factor (fvf)To simplify calculations of FV use table of fvf.

Years 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

1 1,01 1,02 1,03 1,04 1,05 1,06 1,07 1,08 1,09 1,1

2 1,02 1,04 1,06 1,08 1,10 1,12 1,14 1,17 1,19 1,21

3 1,03 1,06 1,09 1,12 1,16 1,19 1,23 1,26 1,295 1,33

4 1,04 1,08 1,13 1,17 1,22 1,26 1,31 1,36 1,41 1,46

5 1,05 1,1 1,16 1,22 1,28 1,34 1,40 1,47 1,54 1,61

6 1,06 1,13 1,19 1,27 1,34 1,42 1,50 1,59 1,68 1,77

7 1,07 1,15 1,23 1,32 1,41 1,50 1,61 1,71 1,83 1,94

8 1,08 1,17 1,27 1,37 1,48 1,59 1,72 1,85 1,99 2,14

9 1,09 1,20 1,30 1,42 1,55 1,69 1,84 1,999 2,17 2,36

10 1,1 1,22 1,34 1,48 1,63 1,79 1,97 2,16 2,37 2,59

In Microsoft Excel: Use FV function. FV(rate,nper,pmt,pv,type)Where: rate= interest rate. nper= n periods,pmt= annuity value, pv= present value, Type=1 for beginning of the period and 0 for

end for end of period.

Present value of a future cash flow (inflow or

outflow) is the amount of current cash that is of equivalent value to the decision-maker.

Discounting is the process of determining present value of a series of future cash flows.

The interest rate used for discounting cash flows is also called the discount rate.

How much the investor give up now to get the amount of Re.1 at the end of one year, assuming discount rate is 10% ?

Formula to calculate the present value of a lump sum to be received after some future periods

nn i

FVi

FVPV

1

1

1

Table of present value factorYears 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

1 0,99 0,98 0,97 0,96 0,95 0,94 0,935 0,93 0,92 0,91

2 0,98 0,96 0,94 0,92 0,91 0,89 0,87 0,86 0,84 0,83

3 0,97 0,94 0,92 0,89 0,86 0,84 0,82 0,79 0,77 0,75

4 0,96 0,92 0,89 0,85 0,82 0,79 0,76 0,74 0,71 0,68

5 0,95 0,91 0,87 0,82 0,78 0,75 0,71 0,68 0,65 0,62

6 0,94 0,89 0,84 0,79 0,75 0,70 0,67 0,63 0,596 0,56

7 0,93 0,87 0,81 0,76 0,71 0,67 0,62 0,58 0,55 0,51

8 0,92 0,85 0,79 0,73 0,68 0,63 0,58 0,54 0,50 0,47

9 0,914 0,84 0,77 0,70 0,64 0,59 0,54 0,50 0,46 0,42

10 0,905 0,82 0,74 0,68 0,61 0,56 0,51 0,46 0,42 0,39

Ordinary AnnuityOrdinary Annuity: Payments or receipts occur at the end of each period.

Annuity DueAnnuity Due: Payments or receipts occur at the beginning of each period.

An AnnuityAn Annuity represents a series of equal payments (or receipts) occurring over a specified number of equidistant periods.

Student Loan Payments Car Loan Payments Insurance Premiums Mortgage Payments Retirement Savings

0 1 2 3

$100 $100 $100

(Ordinary Annuity)EndEnd ofPeriod 1

EndEnd ofPeriod 2

Today EqualEqual Cash Flows Each 1 Period Apart

EndEnd ofPeriod 3

0 1 2 3

$100 $100 $100

(Annuity Due)BeginningBeginning ofPeriod 1

BeginningBeginning ofPeriod 2

Today EqualEqual Cash Flows Each 1 Period Apart

BeginningBeginning ofPeriod 3

FVAFVAnn = A(1+i)n-1 + A(1+i)n-2 + ... + A(1+i)1 + A(1+i)0

A A A

0 1 2 n n n+1

A = Periodic Cash Flow

Cash flows occur at the end of the period

i%

Overview of an Ordinary Annuity

The future compound value of an ordinary annuity is as follows:

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

The term within the curly brackets {} is the compound value factor for an annuity of Re.1, and A is the annuity.

Suppose you have decided to deposit Rs.30,000 per year in your Public Provident Fund Account for 15 years. What will be the accumulated amount in your Public Provident Fund Account at the end of 15 years if the interest rate is 8 percent ?

FVADFVADnn = A(1+i)n + A(1+i)n-1 + ... + A(1+i)2 +

A(1+i)1 = FVAFVAn n (1+i)

A A A A A

0 1 2 3 n-1n-1 ni% . . .

Cash flows occur at the beginning of the period

The future compound value of an annuity due is as follows:

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

Suppose you have decided to deposit Rs.30,000 in an investment for 15 years at the beginning of each year. What will be the accumulated amount at the end of 15 years if the interest rate is 8 percent ?

Hint: On the time line, each payment would be shifted to the left one year, so each payment would be compounded for one extra year.

PVAPVAnn = A/(1+i)1 + A/(1+i)2 + ... + A/(1+i)n

A A A

0 1 2 nn n+1

PVAPVAnn

A= Periodic Cash Flow

i% . . .

Cash flows occur at the end of the period

The present value of an annuity hence will be PV = A {[1 - 1/(1+r)^n]/r}

The term within the curly brackets {} is the Present value factor for an annuity of Re.1, and A is the annuity.

Suppose you have decided to deposit Rs.30,000 per year in your Public Provident Fund Account for 15 years. What will be the present value of your Public Provident Fund Account if the interest rate is 8 percent ?

PVADPVADnn = A/(1+i)0 + A/(1+i)1 + ... + A/(1+i)n-1 = PVAPVAn n (1+i)

A A A A

0 1 2 n-1n-1 n

PVADPVADnn

A: Periodic Cash Flow

i% . . .

Cash flows occur at the beginning of the period

The present value of an annuity due hence will be

PV = A {[1 - 1/(1+r)^n]/r} (1+r)

Suppose you have decided to deposit Rs.30,000 in an investment for 15 years at the beginning of each year. What will be the present value of your investment if the interest rate is 8 percent ?

How much should you save annually

You want to buy a house after 5 years when it is expected to cost Rs.2 lacs. How much should you save annually if your savings earn a compound return of 12 percent ?

Finding the interest rate

A finance company advertises that it will pay a lump sum of Rs.10,00,000 at the end of 6 years to investors who deposit annually Rs.1,00,000 for 6 years. What interest rate is implicit in this offer?

Suppose you can lease a computer from its manufacturer for Rs. 1000 per month. The lease runs for 36 months, with payments due at the end of the month. As an alternative, you can buy it for Rs. 20000. In either case, at the end of the 36 months the computer will be worth zero.

If you save Rs. 2,00,000 per year at an interest rate of 8 percent, how long will it take for you to accumulate Rs. 25,00,000?

If you earn an interest rate of 8 percent, how much must you save for each of the next 20 years to accumulate the Rs. 25,00,000?

Nirmal had invested Rs. 12000 per year for last three years and has extended his investment for another 15 years but this time he has chosen a less risk fund which is expected to give 12% per annum, earlier investment gave the return of 18% per annum. What will be the amount at the end of 15 years from now?

Important equation to remember

PV(1+i)^n + PMT {[(1+i)^n - 1]/ i} + FV = 0

Perpetuity is an annuity that occurs indefinitely. In perpetuity, time period, n, is so large (mathematically n approaches infinity) that the expression (1+r)^n in the present value equation tends to become zero, and the formula for a perpetuity simply condenses into:

PV = A/rwhere A is the annuity amount occurring

indefinitely and r is the interest rate.

Present value of a growing annuityA cash flow that grows at a constant rate for a specified period of time is a growing annuity. The time line of a growing annuity is shown below:

A A(1 + g) A(1 + g)n-1

0 1 2 3 n

The present value of a growing annuity can be determined using the following formula :

(1 + g)n

(1 + r)n

PV of a Growing Annuity = A

r – g

The above formula can be used when the growth rate is less than the discount rate (g < r) as well as when the growth rate is more than the discount rate (g > r). However, it does not work when the growth rate is equal to the discount rate (g = r) – in this case, the present value is simply equal to n A/(1+i).

1 –

Present value of a growing annuity

For example, suppose you have the right to harvest a teak plantation for the next 20 years over which you expect to get 100,000 cubic feet of teak per year. The current price per cubic foot of teak is Rs 500, but it is expected to increase at a rate of 8 percent per year. The discount rate is 15 percent. The present value of the teak that you can harvest from the teak forest can be determined as follows:

1.0820

1 – 1.1520

PV of teak = Rs 500 x 100,000 (1.08) 0.15 – 0.08

= Rs.551,736,683

An annuity of Rs.100 is expected to grow at a rate of 2% every year. Assuming the interest rate as 10% per annum

o Calculate the present value for this growing annuity for a 5 year duration

o Calculate the future value for this growing annuity for a 5 year duration

PV = A/ (r-g)

The PV of an uneven cash flow stream is found as the sum of the PVs of the individual cash flows of the stream.

Present value of an uneven series A1 A2 An

PVn = + + …… + (1 + r) (1 + r)2 (1 + r)n

n = At t =1 (1 + r)t

Year Cash Flow PVIF12%,n Present Value of Rs. Individual Cash Flow

1 1,000 0.893 893 2 2,000 0.797 1,594 3 2,000 0.712 1,424 4 3,000 0.636 1,908 5 3,000 0.567 1,701 6 4,000 0.507 2,028 7 4,000 0.452 1,808 8 5,000 0.404 2,020

Present Value of the Cash Flow Stream 13,376

Future value of an uneven series

The future value of an uneven cash flow stream (sometimes called the terminal value) is found by compounding each payment to the end of the stream and then summing the future values

FVn = A1 (1 + r) + A2 (1 + r)2 +……+ An (1 + r)n-1

n = At (1 + r)n-t

t =1

Shorter compounding period

Future value = Present value 1+ r mxn

m

Where r = nominal annual interest rate

m = number of times compounding is done in a year

n = number of years over which compounding is done

Example : Rs.5000, 12 percent, 4 times a year, 6 years

5000(1+ 0.12/4)4x6 = 5000 (1.03)24

= Rs.10,164

EFFECTIVE VERSUS NOMINAL RATE

r = (1+k/m)m –1

r = effective rate of interest

k = nominal rate of interest

m = frequency of compounding per year

Example : k = 8 percent, m=4

r = (1+.08/4)4 – 1 = 0.0824

= 8.24 percent Nominal and Effective Rates of InterestNominal and Effective Rates of Interest

Effective Rate %

  Nominal Annual Semi-annual Quarterly Monthly

Rate % Compounding Compounding Compounding Compounding

8 8.00 8.16 8.24 8.30

12 12.00 12.36 12.55 12.68