AMB111F Financial Maths Notes

AMB111F Financial Maths Notes Compound Inteest and Depeciation Compound Inteest: Inteest computed on the cuent amount that inceases at egula intevals. Simple inteest: Inteest computed on the oiginal fixed amount (pincipal). Suppose an amount of P ands is invested (in a savigs bank o insuance) at the ate of % pe annum compound inteest, then afte 1 yea the new amount is A 1 = P + P (/100) = P (1 + /100). Afte 2 yeas the new amount is A 2 = P (1+/100)+P (1+/100)(/100) = P (1+/100) 2. Afte 3 yeas the new amount is A 3 = P (1 + /100) 2 + P (1 + /100) 2 (/100) = P (1 + /100) 3, and so foth. Thus the amount afte n yeas at % p.a. compound inteest is P (1 + /100) n. The initial amount invested, denoted by P, is usually called the P incipal; is the ate at which the money eans compound inteest. The numbe n of yeas duing which P is invested is called the peiod, which may also be given in months o any faction of a yea. NOTE: Simple inteest is computed on the same fixed pincipal evey yea (o faction of the yea) and the esulting amount is A = P (1 + n/100), wheeas compound inteest is computed on the new amount. Example : 1. A student invests R500 fo 5 yeas at 12,5 % pe yea compound inteest. Calculate how much money will he have at the end of 5 yeas. Soln. Amount afte n yeas is A = P (1 + /100) n. Hee P = R500; n = 5 and =12, 5. Thus A = 500(1 + 0, 125) 5 = 500(1, 125) 5 = R905, 02. Example : 1. A student invests R500 fo 5 yeas at 12,5 % pe yea simple inteest. Calculate how much money will he have at the end of 5 yeas. Soln. Amount afte n yeas is A = P (1 + n/100). Hee P = R500; n = 5 and =12, 5. Thus A = 500(1 + 5(0, 125)) = 500(1, 625) = R812, 50. 2. A student invests R500 fo 5 yeas at 12, 5 % compound inteest p.a. calculated quately. Find the amount afte 5 yeas. Soln. Hee we need to convet the yealy ate to the quately ate. Thee ae 4 quates in a yea, thus the quately ate is 12, 5 4 %. Also 5 yeas becomes 5 4 = 20 quates. So the amount is A = P (1 + /100) n = 500(1 + 12,5 400 )20 = 1

R925, 23 3. R1000 amounts to R1500 afte 2,5 yeas compound inteest. The inteest was calculated monthly. Find the inteest ate pe yea. Soln. A = P (1+/100) n implies ( A P )1/n =1+/100. Thus = 100[( A P )1/n 1] pe month. so n =2, 5 12 = 30 months. Theefoe = 100[( 1500 1000 )1/30 1] = 100[(1.5) 1/30 1]=1.360725 pe month. Hence pe yea (12 months) the ate is =1.360725 12 = 16.3287. Calculating the peiod n: Let A = P (1 + /100) n. Then (A/P )=(1+/100) n implies log(a/p )= n log(1 + /100), thus n = log(a/p ). log(1+/100) 4. Calculate how long it will take an investo to make a gain of R1000 on an initial investment of R3000 if the compound inteest ate is 11,5 % p.a. calculated monthly. Soln. Hee the total inteest to be made is R1000. So the final amount must be R4000. =11, 5 12 pe month. Thus the peiod in months is n = log(a/p ) = log(4/3) = 301626 months, tanslating to about 2, 51 log(1+/100) log(1+(11.5)/1200) yeas. Depeciation Depeciation is the evese of compound inteest. An item loses value the moment it leaves a stoe, and we say it depeciates. Suppose its initial value is P ands and depeciates at the ate of % p.a. What is its value afte n yeas? Afte 1 yea: value V 1 = P P (/100) = P (1 /100) Afte 2 yeas: value V 2 = P (1 /100) P (1 /100)(/100) = P (1 /100) 2, and so foth. Thus its value afte n yeas is V n = P (1 /100) n. Example 1. A ca costs R180 000 and depeciates at a ate of 8 % p.a. calculated on the diminishing balance. Calculate the value of the ca afte 5 yeas. Soln. Its value is V 5 = P (1 /100) n = 180000(1 8/100) 5 = R118634, 67. Calculating the peiod n: Let V n = P (1 /100) n. Then (V n /P )=(1 /100) n implies log(v n /P )= n log(1 /100), thus n = log(vn/p ) log(1 /100). 2

Annuities. An annuity is a seies of fixed amounts paid at egula intevals, and accumulating compound inteest in the pocess. A value of the annuity afte a given inteval is the amount available at the end of the inteval with due inteest also added. If I pay fixed amounts of R2000 quately into a savings account fo 10 yeas, then the seies of those fixed amounts is an annuity. The value of the annuity is the amount available afte the specified peiod. Futue value annuity: This is the value of an annuity afte the specified peiod, fo example in investing fo etiement, the amount that should be available is the futue value. Altenatively the amout to which the annuity gows is the futue value of the annuity. Pesent value annuity: the single sum that needs to be invested now to get a pecalculated futue value afte a cetain peiod of time, is the pesent value of the annuity. Sinking Fund: This is an investment made to eplace valuable and expensive items at a late stage. E.g. a ca, tacto o funitue. Example 1. Calculate the value of an annuity in which R1000 is invested at the end of each yea at 10 % pe annum fo 5 yeas. soln. Annuities involve compound inteest, see the following table: Amount/Value end yea 1 1000 end yea 2 1000 + 1000(1 + 10/100) end yea 3 1000 + 1000(1 + 10/100) + 1000(1 + 10/100) 2 end yea 4 1000 + 1000(1 + 10/100) + 1000(1 + 10/100) 2 + 1000(1 + 10/100) 3 end yea 5 1000 + 1000(1 + 10/100) + 1000(1 + 10/100) 2 + 1000(1 + 10/100) 3 + 1000(1 + 10/100) 4 = R6105, 10 So the equied value of the annuity afte 5 yeas is R6105, 10 (futue value). Now let P = fixed deposit of an annuity, A = futue value of the annuity, n = peiod of the annuity, and = inteest ate pe peiod. If the deposits ae made at the end of the yea, then the above example shows that A = n 1 k=0 P (1 + /100)k. Now fom the summation of a geometic seies, A = P 100[(1 + 100 )n 1]. NOTE: When money is invested at the beginning of the yea, the futue value is A = P 100 (1 + /100)[(1 + 100 )n 1] because inteest is added at the end of the yea. 3

Example 1. Calculate the value of an annuity in which R1000 is invested at the beginning of each yea at 10 % pe annum fo 5 yeas. Soln. Futue value is FV = A = P 100 (1 + /100)[(1 + 100 )n 1] = 1000 100 (1 + 10 10/100)[(1 + 10 100 )5 1] = R6715.61 Example 2. Themba invests R500 evey 6 months at 9% p.a. inteest, compounded half-yealy fo 10 yeas. Calculate the value of the annuity afte 10 yeas. soln. Convet to half-yeas: P = R500; 10 yeas = 20 half yeas; ate = 9/2 pe half-yea. Using A = P 100 [(1+, we have A = 500 200 [(1+ 9 9 200 )20 1] = R15685.71. Now conside A = P 100[(1 + amount payable peiodically in an annuity: Soln. P = A 100[(1+ 100 )n 1] and find a fomula fo the fixed Example 3. John wants to have an amount of R10000 at the end of 5 yeas. The best compound inteest ate that he can have is 8 % p.a.. Calculate his yealy fixed deposits (deposited at the end of the yea) in ode to achieve his goal. Soln. Fixed deposit = P = A 100[(1+ = 8(10000) 100[(1+ 8 100 )5 1] = R1704.56 Pesent value: This is a single sum of money invested unde the same tems as an annuity. Instead of paying fixed deposits at egula intevals, only one lage sum is invested. The financial institution then woks out fixed deposits as in the futue value. Let PV = pesent value to be invested; FV = futue value; n = peiod of investement, = inteest ate. Assume the lump sum is deposited at the beginning of the yea. Afte 1 yea, FV = PV(1 + /100) 1. Afte 2 yeas, FV = PV(1 + /100) 2. It is clea that afte n yeas FV = PV(1 + /100) n. Example 4. Tshidi has an annuity to which she contibutes R1000 p.a. (at the end of the yea) at 6% annual compound inteest. The annuity will matue in 25 yeas. Calculate the pesent value of the annuity. Soln. FV = P 100 [(1 + 100 )n 1] = 1000 100 [(1 + 6 6 100 )25 1] = 54864.51. Now 4

FV = PV(1 + /100) n implies PV = FV = 54864.51 = R12783.36. (1+/100) n (1+6/100) 25 Example 4. Tshidi has an annuity to which she contibutes R1000 p.a. (at the beginning of the yea) at 6% p.a compound inteest annually. The annuity will matue in 25 yeas. Calculate the pesent value of the annuity. Soln. FV = P 100 (1+/100)[(1+ = 1000 100 (1+6/100)[(1+ 6 6 100 )25 1] = (54864.51((1.06) = 58156.38. Now FV = PV(1 + /100) n implies PV = FV = 58156.38 = R13550.36. (1+/100) n (1+6/100) 25 Fomula involving pesent value and fixed deposits: Recall: FV = P 100 [(1 + 100 )n 1] whee P = fixed deposit caied out peiodically at the end of the yea. FV = PV(1 + /100) n. Thus PV(1 + /100) n = P 100[(1 + 100 )n 1]. Solving fo PV, we have PV = P 100 [(1+ 100 )n 1] 100P [(1+ = (1+/100) n. (1+/100) n Similaly if deposits ae made at the beginning of the yea. Example 5. Which of the following will give geate financial etun? (a) An annuity of R100 deposited pe month (at the end) fo 20 yeas at 12 % p.a. inteest compounded half-yealy. (b) A single deposit of R10000 (at the beginning) invested fo 20 yeas at 12 % p.a. inteest compounded half-yealy. Soln. (a) Conveting to half yea peiod, 1 month = 1/6 half yea, 20 yeas = 40 half yeas; R100/month = R600/half yea. Theefoe FV = P 100 [(1 + 100 )n 1] = 600 200 12 [(1 + 12 200 )40 1] = R92857.18. (b) FV = PV(1 + /100) n = 10000(1 + 12/200) 40 = R102857.18. Theefoe (b) yields a bigge etun. Loan Repayments. Loan epayment woks just like annuity (paid at the end of an inteval). The diffeence is that you pay the financial institution inteest. Example 6. Calculate the total cost of epaying a ca loan of R100 000 at 9 % p.a. in equal monthly epayments ove a 25-yea peiod. 5

Soln. The total cost of epayment is just like a futue value in an annuity with pesent value PV = 100 000. Now PV = P 100 [(1+ implies (1+/100) n P = PV((1+/100)n ) = R839.20 = 100 [(1+ = PV ((1+/100)n ) 100[(1+ = (9/12)100000((1+9/1200)300 ) 100[(1+ 9 1200 )300 1] monthly epayments. Thus the total payable = R839.20 300 = R251758.91. (Note that 25 yeas = 300 months). Sinking Fund. This is an investment made to eplace expensive items in a few yeas time. Scap value is the value of the depeciated item afte a peiod of time. Example. Machiney is puchased at a cost of R550 000 and is expected to ise in cost at 15 % p.a. compound inteest and depeciate in value at 8 % pe annum compounded annually. A sinking fund is stated to make povision fo eplacing the old machine. The sinking fund pays 16 % inteest p.a. compounded monthly, and you make (fixed) monthly payments into this account fo 10 yeas, stating immediately and ending one month befoe the puchase of the new machine. Detemine: (a) the eplacement cost of a new machine 10 yeas fom now; (b) the scap value of the machine in 10 yeas time; (c) the monthly payments into the sinking fund that will make povision fo the eplacement of the new machine. Soln. (a) Let PV = R550000. Then the futue value of the machine is FV = PV(1 + /100) n = 550000(1 + 15/100) 10 = R2225056.76 = Replacement Cost (b) Scap value is SV = PV(1 /100) n = 550000(1 8/100) 10 = R238913.65 (c) Now FV - SV = R2 225 056.76 - R238 913.65 = 1986143.11 is the equied value (SF) of the sinking fund. Conside SF = P 100 (1 + /100)[(1 + 100 )n SF 1] whee P = fixed instalment. Thus P = = 100 [(1+ (1+/100) = R6699.27 (monthly instalments payable). 1986143.11 1200[(1+ 16 1200 )120 1](1+16/1200) 6