# The Time Value of Money

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1 he ime Value of Money Inteest Rates and Futue Value Inteest ates ae a facto in the valuation of vitually all financial instuments. While all money maket ates () ae quoted on an annual basis (PR nnual Pecentage Rate), depending on how inteest is compounded, the effective ate (ER nnual Effective Rate) can diffe significantly. Discete Compounding With discete compound inteest calculations, the inteest is assumed to be peiodically (annually, semi-annually etc.) einvested at the same inteest ate and, thus, futhe inteest is eaned on the inteest assumed to be einvested. he fomula used fo calculating the amount to be ealized on matuity (futue value) with discete compounding is: 1 m amount invested numbe of compounding peiods pe yea nnual Compounding 50,000 is invested fo 3 yeas at 12% pe annum. Calculate the value of the investment at the end of 3 yeas, if inteest is compounded annually. 50, /1 70, ,000 is invested fo 6 months at 12% pe annum. Calculate the value of the investment at the end of 6 months, if inteest is compounded annually. Semi-nnual Compounding 50, /1. 52, ,000 is invested fo 3 yeas at 12% pe annum. Calculate the value of the investment at the end of 3 yeas, if inteest is compounded semi-annually. 50, /2 70, ,000 is invested fo 6 months at 12% pe annum. Calculate the value of the investment at the end of 6 months, if inteest is compounded semi-annually.

2 50, /2 53,000 he effect of inceasing the compounding fequency on the teminal values of an investment, and the diffeence between the PR and ER, can be obseved fom the following example. 100 is boowed fo 1 yea at 10% annual inteest. Calculate the teminal values of the loan at the end of the yea, inteest is compounded annually, semi-annually, quately, monthly, weekly and daily. Compounding Fequency m n Value of 100 at end of yea PR ER nnually % 10% Semi-annually % 10.25% Quately % 10.38% Monthly % 10.47% Weekly % 10.51% Daily % 10.52% Continuous Compounding So fa we have assumed that inteest is compounded at discete intevals. Instead, we may assume that inteest is compounded continuously, then m tends to infinity. In the case of continuous compounding, the teminal value of investment is aived at using the following fomula: e e amount invested exponential 1000 is invested fo 1 yea at 12% annual inteest. Calculate the teminal value of the investment at the end of the yea, assuming continuous compounding of inteest. 1000e 0.12x1 = 1, It may be noted that the teminal value of investment aived at though continuous compounding is quite close to the teminal value with daily compounding ( 1,127.47). Hence, fo pactical puposes, continuous compounding may be teated as equivalent to daily compounding. Continuous compounding of inteest is used extensively in the picing of deivatives. Hence, undestanding the concept and application of continuous compounding is peequisite fo the study of deivatives.

3 Pesent Value ssume that an investo wishes to have a cetain amount of money afte yeas of investment with the going annual inteest ate equal to. he fomula used fo calculating the amount of money he would have to invest now (pesent value) to obtain the amount in the futue, with discete compounding is: 1 m amount obtained in the futue numbe of compounding peiods pe yea gandpaent wishes to set up an endowment fund that will allow thei gandchild to eceive 100,000 once they each the age of 18. ssume at the moment that the gandchild is 7 and that the going inteest ate is 4 pecent pe annum. ssuming annual compounding, how much will the gandpaent have to invest today to ensue that the child will eceive the endowment in 11 yeas time? 100, , What if inteest on the investment was compounded quately? 100, , he fomula used fo calculating the amount of money he would have to invest now (pesent value) to obtain the amount in the futue, with continuous compounding is: e - ssuming that inteest is compounded continuously, and the cuent ate of inteest is 2%, how much would you have to invest today to guaantee a lump sum of 10,000 in 5 yeas time? 10,000e -0.02x5 = 9, In cases you eceive expected cash flows that ae unequal in size, in peiodic payments, then the pesent value is calculated as:

4 fou yea investment will geneate payments of 600, 700, 800 and 600 in yeas 1, 2, 3 and 4 of the investment peiod espectively. ssuming annual compounding and an inteest ate of 5% pe annum, what is the pesent value of the investment? PV = 600(1.05) (1.05) (1.05) (1.05) -4 = 2, nnuity his is the case you eceive a constant steam of equal sized cash flows, fo a fixed peiod. he pesent value of an annuity is calculated as: Note: s tends to (no end) then an annuity becomes a pepetuity. lso, eaanging the above equation we get Some common uses of annuities in eveyday life ae loan payments. Loan payments ae calculated using the annuity fomulas to ensue that you pay the same payment evey month fo a cetain numbe of months o yeas, so that you will have paid back the entie loan by the end of the peiod, along with whateve inteest is necessay. 1 If an annuity is to pay egula cash flows of 100 each yea fo the next twenty yeas and the inteest ate is 12%, what is the pesent value of the annuity? ssume annual compounding Suppose you wish to boow 1,000 at 12% pe annum compounded monthly and you want to know what equal monthly payments would be needed to amotise (pay off) ove the next 5 yeas. = 0.12/12 = 0.01 = 60 months PV = pe month

5 3 he puchase of a new machine fo 100,000 today is expected to geneate additional evenues of 25,000 pe yea fo the next 10 yeas. If the discount ate is 16%, is this new machine a pofitable investment? , ,690 he machine will cost 20,690 pe yea fo the next 10 yeas but will geneate evenues of 25,000 pe yea fo the next 10 yeas. Pepetuity Is you eceive a constant steam of equal sized cash flows, with no end. he pesent value of an annuity is calculated as: Remembe, s tends to (no end) then an annuity becomes a pepetuity. hese types of payment stuctues ae less common than annuities, because individuals aely set up payment stuctues that last foeve. Howeve, chaities often set up this kind of stuctue. ake the example of a scholaship, fom now until the end of time, one student a yea will get a 5000 scholaship. he chaity must know how much money to put aside so this payment is made each yea. hat is calculated as a pepetuity.

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