STUDY GUIDE FOR TOPIC 1: FUNDAMENTAL CONCEPTS OF FINANCIAL MATHEMATICS. Learning objectives

Transcription

1 Study Gude for Topc 1 1 STUDY GUIDE FOR TOPIC 1: FUNDAMENTAL CONCEPTS OF FINANCIAL MATHEMATICS Learnng objectves After studyng ths topc you should be able to: apprecate the ever-changng envronment n whch corporates operate and the major nnovatons n the theory of fnance. Prudent applcaton of these nnovatons can help the fnancal manager better utlse the scarce economc resources of the frm; explan the objectves of the frm and the potental costs mposed on the frm by the conflctng objectves of management and shareholders; dfferentate between smple and compound nterest; and demonstrate an understandng of the concepts of present value and future value as appled to practcal fnancal stuatons. Introducton As fnance s concerned wth money, and because the nterest rate s the prce of money, most fnancal decsons nvolve nterest rate consderatons. Ths lecture deals wth the mathematcs of nterest and present value whch s of vtal mportance for the comprehenson of almost all the materals subsequently covered n ths subject. The key content areas covered n ths topc are: A. Overvew of Fnance External Envronment and the Fnancal Manager Evoluton of Fnance Objectves of the Frm B. Fnancal Mathematcs Smple Interest Compound Interest Effectve Rate of Interest Present Value Ordnary Annutes Annuty Due Deferred Annutes Perpetutes Contnuous Compoundng Determnng Interest Rates 1

2 2 Study Gude for Topc 1 A. Overvew of fnance External envronment and the fnancal manager Corporate managers today operate n a dynamc and volatle fnancal envronment. Ths envronment s eptomsed by the followng characterstcs: deregulaton of fnancal markets; greater nterest rate and exchange rate volatlty; rapd technologcal advance; fnancal nnovaton; and changng taxaton regmes. Evoluton of fnance Academc theory has evolved to explan ths ever-changng fnancal envronment, and has been charactersed by a number of major breakthroughs. The academc theory of fnance whch has developed over recent years helps us better understand ths ever-changng fnancal envronment. It also asssts the fnancal manager to manage the busness more effectvely n such an envronment. Some of the more sgnfcant developments n academc theory related to fnance whch we wll be explorng n ths subject are as follows: captal budgetng usng the dscounted cash flow technques; asset valuaton models; portfolo theory and the captal asset prcng model; the effcent market hypothess; agency theory; captal structure and cost of captal optmsaton; and dvdend polcy optmsaton. Objectves of the frm It s generally accepted by academcs and practtoners alke, that the prmary objectve of the management of an organsaton should be to maxmse shareholders wealth. However, a number of conflcts wll arse wthn the frm whch operate to effectvely move the organsaton away from ths objectve. The frst of these conflcts s the competng nterests of short-term proft maxmsaton and long-term wealth maxmsaton. A good example of ths conflct s research and development expendture. A manager could readly boost a frm s short term profts by reducng ths type of expendture, however such an acton could be to the detrment of the organsaton s long term growth. Another conflct arsng wthn the organsaton s that between the competng nterests of management and shareholders. Management do not always naturally act n the best nterests of shareholders. For example, they may purchase expensve vehcles for themselves wth company funds. The cost of such extra expendture s clearly an opportunty cost for shareholders and contrary to the prmary objectve of the frm - maxmsng shareholders wealth. 2

3 Study Gude for Topc 1 3 B. Fnancal mathematcs Smple nterest Smple nterest can be calculated usng the followng easy formulae: I = P. r. t and S = P. (1 + r. t) where I = Amount of nterest P = Prncpal r = Smple nterest rate per annum t = Number of years S = Accumulated sum For example, f you nvest $1,000 at an nterest rate of 10% per annum, nterest wll be payable as follows: Interest for half a year = $1, = $50 Interest for one year = $1, = $100 Usng the second formula, how much wll you repay on a loan of $1,000 at smple nterest of 10% per annum, f you repay the debt n full (ncludng nterest) after two years? Answer: S = $1000 [1 + (0.1 2)] = $1200 Compound nterest Compound nterest arses where the nvestor does not wthdraw the nterest receved from the nvestment account, and thus effectvely receves nterest on nterest earned n prevous perods. Compound nterest can be ascertaned as follows: FV n = P 0 (1 + ) n where FV n = Compound value at the end of n perods P 0 = Amount of prncpal at the start of the frst perod = Interest rate per perod For example, how much wll you have after one and a half years f nterest s compounded at 5% per half year and you commence wth $100.00? Answer: FV 3 = $100 ( ) 3 = $

4 4 Study Gude for Topc 1 Effectve rate of nterest One must be careful to dstngush between nomnal nterest rates and effectve nterest rates. The former s smply the quoted annual rate whle the latter s defned as that rate whch would produce the same endng (future) value f annual compoundng were used (Brgham, pp. 230). The effectve rate wll always be hgher than the nomnal rate where nterest s compoundng more frequently than once per year. Effectve nterest rate per annum = 1+ j m m 1 where j = Nomnal nterest rate per annum m = Frequency of compoundng For example, what wll be your effectve nterest rate f you nvest $1 at 1% per month for 12 months? Answer: Effectve Interest Rate = 1+ 12% = 12.68% Present value The present value of a seres of future Cash flows s calculated usng the compound nterest formula dscussed earler: FV n = P 0 (1 + ) n P 0 = FV n /(1 + ) n PV = A n (1 + ) -n Where PV = Present value of future amount A n = Amount we wsh to have avalable at the end of n years For example, a loan s repad by a lump sum (ncludng nterest) of $ after 3 years. If nterest s at 5% per annum, how much was borrowed. Answer: PV = $ (1.05) -3 = $ Apart from usng ths formula, the present value can be calculated by students usng ether tables for the tme value of money or by usng fnancal calculators. 4

5 Study Gude for Topc 1 5 For example, usng tables for the tme value of money, what s the present value of $ payable n 5 years, dscounted at 5% per annum? Answer: PV = FV 5 (PVIF 5%, 5 ) = $ (0.7835) = $ Ordnary annutes An annuty s a stream of lke cash flows. The followng formulae are approprate for calculatng the present or future value of an annuty: PV = R[1 - (1 + )-n ] FV n = R[(1 + )n 1] where R s the annuty cash flow. For example, f you open a savngs account wth $300 and depost $200 at the end of each quarter for the rest of the year, what amount wll you have accumulated? Assume nterest s payable at 8% per annum, compoundng quarterly (.e. nterest s 2% each quarter). Answer: FV n = PV (1 + ) n = $300 (1.02) 4 = $ FV n = = R[(1 + ) n 1 $200[(1.02) 4 1] 0.02 = $ Therefore, savngs wll total $1, after one year. Annuty due If the annuty s pad (receved) at the start of each perod (and not the end), ths s called an annuty due. The formula for ths stuaton s as follows: FV n = R[(1+ )n 1. (1+ ) 5

6 6 Study Gude for Topc 1 The extra perod of compoundng s added because the cash flows occur earler (at the start of the perod) and therefore each cash flow s earnng nterest for an addtonal perod. For example, f the $200 payments n the above example are made at the start of each quarter and not the end: Answer: FV n = FV n = R[(1+ )n 1 = $ (1.02). (1+ ) = $ (gnorng the ntal $300 payment) Deferred annutes If the annuty has a delay of more than one perod before t commences, then ths s termed a deferred annuty. The present value of such an annuty can be determned by frstly obtanng the present value of the annuty at the start of the perod n whch the frst cash flow occurs, and then dscountng ths amount back to the present day. For example, the present value of $200 per perod for ten years startng n two years tme at an nterest rate of 8% per annum s as follows: PV = $200 (PVIFA 8%, 10 )(PVIF 8%,2 ) = $200 (6.7101) (0.8573) = $1, Perpetutes A perpetuty s a stream of equal cash flows whch are expected to contnue ndefntely. Ths can be contrasted wth an annuty whch has a fnte lfe. You wll recall that the present value of an annuty was calculated as follows: PV = R[1 (1+ ) -n Now, as n approaches nfnty, the formula becomes: PV = R For example, what s the present value of a perpetuty of $200 per annum at an nterest rate of 10% per annum? Answer: PV = $ = $2000 6

7 Study Gude for Topc 1 7 Contnuous compoundng Ths s an nvestment n whch nterest s added contnuously rather than at dscrete ponts n tme. FV 1 = P 0 (1+ j m )m As m ncreases toward nfnty, the above equaton becomes: FV n = P 0 e jn where lm (1+ j m )m = e j m and n = duraton of nvestment n years For example, what s the accumulated sum of $100 after payable at 10% compoundng contnuously? years f nterest s Answer: FV 2.5 = $100e (0.1)(2.5) = $100e 0.25 = $ = $ Determnng nterest rates If one s gven the present and future values of an amount (or seres of amounts), the nterest rate necessary to acheve that growth can be determned n a number of ways. Frstly, the student can solve for the value of usng the present or future value formulae. Secondly, the student can use tables for the tme value of money and agan solve for the approprate nterest rate. Fnally, the student can nput the present and future values and the number of perods nto a fnancal calculator and solve for the mssng nterest rate. For example, the bank offers to lend you $1000 f you agree to pay $1, after fve years. What rate of nterest are you beng charged? Answer: PV = $1000 = $1, (PVIF k, 5 ) PVIF k,5 = $1000/$1, = Usng tables for the tme value of money, the soluton s 10%. 7

8 8 Study Gude for Topc 1 Self study problems You should work through the self study problems as a means of consoldatng your understandng of each topc before attemptng the problem solvng exercses. 1. Bll and Ben purchased a new house 15 years ago for $40,000. They borrowed $30,000 for twenty years at a fxed nterest rate of 8% per annum, compoundng annually, repayable by equal annual nstalments due at the end of each year. (a) (c) Calculate the amount of the annual nstalments due on the loan. How much prncpal would be owng on the loan mmedately after the payment of the ffteenth nstalment (assumng all nstalments were pad as scheduled)? If the loan was repad as scheduled over the twenty years, how much would be pad (n total) as nterest? 2. (Adopted from Brgham) A father s plannng a savngs program to put hs daughter through unversty. Hs daughter s now 13 years old. She plans to enrol at the unversty n 5 years, and t should take her 4 years to complete her educaton. Currently, the cost per year (for everythng - food, clothng, tuton, books, transportaton, and so forth) s $12,500, but a 5 percent nflaton rate n these costs s forecasted. The daughter recently receved $7,500 from her grandfather s estate; ths money, whch s nvested n a bank account payng 8 percent nterest compounded annually, wll be used to help meet the costs of the daughter s educaton. The rest of the costs wll be met by money the father wll depost n the savngs account. He wll make 6 equal deposts to the account n each year from now untl hs daughter starts unversty. These deposts wll begn today and wll also earn 8 percent nterest. (a) (c) What wll be the present value of the cost of four years of educaton at the tme the daughter becomes 18? (Hnt: Calculate the future value of the cost (at 5%) for each year of her educaton and then dscount three of these costs back (at 8%) to the year n whch she turns 18, then sum the four costs. What wll be the value of the $7,500 whch the daughter receved from her grandfather s estate when she starts college at age 18? (Hnt: 5 years at 8%.) If the father s plannng to make the frst of 6 deposts today, how large must each depost be for hm to be able to put hs daughter through college? 8

9 Study Gude for Topc Vctor has just nvested $5,000 for 5 years. Calculate the future value of ths nvestment assumng that the nvestment wll earn nterest at: (a) (c) an annual nomnal rate of 12%, compounded quarterly; an annual nomnal rate of 12% compounded monthly; and an annual nomnal rate of 14% compounded quarterly. 4. John s now ffty years old, and has just nherted $200,000 from the estate of a recently deceased uncle. He and hs wfe expect to retre n fve years tme. John ntends to nvest an amount whch wll produce an ncome of $15,000 per annum for twenty years, commencng n fve years tme. Requred: Calculate the amount of the requred nvestment to be made now to produce an annual ncome of $15,000 payable n a lump sum at the end of each year for 20 years, commencng n fve years tme, assumng an annual compound nterest rate of 12%. 5. How much would need to be nvested at the end of each compoundng perod to produce a future value of $10,000 n fve years tme f the nvestments wll earn nterest at: (a) (c) an annual nomnal rate of 16%, compounded quarterly; an annual nomnal rate of 15%, compounded monthly; and an annual nomnal rate of 17%, compounded half-yearly. 6. (a) Peter s now 45 years old, s self employed, and ntends to retre at the age of 55 years. He ntends to nvest $10,000 per annum, at the end of each year for ten years at an annual compound rate of 17%. What wll be the future value of hs nvestments n ten years tme? If the annual compound nterest rate on Peter s nvestments for part (a) was 17.5%, what would be the future value of hs nvestments n ten years tme? Solutons to self study problems 1. (a) PV = R PVIFA 8%,20 30,000 = R R = 30, R = $3,055 PV 15 = 3,055 PVIFA 8%,5 = 3, PV 15 = $12,197 9

10 10 Study Gude for Topc 1 (c) Total Amount Pad = 3, = $61,100 Interest Pad = 61,100 30,000 = $31, (a) Frst, determne the annual cost of unversty. The current cost s $12,500 per year, but that s escalatng at a 5 percent nflaton rate: Unversty Year Current Cost Years from Now Inflaton Adjustment Cash Requred $12,500 12,500 12,500 12, (1.05) 5 (1.05) 6 (1.05) 7 (1.05) 8 $15,954 16,751 17,589 18,468 Now put these costs on a tme lne: ,954 16,751 17,589 18,468 How much must be accumulated by Age 18 to provde these payments at Ages 18 through 21 f the funds are nvested n an account payng 8 percent, compounded annually? $15,954 (PVIF 8%, 0 ) = $15,954 $16,751 (PVIF 8%, 1 ) = 15,510 $17,589 (PVIF 8%, 2 ) = 15,079 $18,468 (PVIF 8%, 3 ) = 14,660 $61,203 Thus, the father must accumulate $61,203 by the tme hs daughter reaches age 18. She has $7,500 now (age 13) to help acheve that goal. Fve years hence that $7,500, when nvested at 8 percent, wll be worth $11,020: $7,500 (1.08) 5 = $11,020. (c) The father needs to accumulate only $61,203 - $11,020 = $50,183. If we assume equal annual deposts today (age 13) and at ages 14, 10

11 Study Gude for Topc , 16, 17, and 18, he must make 6 equal payments to accumulate $50,183. Each payment, therefore, must be $6,841: $50,183 = PMT (FVIFA 8%, 6 ). PMT = $6, (a) = 12% quarterly FV = PV = FV = $9,031 = 12% monthly FV = PV = FV = $9,083 (c) = 14% quarterly FV = PV = 5000 (1.035) 20 = FV = $9, PV = CF PVIFA 12%,20 = 15, PV = $112,041 PV = FV PVIF 12%,5 = 112, PV = $63, (a) $336 $113 11

12 12 Study Gude for Topc 1 (c) $ (a) $223,931 $229,500 12