06-22-2015, 07:25 PM

Time Value of Money

When cash flows take place at different time intervals, their value is different for a no. of reasons such as interest factor, inflation,

uncertainty ettz. In order to ascertain their value, these flows are required to be compared. The comparison is possible by using a time

line that shows the value and timing of cash flows.

Terms associated with cash flow:

When cash flows take place at different time intervals, their value is different for a no. of reasons such as interest factor, inflation,

uncertainty ettz. In order to ascertain their value, these flows are required to be compared. The comparison is possible by using a time

line that shows the value and timing of cash flows.

Terms associated with cash flow:

Positive cash flow : When a firm receives cash which is called inflow, it is a positive cash flow. Negative cash flow: When a firm makes

payments, this is outflow of cash and is called negative cash flow.Future cash flow & present cash flow : Future cash flows are worth less than the present cash flow. Rs.ioo available with a person presently and Rs..100 to be received after a year, carry higher present value. It is becausethe people prefer present consumption rather than in future and to put off their consumption for future, more value requires to be offered.Similarly due to inflation, there is erosion in money value.Further there is uncertainty about receipt of money in

future.Discounting: It is a process by which the future cash inflows are adjusted/discounted according the their present value. (say

Rs.ioo received at the end of a year, if discounted at say io%, will give a value around Rs.9o). For this purpose, the rate that used is

called discount rate.

Compounding : It is a process by which the present value is converted into future value by using a given rate of interest.

Trade-off: When present consumption is put-off, for some future time period in return for some monetary return such as interest, this is

called exchange or Trade Off.

Effect of Inflation and Risk on the discount rate.

The inflation reduces the purchasing power of future cash flows and it reduces the value of future cash flows. Hence, if the inflation is higher, the discount rate will be higher to calculate the present value.Risk is due to uncertainty. If risk is higher, the discount rate would be higher.

Importance of discounting of the cash flows.The comparison becomes easier for cash inflows of different periods.The user becomes

indifferent for present and future cash inflows and can take more rationale decisions.

Other kinds of cash flows: The cash flows can be simple cash flows, annuities, growing annuities, perpetuities and growing perpetuities.

Simple cash flow: It is single cash flow to occur at a specified future time. Say Rs.i000 to be received at the end of one year. Its

present value can be worked out by discounting back by using an appropriate discount rate.

CALCULATION OF PRESENT AND FUTURE VALUES OF CASH FLOWS Calculation of present value of a future (simple) cash

flow: The present value can be calculated by using the formula

PV = Discount Factor x CI OR PV = 1 / (1+r) x C1 (where CI is the expected cash flow at the end of period .1)

Example : A person wants to get Rs.10 lac after one year. With prevailing interest rate of 9%, how much he should invest.

= 10,00,000 / 1.09 = 917531.19

Calculation of net present value

It can be calculatdd as NPV= PV of future cash flow -required investment OR it can be calculated as

= [(Ci / (t+r) + (Ci / (i+r) + (Ci / (i+r) + (Cn (i+r)] - Co Co = Cash flow at time o i.e. today.

r' means rate of discount.

Ci = Present value of cash flow at end of period one.

You proposes to invest Rs.8 lac in a house and expect that it will fetch Rs.io lac at the end of the year. At 9% interest rate,

what is the net present value. Use the data in the above problem.

NPV= PV-required investment

= 917531.19 - 800000 = 117531.19

Example-2 : X paid Rs.l00000 to Y on Jan 01, 2009. Y returned Rs.65000 after one year, Its.g0000 at end of 2nd year.

What is the present value of amount paid by Y to X and what is net present value of the cash flows for X at 10% discount rate ? Present value of amount paid by X = 100000

Present value of amount paid by after one year = 6500o / 1.1 = 59090 Present value of amount paid by Y after 2 years = 90000 / 1.1 x 1.1 = 74380 Present value of total amount paid by Y = 59090 + 74380 = 133470

Net present value of cash flows = 133470 - 100000 = Rs. 33470

Effect of Inflation and Risk on the present value.

The inflation reduces the purchasing power of future cash flows and it reduces the present value of future cash flows.

Risk is due to uncertainty. If risk is higher, the discount rate would be higher and present value of future cash flows would be lower.

Rule to take decision on making an investment:

If net present value at an acceptable discount rate is zero or positive, the investment can be made.

If internal rate of return (i.e. acceptable discount rate at which net present value is zero) is greater than the opportunity cost of capital, the investment can be made.

Ibbotson and Sinquefield's Study:

Ibbotson and Sinquefield conducted a study of returns on investment in bonds and stock between the period 1926 - 1992. It was found

that average return on stocks was 12.4%, on treasury bonds 5.2%, on treasury bills 3.6%. If investment is made in these 3 instruments at above rates of return, with a time horizon of 40 years, the future value of investment at 12.4% in stock will be 12 times more than

investment in treasury bonds at 5.2%

at 12.4% in stock will be 25 times more than investment in treasury bills at 3.6% This snows that if the time horizon is longer, the gap between the return will be greater.

Conversion of nominal cash flows of future period, to real cash flows. Real cash flow = (Nominal cash flow) / (1+ inflation rate) If the inflation rate is 10% per annum and an investor expects Rs.io lac, what will be his real cash flow. Real cash flow = (Nominal cash flow) / (1+ inflation rate) 10,00,000 / 1+10% = 10 lac / 1.10 = Rs. 909091

Calculation of future value of a present (simple) cash flow: This can be done by way of compounding by using the formula

CF o (1+0 where (1 = at the end of period 0 (CF o = present cash flow) and ( r = rate of discount)

Example : X deposited Rs.i0000 with bank at 10% rate of interest for 2 years. What is the future value (maturity value) of this amount: =

10000 (1+10)2 = 10000 x 1.10 x 1.10 = 12100

Rule of 72

By using this method, we can find out as to, in how much period (appx), an amount will become double at a particular interest rate. If 72 is divided by the interest rate, the resultant product, is the time period during which the amount will be doubled. For example if the

interest rate is 8%, the amount will double in 9 years.

Concept of effective rate of interest

It is the actual rate of interest that takes into account the compounding effect of interim interest payment, if any. At more frequent

compounding, the effective rate of interest would be more.

Determination of effective rate of interest It can be done by using the following formula:

(1+ given annual RoI /N )n - 1 (where N = no. of compounding periods. Say42= months).

Where the annual interest rate is 20%, what will be effective rate, where the compounding is on half-yearly (semi-annual) basis.

(1+ 'given annual Rol /N )n - 1

(1.10 2 - 1) = 1.21 — 1.00 = 21%

Effective rates for various compounding frequencies for 10% RoI:

Where compounding frequency is annual Effective Rol = 0.10 = 10% Where the compounding frequency is semi annual: Effective Rol = (1 + 10/2) 2 - 1) = 10.25% Where the compounding frequency is monthly : Effective RoI = (1 + 10/12) 12 -1) = 10.47% Where the

compounding frequency is daily : Effective Rol = (i + 10/365)3°5 - 1) = 10.5156% Where the compounding frequency is on continuous

basis: Effective Rol = e,1° - 1) = 10.5171% Continuous compounding & calculation of effectiveinterest rate :It can be computed as = exp r - 1 (where 'exp' means exponential number and 'r' means given interest rate).

Impact of frequency of compounding on effective rate of interest:

Effective rate increases and present value of future cash flows decreases if compounding is more frequent. ANNUITYIt is a constant cash

flow, accruing at regular intervals of time, for a pre-fixed period.It can occur at the beginning of each period (called annuity due) or at the

end of each period (called ordinary annuity)

Present value of amend-of-the period annuity (Ordinary ann ty):

= PV (A,r,n) = A fi-(1/ (1+r) n) r

(Alternative formulae : PV = A / r [{(1+r)n -1} / (1+r)n j

(A = annuity r=discount rate n=no. of years.

Example-1: Calculate the PV of Rs.9000 each year for 5 year/5 where R41$12%.

= PV (A,r,n) = A {1-(1/ (i+r) n r \ C) ‘'k

= 9000 {141/ (1+12)s) / 0.12 = 32442.98

Example-2 : X deposits Rs.i000 at end of each year for 4 y ars at io rate of Hite t. is the present value of the amount, +.1D be deposited:

= A + A/r [1-1/(i+r) n-i /(i+r) = moo+ 1000 / 0.1 [1-1/(1+0.10)4-1] / (Li) = 1000+ moo / 0.1 [1-1/(1.331)] (1.1) = woo+ moo / 0.1 [1-0.7513)] (1.1) = 1000+ 10000 [0.2487] (1.1) = woo + 2487 = 3487 (Li) = 317

Future value of an end-of-the period annuity (Ordinary annuity):

= FV (A,r,n) = A {(i+r)n -1} r

(Alternative formulae : FV = A / r {(1+1)n --1} (A = annuity r=discount rate n=uo. of years.

Example - Z deposits Rs.i0000 annually for next 5 years. At 10% rate of interest, what will he be getting?

= A [ (1 +r) n - 1] / r = 10000 [ (1 +0.10) 5 / 0.10]

= 10000 [1.61051-1/ 0.10] = 10000 [0.61051 / 0.10] = Rs.61051

P r e s e n t v a l u e o f b e g i n n i n g - o f - a - p e r i o d a n n u i t y ( A n n u i t y d u e ) o v e r n y e a r s : = A + A / r { [ 1 - 1 ( i + r ) n -

i ] (A = c a s h f l o w p e r p e r i o d o r a n n u i t y , r = r a t e o f i n t e r e s t , n = n o . o f p a yme n t s )

Example : X deposits Rs.i000 at beginning of each year for 4 years at 10% rate of interest. What is the present value of the amount, to be

deposited:

= A + A/r [1-1/(t+r) n-1] = 1000+ 1000 / 0.1 [1-1/(1+0.10)4-1]

= 1000+ 1000 / 0.1 [1-1/(1.331) = 1000+ 1000 / 0.1 [1-0.7513)]

= 1000+ 10000 [0.2487] = 1000 + 2487 = 3487

Future value of beginning-of-a-period annuity (Annuity due):

= C {[(i+r) n -1] r } x (i+r) OR A / r [(i+r) {(1+r)n-1)11

Example - Z wants to deposit Rs.10000 in a recurring deposit account for 3 years in the

beginning of the year. At io% p.a., how much will he be getting? =A/r{(1+r)[(i+r)n-i]}

10000/0.10 +0.0 [(1 +0.1) 3 - = 10000 / 0.10 [(1 +0.1) (1.331) - 1]

= 10000 / 0.10 [1.1 x .331] = 10000 / 0.10 x 0.3641 = Rs.36410

C a l c u l a t i o n o f A n n u i t y g i v e n f u t u r e v a l u e : = A ( F V , r , n ) = F V { r / ( i + r ) n - 1 } Z w a n t s t o

r e c e i v e R s . 6 1 0 5 1 a n d w a n t s t o d e p o s i t a n e q u a l a m o u n t f o r 5 y e a r s a t 1 0 % p . a . . H o w

much should deposit regularly?

= 61051{0.10 / (i+0.10)5- 1}

= 61051{0.10 / (1.61051)- 1}

Compiled by Mr. Sanjay Kumar Trivedy, Sr. Mgr., RSTC, mumbai

26

= 61051{0.10 / 0.61051 = Rs.i0000

Baloon repayment loan

it is a loan in which repayment of interest only is made during life period of the loan. The amount o loan is repaid in one instalment

at the end of repayment period, such as debenturesand bond's. The companies raising bonds or debentures set aside funds in a

sinking fundsmaturing with maturity period of the loan, so that do not face any problem at the time ofrepayment.

Sinking fund -It is fund created by companies for meeting special repayment obligations in which regularcontributions are made on

annual or other basis.

Growing annuity- It a cash flow which grows at a constant rate for a specified period of time.If A is the current cash flow and g is the

expected growth rate, the value can be calculated asunder:

Present value of a growing annuity : =A [(i+g) / (r — g)] {(1-(1+g) n / (i+r) " ) (g stands for expected growth rate)The present value

can be estimated in cases except when the growth rate is equal to discountrate, the present value is equal to the nominal sums of

annuities over the period, without the

growth effect.

Present value of a growing annuity for n years be calculated when r = g It is = nA

PERPETUITY

It is a constant cash flow paid or received at regular time intervals forever, such as a life time pension or rentals received from use of land.

Calculation of present value of perpetuity : It can be calculated by use of the formula : A / rWhen the coupon rate is equal to interest

rate, the value will be equal to face value.

Growing perpetuity- It is a cash flow which forever, is expected to grow at a constant rate. The present value of a growing perpetuity can

be written as.. Ci / (r-g)

Console Bond- It is a bond which does not have maturity. On such bond fixed coupon or RoI is paid.An investor has a console bond of

Rs.i0000 with 8% fixed coupon. If the interest rate is 10%,the current value of the bond would be: When a firm receives cash which is called inflow, it is a positive cash flow. Negative cash flow: When a firm makes

payments, this is outflow of cash and is called negative cash flow.Future cash flow & present cash flow : Future cash flows are worth less than

the present cash flow. Rs.ioo available with a person presently and Rs..100 to be received after a year, carry higher present value. It is

becausethe people prefer present consumption rather than in future and to put off their consumption for future, more value requires to be

offered.Similarly due to inflation, there is erosion in money value.Further there is uncertainty about receipt of money in

future.Discounting: It is a process by which the future cash inflows are adjusted/discounted according the their present value. (say

Rs.ioo received at the end of a year, if discounted at say io%, will give a value around Rs.9o). For this purpose, the rate that used is

called discount rate.

Compounding : It is a process by which the present value is converted into future value by using a given rate of interest.

Trade-off: When present consumption is put-off, for some future time period in return for some monetary return such as interest, this is

called exchange or Trade Off.

Effect of Inflation and Risk on the discount rate.

The inflation reduces the purchasing power of future cash flows and it reduces the value of future cash flows. Hence, if the inflation is higher,

the discount rate will be higher to calculate the present value.Risk is due to uncertainty. If risk is higher, the discount rate would be higher.

Importance of discounting of the cash flows.The comparison becomes easier for cash inflows of different periods.The user becomes

indifferent for present and future cash inflows and can take more rationale decisions.

Other kinds of cash flows: The cash flows can be simple cash flows, annuities, growing annuities, perpetuities and growing perpetuities.

Simple cash flow: It is single cash flow to occur at a specified future time. Say Rs.i000 to be received at the end of one year. Its

present value can be worked out by discounting back by using an appropriate discount rate.

CALCULATION OF PRESENT AND FUTURE VALUES OF CASH FLOWS Calculation of present value of a future (simple) cash

flow: The present value can be calculated by using the formula

PV = Discount Factor x CI OR PV = 1 / (1+r) x C1 (where CI is the expected cash flow at the end of period .1)

Example : A person wants to get Rs.10 lac after one year. With prevailing interest rate of 9%, how much he should invest.

= 10,00,000 / 1.09 = 917531.19

Calculation of net present value

It can be calculatdd as NPV= PV of future cash flow -required investment OR it can be calculated as

= [(Ci / (t+r) + (Ci / (i+r) + (Ci / (i+r) + (Cn (i+r)] - Co Co = Cash flow at time o i.e. today.

r' means rate of discount.

Ci = Present value of cash flow at end of period one.

Example-i : You proposes to invest Rs.8 lac in a house and expect that it will fetch Rs.io lac at the end of the year. At 9% interest rate,

what is the net present value. Use the data in the above problem.

NPV= PV-required investment

= 917531.19 - 800000 = 117531.19

Example-2 : X paid Rs.l00000 to Y on Jan 01, 2009. Y returned Rs.65000 after one year, Its.g0000 at end of 2nd year.

What is the present value of amount paid by Y to X and what is net present value of the cash flows for X at 10% discount rate ? Present value

of amount paid by X = 100000

Present value of amount paid by after one year = 6500o / 1.1 = 59090 Present value of amount paid by Y after 2 years = 90000 / 1.1 x 1.1 =

74380 Present value of total amount paid by Y = 59090 + 74380 = 133470

Net present value of cash flows = 133470 - 100000 = Rs. 33470

Effect of Inflation and Risk on the present value.

The inflation reduces the purchasing power of future cash flows and it reduces the present value of future cash flows.

Risk is due to uncertainty. If risk is higher, the discount rate would be higher and present value of future cash flows would be lower.

Rule to take decision on making an investment:

If net present value at an acceptable discount rate is zero or positive, the investment can be made.

Compiled by Mr. Sanjay Kumar Trivedy, Sr. Mgr., RSTC, mumbai

25

If internal rate of return (i.e. acceptable discount rate at which net present value is zero) is greater than the opportunity cost of capital, the

investment can be made.

Ibbotson and Sinquefield's Study:

Ibbotson and Sinquefield conducted a study of returns on investment in bonds and stock between the period 1926 - 1992. It was found

that average return on stocks was 12.4%, on treasury bonds 5.2%, on treasury bills 3.6%. If investment is made in these 3 instruments at

above rates of return, with a time horizon of 40 years, the future value of investment at 12.4% in stock will be 12 times more than

investment in treasury bonds at 5.2%

at 12.4% in stock will be 25 times more than investment in treasury bills at 3.6% This snows that if the time horizon is longer, the gap between

the return will be greater.

Conversion of nominal cash flows of future period, to real cash flows. Real cash flow = (Nominal cash flow) / (1+ inflation rate) If the inflation

rate is lo% per annum and an investor expects Rs.io lac, what will be his real cash flow. Real cash flow = (Nominal cash flow) / (1+ inflation

rate) io,00,000 / 1+1o% = 10 lac / 1.10 = Rs. 909091

Calculation of future value of a present (simple) cash flow: This can be done by way of compounding by using the formula

CF o (1+0 where (1 = at the end of period 0 (CF o = present cash flow) and ( r = rate of discount)

Example : X deposited Rs.i0000 with bank at 10% rate of interest for 2 years. What is the future value (maturity value) of this amount: =

10000 (1+10)2 = 10000 x 1.10 x 1.10 = 12100

Rule of 72

By using this method, we can find out as to, in how much period (appx), an amount will become double at a particular interest rate. If 72 is

divided by the interest rate, the resultant product, is the time period during which the amount will be doubled. For example if the

interest rate is 8%, the amount will double in 9 years.

Concept of effective rate of interest

It is the actual rate of interest that takes into account the compounding effect of interim interest payment, if any. At more frequent

compounding, the effective rate of interest would be more.

Determination of effective rate of interest It can be done by using the following formula:

(1+ given annual RoI /N )n - 1 (where N = no. of compounding periods. Say42= months).

Where the annual interest rate is 20%, what will be effective rate, where the compounding is on half-yearly (semi-annual) basis.

(1+ 'given annual Rol /N )n - 1

(1.10 2 - 1) = 1.21 — 1.00 = 21%

Effective rates for various compounding frequencies for 10% RoI:

Where compounding frequency is annual Effective Rol = 0.10 = 10% Where the compounding frequency is semi annual: Effective Rol = (1

+ 10/2) 2 - 1) = 10.25% Where the compounding frequency is monthly : Effective RoI = (1 + 10/12) 12 -1) = 10.47% Where the

compounding frequency is daily : Effective Rol = (i + 10/365)3°5 - 1) = 10.5156% Where the compounding frequency is on continuous

basis: Effective Rol = e,1° - 1) = 10.5171% Continuous compounding & calculation of effectiveinterest rate :It can be computed as = exp r

- 1 (where 'exp' means exponential number and 'r' means given interest rate).

Impact of frequency of compounding on effective rate of interest:

Effective rate increases and present value of future cash flows decreases if compounding is more frequent. ANNUITYIt is a constant cash

flow, accruing at regular intervals of time, for a pre-fixed period.It can occur at the beginning of each period (called annuity due) or at the

end of each period (called ordinary annuity)

Present value of amend-of-the period annuity (Ordinary ann ty):

= PV (A,r,n) = A fi-(1/ (1+r) n) r

(Alternative formulae : PV = A / r [{(1+r)n -1} / (1+r)n j

(A = annuity r=discount rate n=no. of years.

Example-1: Calculate the PV of Rs.9000 each year for 5 year/5 where R41$12%.

= PV (A,r,n) = A {1-(1/ (i+r) n r \ C) ‘'k

= 9000 {141/ (1+12)s) / 0.12 = 32442.98

Example-2 : X deposits Rs.i000 at end of each year for 4 y ars at io rate of Hite t. is the present value of the amount, +.1D be deposited:

= A + A/r [1-1/(i+r) n-i /(i+r) = moo+ 1000 / 0.1 [1-1/(1+0.10)4-1] / (Li) = 1000+ moo / 0.1 [1-1/(1.331)] (1.1) = woo+ moo / 0.1 [1-0.7513)]

(1.1) = 1000+ 10000 [0.2487] (1.1) = woo + 2487 = 3487 (Li) = 317

Future value of an end-of-the period annuity (Ordinary annuity):

= FV (A,r,n) = A {(i+r)n -1} r

(Alternative formulae : FV = A / r {(1+1)n --1} (A = annuity r=discount rate n=uo. of years.

Example - Z deposits Rs.i0000 annually for next 5 years. At 10% rate of interest, what will he be getting?

= A [ (1 +r) n - 1] / r = 10000 [ (1 +0.10) 5 / 0.10]

= 10000 [1.61051-1/ 0.10] = 10000 [0.61051 / 0.10] = Rs.61051

P r e s e n t v a l u e o f b e g i n n i n g - o f - a - p e r i o d a n n u i t y ( A n n u i t y d u e ) o v e r n y e a r s : = A + A / r { [ 1 - 1 ( i + r ) n -

i ] (A = c a s h f l o w p e r p e r i o d o r a n n u i t y , r = r a t e o f i n t e r e s t , n = n o . o f p a yme n t s )

Example : X deposits Rs.i000 at beginning of each year for 4 years at 10% rate of interest. What is the present value of the amount, to be

deposited:

= A + A/r [1-1/(t+r) n-1] = 1000+ 1000 / 0.1 [1-1/(1+0.10)4-1]

= 1000+ 1000 / 0.1 [1-1/(1.331) = 1000+ 1000 / 0.1 [1-0.7513)]

= 1000+ 10000 [0.2487] = 1000 + 2487 = 3487

Future value of beginning-of-a-period annuity (Annuity due):

= C {[(i+r) n -1] r } x (i+r) OR A / r [(i+r) {(1+r)n-1)11

Example - Z wants to deposit Rs.10000 in a recurring deposit account for 3 years in the

beginning of the year. At io% p.a., how much will he be getting? =A/r{(1+r)[(i+r)n-i]}

10000/0.10 +0.0 [(1 +0.1) 3 - = 10000 / 0.10 [(1 +0.1) (1.331) - 1]

= 10000 / 0.10 [1.1 x .331] = 10000 / 0.10 x 0.3641 = Rs.36410

When cash flows take place at different time intervals, their value is different for a no. of reasons such as interest factor, inflation,

uncertainty ettz. In order to ascertain their value, these flows are required to be compared. The comparison is possible by using a time

line that shows the value and timing of cash flows.

Terms associated with cash flow:

When cash flows take place at different time intervals, their value is different for a no. of reasons such as interest factor, inflation,

uncertainty ettz. In order to ascertain their value, these flows are required to be compared. The comparison is possible by using a time

line that shows the value and timing of cash flows.

Terms associated with cash flow:

Positive cash flow : When a firm receives cash which is called inflow, it is a positive cash flow. Negative cash flow: When a firm makes

payments, this is outflow of cash and is called negative cash flow.Future cash flow & present cash flow : Future cash flows are worth less than the present cash flow. Rs.ioo available with a person presently and Rs..100 to be received after a year, carry higher present value. It is becausethe people prefer present consumption rather than in future and to put off their consumption for future, more value requires to be offered.Similarly due to inflation, there is erosion in money value.Further there is uncertainty about receipt of money in

future.Discounting: It is a process by which the future cash inflows are adjusted/discounted according the their present value. (say

Rs.ioo received at the end of a year, if discounted at say io%, will give a value around Rs.9o). For this purpose, the rate that used is

called discount rate.

Compounding : It is a process by which the present value is converted into future value by using a given rate of interest.

Trade-off: When present consumption is put-off, for some future time period in return for some monetary return such as interest, this is

called exchange or Trade Off.

Effect of Inflation and Risk on the discount rate.

The inflation reduces the purchasing power of future cash flows and it reduces the value of future cash flows. Hence, if the inflation is higher, the discount rate will be higher to calculate the present value.Risk is due to uncertainty. If risk is higher, the discount rate would be higher.

Importance of discounting of the cash flows.The comparison becomes easier for cash inflows of different periods.The user becomes

indifferent for present and future cash inflows and can take more rationale decisions.

Other kinds of cash flows: The cash flows can be simple cash flows, annuities, growing annuities, perpetuities and growing perpetuities.

Simple cash flow: It is single cash flow to occur at a specified future time. Say Rs.i000 to be received at the end of one year. Its

present value can be worked out by discounting back by using an appropriate discount rate.

CALCULATION OF PRESENT AND FUTURE VALUES OF CASH FLOWS Calculation of present value of a future (simple) cash

flow: The present value can be calculated by using the formula

PV = Discount Factor x CI OR PV = 1 / (1+r) x C1 (where CI is the expected cash flow at the end of period .1)

Example : A person wants to get Rs.10 lac after one year. With prevailing interest rate of 9%, how much he should invest.

= 10,00,000 / 1.09 = 917531.19

Calculation of net present value

It can be calculatdd as NPV= PV of future cash flow -required investment OR it can be calculated as

= [(Ci / (t+r) + (Ci / (i+r) + (Ci / (i+r) + (Cn (i+r)] - Co Co = Cash flow at time o i.e. today.

r' means rate of discount.

Ci = Present value of cash flow at end of period one.

You proposes to invest Rs.8 lac in a house and expect that it will fetch Rs.io lac at the end of the year. At 9% interest rate,

what is the net present value. Use the data in the above problem.

NPV= PV-required investment

= 917531.19 - 800000 = 117531.19

Example-2 : X paid Rs.l00000 to Y on Jan 01, 2009. Y returned Rs.65000 after one year, Its.g0000 at end of 2nd year.

What is the present value of amount paid by Y to X and what is net present value of the cash flows for X at 10% discount rate ? Present value of amount paid by X = 100000

Present value of amount paid by after one year = 6500o / 1.1 = 59090 Present value of amount paid by Y after 2 years = 90000 / 1.1 x 1.1 = 74380 Present value of total amount paid by Y = 59090 + 74380 = 133470

Net present value of cash flows = 133470 - 100000 = Rs. 33470

Effect of Inflation and Risk on the present value.

The inflation reduces the purchasing power of future cash flows and it reduces the present value of future cash flows.

Risk is due to uncertainty. If risk is higher, the discount rate would be higher and present value of future cash flows would be lower.

Rule to take decision on making an investment:

If net present value at an acceptable discount rate is zero or positive, the investment can be made.

If internal rate of return (i.e. acceptable discount rate at which net present value is zero) is greater than the opportunity cost of capital, the investment can be made.

Ibbotson and Sinquefield's Study:

Ibbotson and Sinquefield conducted a study of returns on investment in bonds and stock between the period 1926 - 1992. It was found

that average return on stocks was 12.4%, on treasury bonds 5.2%, on treasury bills 3.6%. If investment is made in these 3 instruments at above rates of return, with a time horizon of 40 years, the future value of investment at 12.4% in stock will be 12 times more than

investment in treasury bonds at 5.2%

at 12.4% in stock will be 25 times more than investment in treasury bills at 3.6% This snows that if the time horizon is longer, the gap between the return will be greater.

Conversion of nominal cash flows of future period, to real cash flows. Real cash flow = (Nominal cash flow) / (1+ inflation rate) If the inflation rate is 10% per annum and an investor expects Rs.io lac, what will be his real cash flow. Real cash flow = (Nominal cash flow) / (1+ inflation rate) 10,00,000 / 1+10% = 10 lac / 1.10 = Rs. 909091

Calculation of future value of a present (simple) cash flow: This can be done by way of compounding by using the formula

CF o (1+0 where (1 = at the end of period 0 (CF o = present cash flow) and ( r = rate of discount)

Example : X deposited Rs.i0000 with bank at 10% rate of interest for 2 years. What is the future value (maturity value) of this amount: =

10000 (1+10)2 = 10000 x 1.10 x 1.10 = 12100

Rule of 72

By using this method, we can find out as to, in how much period (appx), an amount will become double at a particular interest rate. If 72 is divided by the interest rate, the resultant product, is the time period during which the amount will be doubled. For example if the

interest rate is 8%, the amount will double in 9 years.

Concept of effective rate of interest

It is the actual rate of interest that takes into account the compounding effect of interim interest payment, if any. At more frequent

compounding, the effective rate of interest would be more.

Determination of effective rate of interest It can be done by using the following formula:

(1+ given annual RoI /N )n - 1 (where N = no. of compounding periods. Say42= months).

Where the annual interest rate is 20%, what will be effective rate, where the compounding is on half-yearly (semi-annual) basis.

(1+ 'given annual Rol /N )n - 1

(1.10 2 - 1) = 1.21 — 1.00 = 21%

Effective rates for various compounding frequencies for 10% RoI:

Where compounding frequency is annual Effective Rol = 0.10 = 10% Where the compounding frequency is semi annual: Effective Rol = (1 + 10/2) 2 - 1) = 10.25% Where the compounding frequency is monthly : Effective RoI = (1 + 10/12) 12 -1) = 10.47% Where the

compounding frequency is daily : Effective Rol = (i + 10/365)3°5 - 1) = 10.5156% Where the compounding frequency is on continuous

basis: Effective Rol = e,1° - 1) = 10.5171% Continuous compounding & calculation of effectiveinterest rate :It can be computed as = exp r - 1 (where 'exp' means exponential number and 'r' means given interest rate).

Impact of frequency of compounding on effective rate of interest:

Effective rate increases and present value of future cash flows decreases if compounding is more frequent. ANNUITYIt is a constant cash

flow, accruing at regular intervals of time, for a pre-fixed period.It can occur at the beginning of each period (called annuity due) or at the

end of each period (called ordinary annuity)

Present value of amend-of-the period annuity (Ordinary ann ty):

= PV (A,r,n) = A fi-(1/ (1+r) n) r

(Alternative formulae : PV = A / r [{(1+r)n -1} / (1+r)n j

(A = annuity r=discount rate n=no. of years.

Example-1: Calculate the PV of Rs.9000 each year for 5 year/5 where R41$12%.

= PV (A,r,n) = A {1-(1/ (i+r) n r \ C) ‘'k

= 9000 {141/ (1+12)s) / 0.12 = 32442.98

Example-2 : X deposits Rs.i000 at end of each year for 4 y ars at io rate of Hite t. is the present value of the amount, +.1D be deposited:

= A + A/r [1-1/(i+r) n-i /(i+r) = moo+ 1000 / 0.1 [1-1/(1+0.10)4-1] / (Li) = 1000+ moo / 0.1 [1-1/(1.331)] (1.1) = woo+ moo / 0.1 [1-0.7513)] (1.1) = 1000+ 10000 [0.2487] (1.1) = woo + 2487 = 3487 (Li) = 317

Future value of an end-of-the period annuity (Ordinary annuity):

= FV (A,r,n) = A {(i+r)n -1} r

(Alternative formulae : FV = A / r {(1+1)n --1} (A = annuity r=discount rate n=uo. of years.

Example - Z deposits Rs.i0000 annually for next 5 years. At 10% rate of interest, what will he be getting?

= A [ (1 +r) n - 1] / r = 10000 [ (1 +0.10) 5 / 0.10]

= 10000 [1.61051-1/ 0.10] = 10000 [0.61051 / 0.10] = Rs.61051

P r e s e n t v a l u e o f b e g i n n i n g - o f - a - p e r i o d a n n u i t y ( A n n u i t y d u e ) o v e r n y e a r s : = A + A / r { [ 1 - 1 ( i + r ) n -

i ] (A = c a s h f l o w p e r p e r i o d o r a n n u i t y , r = r a t e o f i n t e r e s t , n = n o . o f p a yme n t s )

Example : X deposits Rs.i000 at beginning of each year for 4 years at 10% rate of interest. What is the present value of the amount, to be

deposited:

= A + A/r [1-1/(t+r) n-1] = 1000+ 1000 / 0.1 [1-1/(1+0.10)4-1]

= 1000+ 1000 / 0.1 [1-1/(1.331) = 1000+ 1000 / 0.1 [1-0.7513)]

= 1000+ 10000 [0.2487] = 1000 + 2487 = 3487

Future value of beginning-of-a-period annuity (Annuity due):

= C {[(i+r) n -1] r } x (i+r) OR A / r [(i+r) {(1+r)n-1)11

Example - Z wants to deposit Rs.10000 in a recurring deposit account for 3 years in the

beginning of the year. At io% p.a., how much will he be getting? =A/r{(1+r)[(i+r)n-i]}

10000/0.10 +0.0 [(1 +0.1) 3 - = 10000 / 0.10 [(1 +0.1) (1.331) - 1]

= 10000 / 0.10 [1.1 x .331] = 10000 / 0.10 x 0.3641 = Rs.36410

C a l c u l a t i o n o f A n n u i t y g i v e n f u t u r e v a l u e : = A ( F V , r , n ) = F V { r / ( i + r ) n - 1 } Z w a n t s t o

r e c e i v e R s . 6 1 0 5 1 a n d w a n t s t o d e p o s i t a n e q u a l a m o u n t f o r 5 y e a r s a t 1 0 % p . a . . H o w

much should deposit regularly?

= 61051{0.10 / (i+0.10)5- 1}

= 61051{0.10 / (1.61051)- 1}

Compiled by Mr. Sanjay Kumar Trivedy, Sr. Mgr., RSTC, mumbai

26

= 61051{0.10 / 0.61051 = Rs.i0000

Baloon repayment loan

it is a loan in which repayment of interest only is made during life period of the loan. The amount o loan is repaid in one instalment

at the end of repayment period, such as debenturesand bond's. The companies raising bonds or debentures set aside funds in a

sinking fundsmaturing with maturity period of the loan, so that do not face any problem at the time ofrepayment.

Sinking fund -It is fund created by companies for meeting special repayment obligations in which regularcontributions are made on

annual or other basis.

Growing annuity- It a cash flow which grows at a constant rate for a specified period of time.If A is the current cash flow and g is the

expected growth rate, the value can be calculated asunder:

Present value of a growing annuity : =A [(i+g) / (r — g)] {(1-(1+g) n / (i+r) " ) (g stands for expected growth rate)The present value

can be estimated in cases except when the growth rate is equal to discountrate, the present value is equal to the nominal sums of

annuities over the period, without the

growth effect.

Present value of a growing annuity for n years be calculated when r = g It is = nA

PERPETUITY

It is a constant cash flow paid or received at regular time intervals forever, such as a life time pension or rentals received from use of land.

Calculation of present value of perpetuity : It can be calculated by use of the formula : A / rWhen the coupon rate is equal to interest

rate, the value will be equal to face value.

Growing perpetuity- It is a cash flow which forever, is expected to grow at a constant rate. The present value of a growing perpetuity can

be written as.. Ci / (r-g)

Console Bond- It is a bond which does not have maturity. On such bond fixed coupon or RoI is paid.An investor has a console bond of

Rs.i0000 with 8% fixed coupon. If the interest rate is 10%,the current value of the bond would be: When a firm receives cash which is called inflow, it is a positive cash flow. Negative cash flow: When a firm makes

payments, this is outflow of cash and is called negative cash flow.Future cash flow & present cash flow : Future cash flows are worth less than

the present cash flow. Rs.ioo available with a person presently and Rs..100 to be received after a year, carry higher present value. It is

becausethe people prefer present consumption rather than in future and to put off their consumption for future, more value requires to be

offered.Similarly due to inflation, there is erosion in money value.Further there is uncertainty about receipt of money in

future.Discounting: It is a process by which the future cash inflows are adjusted/discounted according the their present value. (say

Rs.ioo received at the end of a year, if discounted at say io%, will give a value around Rs.9o). For this purpose, the rate that used is

called discount rate.

Compounding : It is a process by which the present value is converted into future value by using a given rate of interest.

Trade-off: When present consumption is put-off, for some future time period in return for some monetary return such as interest, this is

called exchange or Trade Off.

Effect of Inflation and Risk on the discount rate.

The inflation reduces the purchasing power of future cash flows and it reduces the value of future cash flows. Hence, if the inflation is higher,

the discount rate will be higher to calculate the present value.Risk is due to uncertainty. If risk is higher, the discount rate would be higher.

Importance of discounting of the cash flows.The comparison becomes easier for cash inflows of different periods.The user becomes

indifferent for present and future cash inflows and can take more rationale decisions.

Other kinds of cash flows: The cash flows can be simple cash flows, annuities, growing annuities, perpetuities and growing perpetuities.

Simple cash flow: It is single cash flow to occur at a specified future time. Say Rs.i000 to be received at the end of one year. Its

present value can be worked out by discounting back by using an appropriate discount rate.

CALCULATION OF PRESENT AND FUTURE VALUES OF CASH FLOWS Calculation of present value of a future (simple) cash

flow: The present value can be calculated by using the formula

PV = Discount Factor x CI OR PV = 1 / (1+r) x C1 (where CI is the expected cash flow at the end of period .1)

Example : A person wants to get Rs.10 lac after one year. With prevailing interest rate of 9%, how much he should invest.

= 10,00,000 / 1.09 = 917531.19

Calculation of net present value

It can be calculatdd as NPV= PV of future cash flow -required investment OR it can be calculated as

= [(Ci / (t+r) + (Ci / (i+r) + (Ci / (i+r) + (Cn (i+r)] - Co Co = Cash flow at time o i.e. today.

r' means rate of discount.

Ci = Present value of cash flow at end of period one.

Example-i : You proposes to invest Rs.8 lac in a house and expect that it will fetch Rs.io lac at the end of the year. At 9% interest rate,

what is the net present value. Use the data in the above problem.

NPV= PV-required investment

= 917531.19 - 800000 = 117531.19

Example-2 : X paid Rs.l00000 to Y on Jan 01, 2009. Y returned Rs.65000 after one year, Its.g0000 at end of 2nd year.

What is the present value of amount paid by Y to X and what is net present value of the cash flows for X at 10% discount rate ? Present value

of amount paid by X = 100000

Present value of amount paid by after one year = 6500o / 1.1 = 59090 Present value of amount paid by Y after 2 years = 90000 / 1.1 x 1.1 =

74380 Present value of total amount paid by Y = 59090 + 74380 = 133470

Net present value of cash flows = 133470 - 100000 = Rs. 33470

Effect of Inflation and Risk on the present value.

The inflation reduces the purchasing power of future cash flows and it reduces the present value of future cash flows.

Risk is due to uncertainty. If risk is higher, the discount rate would be higher and present value of future cash flows would be lower.

Rule to take decision on making an investment:

If net present value at an acceptable discount rate is zero or positive, the investment can be made.

Compiled by Mr. Sanjay Kumar Trivedy, Sr. Mgr., RSTC, mumbai

25

If internal rate of return (i.e. acceptable discount rate at which net present value is zero) is greater than the opportunity cost of capital, the

investment can be made.

Ibbotson and Sinquefield's Study:

Ibbotson and Sinquefield conducted a study of returns on investment in bonds and stock between the period 1926 - 1992. It was found

that average return on stocks was 12.4%, on treasury bonds 5.2%, on treasury bills 3.6%. If investment is made in these 3 instruments at

above rates of return, with a time horizon of 40 years, the future value of investment at 12.4% in stock will be 12 times more than

investment in treasury bonds at 5.2%

at 12.4% in stock will be 25 times more than investment in treasury bills at 3.6% This snows that if the time horizon is longer, the gap between

the return will be greater.

Conversion of nominal cash flows of future period, to real cash flows. Real cash flow = (Nominal cash flow) / (1+ inflation rate) If the inflation

rate is lo% per annum and an investor expects Rs.io lac, what will be his real cash flow. Real cash flow = (Nominal cash flow) / (1+ inflation

rate) io,00,000 / 1+1o% = 10 lac / 1.10 = Rs. 909091

Calculation of future value of a present (simple) cash flow: This can be done by way of compounding by using the formula

CF o (1+0 where (1 = at the end of period 0 (CF o = present cash flow) and ( r = rate of discount)

Example : X deposited Rs.i0000 with bank at 10% rate of interest for 2 years. What is the future value (maturity value) of this amount: =

10000 (1+10)2 = 10000 x 1.10 x 1.10 = 12100

Rule of 72

By using this method, we can find out as to, in how much period (appx), an amount will become double at a particular interest rate. If 72 is

divided by the interest rate, the resultant product, is the time period during which the amount will be doubled. For example if the

interest rate is 8%, the amount will double in 9 years.

Concept of effective rate of interest

It is the actual rate of interest that takes into account the compounding effect of interim interest payment, if any. At more frequent

compounding, the effective rate of interest would be more.

Determination of effective rate of interest It can be done by using the following formula:

(1+ given annual RoI /N )n - 1 (where N = no. of compounding periods. Say42= months).

Where the annual interest rate is 20%, what will be effective rate, where the compounding is on half-yearly (semi-annual) basis.

(1+ 'given annual Rol /N )n - 1

(1.10 2 - 1) = 1.21 — 1.00 = 21%

Effective rates for various compounding frequencies for 10% RoI:

Where compounding frequency is annual Effective Rol = 0.10 = 10% Where the compounding frequency is semi annual: Effective Rol = (1

+ 10/2) 2 - 1) = 10.25% Where the compounding frequency is monthly : Effective RoI = (1 + 10/12) 12 -1) = 10.47% Where the

compounding frequency is daily : Effective Rol = (i + 10/365)3°5 - 1) = 10.5156% Where the compounding frequency is on continuous

basis: Effective Rol = e,1° - 1) = 10.5171% Continuous compounding & calculation of effectiveinterest rate :It can be computed as = exp r

- 1 (where 'exp' means exponential number and 'r' means given interest rate).

Impact of frequency of compounding on effective rate of interest:

Effective rate increases and present value of future cash flows decreases if compounding is more frequent. ANNUITYIt is a constant cash

flow, accruing at regular intervals of time, for a pre-fixed period.It can occur at the beginning of each period (called annuity due) or at the

end of each period (called ordinary annuity)

Present value of amend-of-the period annuity (Ordinary ann ty):

= PV (A,r,n) = A fi-(1/ (1+r) n) r

(Alternative formulae : PV = A / r [{(1+r)n -1} / (1+r)n j

(A = annuity r=discount rate n=no. of years.

Example-1: Calculate the PV of Rs.9000 each year for 5 year/5 where R41$12%.

= PV (A,r,n) = A {1-(1/ (i+r) n r \ C) ‘'k

= 9000 {141/ (1+12)s) / 0.12 = 32442.98

Example-2 : X deposits Rs.i000 at end of each year for 4 y ars at io rate of Hite t. is the present value of the amount, +.1D be deposited:

= A + A/r [1-1/(i+r) n-i /(i+r) = moo+ 1000 / 0.1 [1-1/(1+0.10)4-1] / (Li) = 1000+ moo / 0.1 [1-1/(1.331)] (1.1) = woo+ moo / 0.1 [1-0.7513)]

(1.1) = 1000+ 10000 [0.2487] (1.1) = woo + 2487 = 3487 (Li) = 317

Future value of an end-of-the period annuity (Ordinary annuity):

= FV (A,r,n) = A {(i+r)n -1} r

(Alternative formulae : FV = A / r {(1+1)n --1} (A = annuity r=discount rate n=uo. of years.

Example - Z deposits Rs.i0000 annually for next 5 years. At 10% rate of interest, what will he be getting?

= A [ (1 +r) n - 1] / r = 10000 [ (1 +0.10) 5 / 0.10]

= 10000 [1.61051-1/ 0.10] = 10000 [0.61051 / 0.10] = Rs.61051

P r e s e n t v a l u e o f b e g i n n i n g - o f - a - p e r i o d a n n u i t y ( A n n u i t y d u e ) o v e r n y e a r s : = A + A / r { [ 1 - 1 ( i + r ) n -

i ] (A = c a s h f l o w p e r p e r i o d o r a n n u i t y , r = r a t e o f i n t e r e s t , n = n o . o f p a yme n t s )

Example : X deposits Rs.i000 at beginning of each year for 4 years at 10% rate of interest. What is the present value of the amount, to be

deposited:

= A + A/r [1-1/(t+r) n-1] = 1000+ 1000 / 0.1 [1-1/(1+0.10)4-1]

= 1000+ 1000 / 0.1 [1-1/(1.331) = 1000+ 1000 / 0.1 [1-0.7513)]

= 1000+ 10000 [0.2487] = 1000 + 2487 = 3487

Future value of beginning-of-a-period annuity (Annuity due):

= C {[(i+r) n -1] r } x (i+r) OR A / r [(i+r) {(1+r)n-1)11

Example - Z wants to deposit Rs.10000 in a recurring deposit account for 3 years in the

beginning of the year. At io% p.a., how much will he be getting? =A/r{(1+r)[(i+r)n-i]}

10000/0.10 +0.0 [(1 +0.1) 3 - = 10000 / 0.10 [(1 +0.1) (1.331) - 1]

= 10000 / 0.10 [1.1 x .331] = 10000 / 0.10 x 0.3641 = Rs.36410