CONCEPT OF TIME AND VALUE OFMONEY Simple and Compound inteest What is the futue value of shs 10,000 invested today to ean an inteest of 12% pe annum inteest payable fo 10 yeas and is compounded; a. Annually (3 maks) b. Semi-quately (2 maks) c. Quately (3 maks) d. Monthly (2 maks) Solving (i) Annually FV = PV (1 +) = 10,000 x (1.12)¹⁰ = 310,584.8 (ii) Semi-annually FV = PV X (1 +)¹⁰(²) 2 = 10,000 (1+0.12)²⁰ = 320,713.5 2 (iii) Quately = FV= PV (1 +)ⁿm 4 = 100,000 (1 +0.12)¹⁰(⁴) = 326,203.77 4
(iv) Monthly FV = PV (1 +)ⁿm 12 Effective Annual Rate = (10,000 (1+0.12)¹² 12 = 330,038.6 QUESTION 2 (c) This is the inteest ate expessed as if it wee compounded once pe yea. The actual ate of inteest eaned (paid) afte adjusting the nominal ate fo factos such as the numbe of compounding peiods pe yea. The effective annual inteest ate is the inteest ate compounded annually but povides the same annual inteest as the nominal ate does when compounded m times pe yea. Example 1 Fo example, suppose you ae offeed 12 pecent compounded monthly. In this case, the inteest is compounded 12 times a yea; so m is 12. You can calculate the effective ate as: Solving EAR = [1 + (Quoted ate)/m]м - 1 = [1 +.12/12]¹² - 1 = 1.01¹² - 1 = 1.126825 1 =12.6825% Example 2 A bank is offeing 12 pecent compounded quately. If you put $ 100 in an account, how much will you have at the end of one yea? What s the EAR? How much will you have at the end of two yeas? Solving The bank is effectively offeing 12%/4 = 3% evey quate. If you invest $ 100 fo fou peiods at 3 pecent pe peiod, the futue value is: Futue value = $ 100 x (1.03)⁴ = $ 100 x 1.1255 = $ 112.55 The EAR is 12.55 pecent [$ 100 x (1 +.1255) = $ 112.55].
We can detemine what you would have at the end of two yeas in two diffeent ways. One way is to ecognize that two yeas is the same as eight quates. At 3 pecent pe quate, afte eight quates, you would have: $ 100 x (1.03)⁸ = $ 100 x 1.2668 = $ 126.68 Altenatively, we could detemine the value afte two yeas by using an EAR of 12.55 pecent; so afte two yeas you would have: $ 100 x (1.1255)² = $ 100 x 1.2688 = $ 126.68 Thus, the two calculations poduce the same answe. This illustates an impotant point. Anytime we do a pesent o futue value calculation, the ate we must be an actual o effective ate. In this case, the actual ate is 3 pecent pe quate. The effective annual ate is 12.55 pecent. It doesn t matte which one we use once we know the EAR. Annuity An annuity is a seies of consecutive payment o eceipts of equal amount ove a defined peiod of time. Usually, the eceipts o payment ae assumed to occu at the end of the yea. Can also be defined as a seies of equal amount payment fo a specified numbe of yeas It is a level steam of cash flow fo a fixed peiod of time. Fo example, a loan epayment plan calls fo the boowe to epay the loan by making a seies of equal payment fo some length of time. A seies of constant o level cash flows that occus at the end of each peiod is called an odinay annuity. Compound Annuities: (ANNUITY FUTURE VALUE) It involves depositing o investing an equal sum of money at the end of each yea fo a cetain numbe of yeas and allowing it to gow Example 1 Assume you want to deposit $500 fo college education at the end of each yea fo the next 5 yeas in a bank. The money will ean 6 pecent inteest. How much money will be thee at the end of 5 th yea. FV5=PMT [1+] ⁿ-1 (ANNUITY FUTURE VALUE) FV5=$500 X(FVIFA) =500x5.637 $2818.5
Example 2 How much must we deposit in an 8 pecent saving accounts at the end of each yea to accumulates $5000 at the end of 10 yeas. FV=PMT [1+]ⁿ-1 5000=PMTX14.4866 PMT=5000 14.4866 $345.15 Example 3 An investo deposits sh. 1000 at the end of each yea fo fou yeas in an account eaning inteest at the ate of 10% pe annum. What is the value at the end of the fouth yea? Futue value of an annuity is given by: FV = [(1+)ⁿ - 1] A= Peiodic annuity amount (1+ )ⁿ = Futue value inteest facto of annuity (FV) = Discount ate/inteest ate Annual annuity Amt (A) = sh. 1000 Numbe of yeas (n) = 4 yeas Theefoe FV = 1000 (1.1⁴ 1) FV= 1000 X 4.6410 = 4641 Example 4 0.1 What would an investo have to deposit at the end of each yea at an inteest ate of 6% if he wishes to accumulate sh. 10,000 in 5 yeas?
Annual Annuity Amount A =? Numbe of yeas = 5 Inteest = 6% 10,000 = A [(1.06)⁵ -1] 0.06 A = 1,774 If an annuity is made at the beginning athe than the end of the peiod, it is efeed to as annuity due. The futue value of an annuity due is elated to a futue value of a nomal annuity by the expession: FV annuity due = FV nomal annuity * (+ ) Example5 Suppose you plan to contibute $ 2,000 evey yea into a etiement account paying 8 pecent. If you etie in 30 yeas, how much will you have? Solution Futue value = Annuity pesent value x (1.08)³⁰ = $ FV=PMT [1+]ⁿ-1 x(1.08)ⁿ Annuity pesent value = $ 2000 x 11.32832 = $ 22, 515.57 The futue value of this amount in 30 yeas is: 22,515.57 x1.08^30 $226,566.4
Pesent value fo annuity cash flow Pension payment, insuance obligation and the inteest owed on bond all involve annuities. To compae these thee types of investments, we need to know the pesent value of each The pesent value of an annuity is given by the expession: PV = A [1- (1/ (1 + ) ⁿ] Example 1 Suppose you ae eceiving $500 at the end of each yea fo the next 5 yeas. The discount ate is 6 pecent. What is the woth of this investment today? PV = A [1- (1/ (1 + ) ⁿ] =500x4.212 $2106 Example 2 What is the pesent value of sh. 10,000 to be eceived at the end of each yea fo 5 yeas at a ate of inteest of 10%? = 10,000 (1-1.1⁵) Annuity Pesent value facto = PMT [ 1 - ( 1/(1 + )t ] 0.1 = 10,000 x 3.7908 = 37,908 Example 3 Afte caefully going ove you budget, you have detemined you can affod to pay $ 632 pe month towads a new spots ca. You call up you local bank and find out that the ate is 1 pecent pe month fo 48 months. How much can you boow? Annuity pesent value = PMT x [pesent value facto)] Annuity Pesent value facto = PMT [ 1 - ( 1/(1 + )t ]
Theefoe pesent value = $ 632 x 37.9740 = $ 24,000 Theefoe, $ 24000 is what you can affod to boow and pay. Amotization of loan An impotant use pesent value concept is in detemining the payment equied fo an installment-type loan. The distinguishing featue of this loan is that it is epaid in equal peiod payment that includes both inteest and pincipal. This payment can be made monthly, quatetely, semi annually o annually. Installment payments ae pevalent in motgages loans, auto loans, consume loans and cetain business loans. Finding the payment/ amotizing a loan Example one Suppose you wished to stat up a new business that specializes in the latest of health food tends, fozen Yak milk. To poduce and maket you poduct, the Yankee Dandy, you need to boow $ 100,000. Because it stikes you as unlikely that this paticula food will be long lived, you popose to pay off the loan quickly by making five equal annual payments. If the inteest ate is 18%, what will be the payment? Annuity pesent value = $ 100,000 = PMTx [1 - pesent value facto] 100,000 = PMT X (1-1/(1.18)⁵ 0.18 PMTX (3.1272) PMT= $ 100,000 = $ 31,977.49 3.1272 Theefoe, you will make a payment of $ 32,000 each. Example two You boow $10,000 at 14 pecent compound annual inteest fo fou yeas. The loan is epayable in fou equal annual installments payable at the end of each yea. a. What is the annual payment that will completely amotize the loan ove fou yeas? (You may wish to ound to the neaest dolla.) b. Of each equal payment, what is the amount of inteest? The amount of loan pincipal? Solving
Annuity Pesent value facto = PMT [ 1 - ( 1/(1 + )t ] USE TABLE FOR ANNUITY a. PV₀ = $10,000 = R(PVIFA₁₄%,₄) = R(2.914) Theefoe R = $10,000 2.914 = $3,432 (to the neaest dolla). (1) (2) (3) (4) (END OF INSTALLMENT ANNUAL PRINCIPAL PRINCIPAL AMOUNT YEAR END PAYMENT INTEREST PAYMENT OWING AT YEAR END (4)t ₁ Χ 0.14 (1) (2) (4) t ₁ - 3) 0 - - - $10,000 1 $3,432 $1,400 $2,032 7,968 2 3,432 1,116 2,316 5,652 3 3,432 791 2,641 3,011 4 3,432 421 3,011 0 $13,728 $3,728 $10,000 Example 2 You boow $10,000 at 14 pecent compound annual inteest fo fou yeas. The loan is epayable in fou equal annual installments payable at the end of each yea. a. What is the annual payment that will completely amotize the loan ove fou yeas? (You may wish to ound to the neaest dolla.) b. Of each equal payment, what is the amount of inteest? The amount of loan pincipal? PV₀ = $10,000 = R(PVIFA₁₄%,₄) = R(2.914) Theefoe R = $10,000 2.914 = $3,432 (to the neaest dolla). (1) (2) (3) (4) (END OF INSTALLMENT ANNUAL PRINCIPAL PRINCIPAL AMOUNT YEAR END PAYMENT INTEREST PAYMENT OWING AT YEAR END (4)t ₁ Χ 0.14 (1) (2) (4) t ₁ - 3) 0 - - - $10,000 1 $3,432 $1,400 $2,032 7,968 2 3,432 1,116 2,316 5,652 3 3,432 791 2,641 3,011 4 3,432 421 3,011 0 $13,728 $3,728 $10,000
Example 3 You an a little shot on you sping beak vacation, so you can put $ 1000 on you cedit cad. You can only affod to make the minimum payment of $ 20 pe month. The inteest ate on the cedit cad is 1.5 pecent pe month. How long will you need to pay off the $ 1000? $ 1000 = $ 20 x 1 pesent value facto 0.015 ($ 1000) x 0.015 = 1- pesent value facto 20 Pesent value facto = 0.25 = 1/ (1+) t (1.015) t = 1/0.25 = 4 This boils down to asking this question How long does it take fo you money to quaduple at 1.5 pecent pe month? (1.015)⁹³ = 3.99 = 4 It will take you about 93/12 = 7.75 yeas at this ate. SUMMARY OF TIME VALUE OF MONEY EQUITION Calculation Futue value of single payment Equation FV=PV (1+) ⁿ Pesent value PV=FV(1 (1+)ⁿ Futue value fo an annuity FV of an annuity=pmt (1+) ⁿ Pesent value of annuity PV of an annuity=pmt (1-(1+)-ⁿ R Futue value of an annuity due FV (annuity due) =futue value of an annuity x (1+) Pesent value of an annuity due PV(annuity due)=pesent value of an annuity x (1+) Thee ae some chaacteistics that should help you to identify and solve the vaious types of annuity poblems:
1. Pesent value of an odinay annuity cash flows occu at the end of each peiod, and pesent value is calculated as of one peiod befoe the fist cash flow. 2. Pesent value of an annuity due cash flows occu at the beginning of each peiod, and pesent value is calculated as of the fist cash flow. 3. Futue value of an odinay annuity cash flows occu at the end of each peiod, and futue value is calculated as of the last cash flow. 4. Futue value of an annuity due cash flows occu at the beginning of each peiod, and futue value is calculated as of one peiod afte the last cash flow. Pactice questions 1. You company poposes to buy an asset fo $335.This investment is vey safe and will be sold in thee yeas time fo $400.You knows that you could invest the $335 elsewhee at 10% with vey little isk. What do you think of the poposed investment? $335 x(1+)^t= 2. You ae consideing a one yea investment. If you put up $1250,you will get back $1350.What is the ate this investment is paying $1250=$1350/(1+)^t 3. You estimate that you will need about $80000 to send you child to college in 8 yeas.you have about $35000 now. If you can ean 20 pecent pe yea, will you make it? At what ate will you just each you goal? - FV=$35000 x(1.2)^8 - FV=$35000 x(1+)^8=$80000 - (1+)=$80000/35000=2.2857 use table 4. You ae offeed an investment that will pay you $200 in one yea, $400 the next yea,$600 the next yea, and $800 at the end of last yea. You can ean 12 pecent on vey simila investment. What is the most will you be willing to pay? $200 x1/1.12^1=178.57 $400 x1/1.12^2=318.88 $600 x1/1.12^3=427.07 $800 x1/1.12^4=508.41 $1432.93 5. An insuance company offes to pay you $1000 pe yea fo 10 yeas if you pay $6710 up font. What ates is implicit in this 10 yeas annuity. $6710=$1000 x(1-pesent annuity value facto)/
6. Suppose you plan to contibute $2000 evey yea into etiement account paying 8 pecent. If you etie in 30 yeas, how much will you have? Annuity pesent value=$2000 x(1-1/1.08^30)/.08 7. Assume you want to deposit $500 fo college education at the end of each yea fo the next 5 yeas in a bank. The money will ean 6 pecent inteest. How much money will be thee at the end of 5 th yea? 8. How much must we deposit in an 8 pecent saving accounts at the end of each yea to accumulate $5000 at the end of 10 yeas. 9. An investo deposits sh. 1000 at the end of each yea fo fou yeas in an account eaning inteest at the ate of 10% pe annum. What is the value at the end of the fouth yea? 10. You boow $10,000 at 14 pecent compound annual inteest fo fou yeas. The loan is epayable in fou equal annual installments payable at the end of each yea. c. What is the annual payment that will completely amotize the loan ove fou yeas? (You may wish to ound to the neaest dolla.) d. Of each equal payment, what is the amount of inteest? The amount of loan pincipal?