## Calculating the Present Value of Future Cash Flows

we already know that money today is more

valuable than money tomorrow we should bear this in mind when adding

sums of money that will be received at different points in time

imagine the following cash flows which one is preferable do we want to

receive 100 and year 2 and 110 in year 3 4

alternatively a hundred and year one and a hundred and year two if we simply do

the math 210 is better than 200 so we should pick the first option this isn’t

necessarily true after we consider the time value of

money in finance we should never add sums of money without considering the

timing when the cash flow occurs this is very very important if I were to deposit

$100 in a bank I would expect the bank to compensate me for the right to use my

money let’s say that the money will stay in

the bank for a year and after one year the bank will repay me the initial

amount plus an interest of three percent so we will have future value equals

present value multiplied by the sum of one plus I where the present value is

the amount we are depositing and I is the interest rate that the bank will pay

us for depositing our money we would have future value equals a hundred

multiplied by the sum of one plus three percent equals one hundred and three

after one year we will receive one hundred and three dollars what if we

wanted to find the present value of a future cash flow how do we find the

present value of a cash flow that we will receive in the future starting from

the equation future value equals present value times the sum of one plus I we can

simply divide by the sum of one plus I and will obtain the present value equals

future value divided by the sum of one plus I

when we account for the time value of money in this way going backwards we

talk about discounting what if we wanted to make another

deposit after year one ends we would have a hundred times the sum of one plus

three percent which is the value at the end of year one and it will be deposited

for another year so at the end of year two we would have the sum of one plus

three percent multiplied by a hundred multiplied by the sum of one plus three

percent equals future value at year two the present value 100 is equal to the

future value divided by 1 plus 3% elevated to the second degree

so in general when we need to discount a future cash flow that is 10 years from

now we need to divide it by 1 plus I elevated to the nth degree

now that we have learned this we can go to our first example and calculate which

one of the two cash flows has a higher present value

here are the two sets of cash flows the first one involves a payment in year

two that is equal to a hundred and the other payment in year three is equal to

one hundred and ten in order to calculate present value

assuming an interest rate of 10% we need to use the formula that we have

here we will have a hundred / the sum of 1 plus 10% to the second

degree we are elevating to the second degree

because the cash flow is in two years from now then we will have a hundred ten

divided by the sum of 1 plus I to the 3rd degree because this is a cash flow

in year three the present value that we will obtain

from this equation is 165 it’s much lower than the simple sum of a hundred

and 110 now let’s calculate the value of the

other set of cash flows we will have a hundred discounted by one

year plus a hundred discounted by two years we obtained that the present value of

the second set of cash flows is 174 as you can see by discounting two sets of

cash flows and obtaining their present value we are able to compare them and

decide which one is preferable the selection of the discount rate is

rather important in this calculation for most people it is their marginal

borrowing rate the interest rate which a bank would apply to them

if we are considering an investor his discount rate will be the rate of return

that he expects on his investments Oh