Financial mathematics

Transcription

1 Chapter 2 Financial mathematics A number of the solutions are shown using both mathematical tables and a Sharp EL-738/735S calculator. The calculator key strokes are shown in a box. 2.1 Deposit in six years: 5500 X ( X 6) = Account balance in five years: (a) 3500 X ( X 5) = 4375 (b) 3500 X ( X 5) = i = X 4 = 7.5% = 8 years 2.5 (a) Total payments (40 X 12 X 2) = 960 (b) i = 800 X 2 = 10% 2 X n X i Ei = 1 + n 2 X 2 X 0.10 Ei = = 13.3% 2.6 (a) (i) 2400 X ( X 5) =

2 (ii) 2400 X ( X 20) = 3600 (b) 3600 Monthly instalments = 60 = $60 (c) Monthly instalments with a deposit = 60 = $ The two disadvantages of simple interest are: - simple interest does not reflect the commercial practice of reinvestment; - simple interest is not time dependent and is inflexible. 2.8 FV = X FVIF,8,10% (2.1436) = /- PV, 8 N, 10 I/Y, COMP FV, answer = (a) FV = 2500 X FVIF,5,5% (1.2763) = (b) FV = 2500 X FVIF,5,12% (1.7623) = (c) FV = 2500 X FVIF,5,18% (2.2878) = (a) /- PV, 5 N, 5 I/Y, COMP FV, the answer = (b) then: 12 I/Y, COMP FV, the answer = (c) then: 18 I/Y, COMP FV, the answer = (a) FV = 6000 X FVIF,3,15% (1.5209) = (b) FV = 6000 X FVIF,6,15% (2.3131) 9

3 = (c) FV = 6000 X FVIF,10,15% (4.0456) = (a) /- PV, 3 N, 15 I/Y, COMP FV, answer = (b) then: 6 N, COMP FV, the answer = (c) then: 10 N, COMP FV, the answer = = X FVIF,8,i% FVIF,8,i% = = i = 18% /- PV, FV, 8 N, COMP I/Y, the answer = 18% = X FVIF,9,i% = = i = 12% /- PV, FV, 9 N, COMP I/Y, the answer = 12% 2.13 interest rate Number of periods (a) 6% 3 X 1 3 (b) 5% 4 X 2 8 (c) 2% 6 X 4 24 (d) 1% 5 X

4 2.14 FV = 6115 X FVIF,8,4% (1.3686) = /- PV, 8 2 = I/Y, 4 X 2 = N, COMP FV, the answer = (a) FV = 4000 X FVIF,4,12% (1.5735) = (b) FV = 4000 X FVIF,8,6% (1.5938) = (c) FV = 4000 X FVIF,16,3% (1.6047) = (a) /- PV, 12 I/Y, 4 N, COMP FV, the answer = (b) then: 12 2 = I/Y, 4 X 2 = N, COMP FV, the answer = (c) then: 12 4 = I/Y, 4 X 4 = N, COMP FV, the answer = FV = 4000 X FVIF,20,3% (1.8061) = /- PV, 12 4 = I/Y, 5 X 4 = N, COMP FV, the answer = Year Deposit Factor FV FVIF,5,8% (1.4693) FVIF,4,8% (1.3605) FVIF,3,8% (1.2597) FVIF,2,8% (1.1664) FVIF,1,8% (1.0800) Clear memory and calculator: 2ndF, M-CLR, 0, /- PV, 5 N, 8 I/Y, COMP FV, M+, then: 300 +/- PV, 4 N, COMP FV, M+, then: 800 +/- PV, 3 N, COMP FV, M+, then: 200 +/- PV, 2 N, COMP FV, M+, then: /- PV, 1 N, COMP FV, M+, RCL, M+, the answer =

5 2.18 PV = X PVIF,6,14% (0.4556) = FV, 6 N, 14 I/Y, COMP PV, the answer = Note - the negative value indicates a payment or outflow of funds PV = X PVIF,6,12% (0.5066) = /- FV, 6 N, 12 I/Y, COMP PV, the answer = Year Cash flow PVIF,18% PV Clear memory and calculator: 2ndF, M-CLR, 0, FV, 1 N, 18 I/Y, COMP PV, M+, then: 2600 FV, 2 N, COMP PV, M+, then: 3800 FV, 3 N, COMP PV, M+, then: 950 FV, 4 N, COMP PV, M+, RCL, M+, the answer = Note - the negative value indicates a payment or outflow of funds. Alternative method: Clear memory: CFi, 2ndF, CA, ON/C 0 ENT, 4700 ENT, 2600 ENT, 3800 ENT, 950 ENT, 18 I/Y, 2ndF, CASH,, COMP, the answer =

6 2.21 Year Cash flow PVIF,8% PV Clear memory and calculator: 2ndF, M-CLR, 0, FV, 1 N, 8 I/Y, COMP PV, M+, then: 1100 FV, 2 N, COMP PV, M+, then: 600 FV, 3 N, COMP PV, M+, then: 2000 FV, 4 N, COMP PV, M+, then: 1500 FV, 5 N, COMP PV, M+, RCL, M+, the answer = Note - the negative value indicates a payment or outflow of funds. Alternative method: Clear memory: CFi, 2ndF, CA, ON/C 0 ENT, 1200 ENT, 1100 ENT, 600 ENT, 2000 ENT, 1500 ENT, 8 I/Y, 2ndF, CASH,, COMP, the answer = Year Cash flow PVIF,15% PV Clear memory and calculator: 2ndF, M-CLR, 0, FV, 1 N, 15 I/Y, COMP PV, M+, then: 2800 FV, 2 N, COMP PV, M+, then: 3400 FV, 3 N, COMP PV, M+, then: 4 N, COMP PV, M+, then: 5 N, COMP PV, M+, then: 2500 FV, 6 N, COMP PV, M+, RCL, M+, the answer =

7 Note - the negative value indicates a payment or outflow of funds. Alternative method: Clear memory: CFi, 2ndF, CA, ON/C, 0 ENT, 2600 ENT, 2800 ENT, 3400 (x,y), 3 ENT, 2500 ENT,15 I/Y, 2ndF, CASH,, COMP, the answer = (a) 3200 = 3890 X PVIF,4,i% 3200 PVIF,4,i% = 3890 = i = 5% (Answer to the nearest whole figure) /- PV, 3890 FV, 4 N, COMP I/Y, the answer = 5.0% (b) PV = 7078 X PVIF,5,8% (0.6806) = FV, 5 N, 8 I/Y, COMP PV, the answer = Decision - as the PV > 4600, accept $7078 in five years (c) = 10,000 X PVIF,n,7% = = n = 4 years FV, /- PV, 7 I/Y, COMP N, the answer = 4 years 14

8 2.24 FV = 800 X FVIFA,12,10% (21.384) = /- PMT, 12 N, 10 I/Y, COMP FV; the answer = (a) FV = 1500 X FVIFA,4,6% (4.3746) = /- PMT, 4 N, 6 I/Y, COMP FV; the answer = (b) FV = 1500 X FVIFA,6,6% (6.9753) = then: 6 N, COMP FV, the answer = (c) FV = 1500 X FVIFA,10,6% (13.180) = then: 10 N, COMP FV, the answer = (a) FV = 800 X FVIFA,5,4% (5.4163) = /- PMT, 5 N, 4 I/Y, COMP FV; the answer = (b) FV = 800 X FVIFA,5,9% (5.9847) = then: 9 I/Y, COMP FV, the answer = (c) FV = 800 X FVIFA,5,12% (6.3528) =

9 then: 12 I/Y, COMP FV, the answer = (a) FV = X FVIFA,5,8% (5.8666) = /- PMT, 5 N, 8 I/Y, COMP FV ; the answer = Decision - accept $82,528 at the end of the fifth year. (b) FV = X FVIFA,5,16% (6.8771) = then: 16 I/Y, COMP FV, the answer = Decision - accept either offer. (c) FV = X FVIFA,5,20% (7.4416) = then: 20 I/Y, COMP FV, the answer = Decision - accept annuity of $12, = A X FVIFA,15,8% (27.152) A = = FV, 8 I/Y, 15 N, COMP PMT, the answer = Note - the negative value indicates a payment or outflow of funds. 16

10 = A X FVIFA,20,10% (57.275) 100,000 A = = FV, 10 I/Y, 20 N, COMP PMT, the answer = Note - the negative value indicates a payment or outflow of funds 2.30 Future Value The 1 st three deposits: 1200 X FVIFA,3,5% X FVIF,2,6% = (3.1525) (1.1236) The last two deposits: 1200 X FVIFA,2,6% (2.0600) = /- PMT, 5 I/Y, 3 N, COMP FV, = then: /- PV, 6 I/Y, 2 N, COMP FV, the answer = FV = 150 X FVIFA,18,1% (19.614) = Note - the 1% comes from the annual rate of 12% divided by 12 (12%/12 = 1%) 150 +/- PMT, = I/Y, 18 N, COMP FV, the answer = (a) PV = 1500 X PVIFA,3,14% (2.3216) = /- PMT, 3 N, 14 I/Y, COMP PV, the answer = (b) PV = 6200 X PVIFA,15,6% (9.7122) =

11 6200 +/- PMT, 15 N, 6 I/Y, COMP PV, the answer = (a) PV = 700 X PVIFA,7,6% (5.5824) = /- PMT, 7 N, 6 I/Y, COMP PV, the answer = (b) PV = 700 X PVIFA,7,12% (4.5638) = then: 12 I/Y, COMP PV, the answer = (c) PV = 700 X PVIFA,7,18% (3.8115) = then: 18 I/Y, COMP PV, the answer = PV = X PVIFA,20,10% (8.5136) = /- PMT, 20 N, 10 I/Y, COMP PV, the answer = Decision - accept $5,000 now ,000 = A X PVIFA,4,10% (3.1699) A = = PV, 4 N, 10 I/Y, COMP PMT, the answer = (outflow) 18

12 2.36 PVIFA,20,11%= (PVIFA,20,10% + PVIFA,20,12%)/2 = = A X PVIFA,20,11% (7.9915) A = = PV, 20 N, 11 I/Y, COMP PMT, the answer = Note - the answer using a calculator is more accurate than using an estimated PVIFA = 3000 X PVIFA,5,i% 8972 PVIFA,5,i% = 3000 = i = 20% 8972 PV, /- PMT, 5 N, COMP I/Y, the answer = 20% = X PVIFA,5,i% PVIFA,5,i% = = i = 16% PV, /- PMT, 5 N, COMP I/Y, the answer = 16% 19

13 = X PVIFA,16,i% PVIFA,16,i% = 21, = i = 12% PV, /- PMT, 16 N, COMP I/Y, the answer = 12% 2.40 (a) = X PVIFA,10,i% PVIFA,10,i% = = i = 10% PV, /- PMT, 10 N, COMP I/Y, the answer = 10% (b) = X PVIFA,10,i% PVIFA,10,i% = = * i = 13% *Note - this is calculated by interpolation using 12% and 14% in the PVIFA Table = ( )/( ) = 0.50 X (14 12) = (or 14 1) = 13 then: /- PMT, COMP I/Y, the answer = 12.97% 20

14 (c) = X PVIFA,10,i% PVIFA,10,i% = = i = 15% then: /- PMT, COMP I/Y, the answer = 15% 2.41 (a) 8000 = A X PVIFA,4,10% (3.1699) 8000 A = = PV, 10 I/Y, 4 N, COMP PMT, the answer = (b) Principal = X PVIFA,2,10% (1.7355) = then: AMRT,, 2 ENT,, the answer (balance) = = A X PVIFA,5,7% (4.1002) A = = PV, 7 I/Y, 5 N, COMP PMT, the answer =

15 Principal at the = X PVIFA,2,7% (1.8080) end of third year = then: AMRT,, 3 ENT,, the answer (balance) = (a) = A X PVIFA,20,12% (7.4694) A = = PV, 12 I/Y, 20 N, COMP PMT, the answer = (b) Principal after = X PVIFA,16,12% (6.9740) 4 th payment = interest in the = X 12% 5 th payment = Principal start 5 th year Instalment Less interest included Principal end of 5 th year then: AMRT, 5 ENT,, 5 ENT,, the answer (balance) = then:,, the answer (interest) = (c) (i) Final Payment = X PVIFA,10,12% (5.6502) = then: ON/C, AMRT,, 10 ENT,, the answer (balance) = (ii) Amount of 10 future payments (10 X 4,016.39)40, less Principal at start of 11 th year Interest saved then: ON/C, COMP PMT, X 10 =,M+, M+, RCL, M+, the answer =

16 2.44 (a) 40,000 = A X PVIFA,20,8% (9.8181) Instalment = Principal after = X PVIFA,10,8% (6.7101) 10 th payment = New instalment calculation: 27, = A X PVIFA,10,10% (6.1446) New instalment = PV, 20 N, 8 I/Y, COMP PMT, AMRT,, 10 ENT,, the balance after the 10 th payment = then: ON/C, PV 10 N, 10 I/Y, COMP PMT, the answer = (outflow) (b) = X PVIFA,n,10% PVIFA,n,10% = n = 12 years then: /- PMT, COMP N, the answer = 12 (years) = A X PVIFA,60,1% ( ) A = PV, 2 nd F P/Y 12 ENT, ON/C,12 I/Y, ON/C, 5 2ndF xp/y, N, COMP PMT, the answer = or, (easier method) PV, = I/Y, 5 X 12 = N, COMP PMT, the answer =

17 = A X PVIFA,30,4% ( ) A = PV, 2 nd F P/Y 2 ENT, ON/C,8 I/Y, ON/C, 15 2ndF xp/y, N, COMP PMT, answer = or, (easier method) PV, 8 2 = I/Y, 15 X 2 = N, COMP PMT, the answer = Note - this calculation can only be done using a financial calculator as the tables do not provide a factor for very high numbers of periods PV, 2 nd F P/Y 12 ENT, ON/C, 20 2ndF xp/y, N, /- PMT, COMP I/Y the answer = (% per annum) or, (easier method) PV, 20 x 12 = N, /- PMT,COMP I/Y the answer = (per period) then: x 12 = the answer = (% per annum) 2.48 PV = 8750 X PVIFA,7,16% X PVIF,4,16% = 8750 X X = /- PMT, 16 I/Y, 7 N, COMP PV, M+, clear calculator 2ndF CA then: RCL, M+, FV, 4 N, 16 I/Y, COMP PV, the answer = OR Clear memory: CFi, 2ndF, CA, ON/C 24

18 0 (x,y), 5 ENT, /- (x,y), 7 ENT, 16 I/Y, 2ndF, CASH,, COMP, the answer = PV = 5,625 X PVIFA,8,14% X PVIF,2,14% = 5625 X X = 20, /- PMT, 14 I/Y, 8 N, COMP PV, M+, clear calculator 2ndF CA then: RCL, M+, FV, 2 N, 14 I/Y, COMP PV, the answer = OR Clear memory: CFi, 2ndF, CA, ON/C 0 (x,y), 3 ENT, /- (x,y), 8 ENT, 14 I/Y, 2ndF, CASH,, COMP, the answer = FV = 1750 X (FVIFA,9,5% - 1) = 1750 X = nd F BGN/END, /- PMT, 5 I/Y, 8 N, COMP FV, the answer = nd F BGN/END FV = 1000 X (FVIFA,6,8% - 1) = 1000 X = nd F BGN/END, /- PMT, 8 I/Y, 5 N, COMP FV, the answer = nd F BGN/END 25

19 2.52 FV = 2400 X (FVIFA,7,3% - 1) = 2400 X = nd F BGN/END, /- PMT, 3 I/Y, 6 N, COMP FV, the answer = nd F BGN/END Initial deposit (FV) = 5000 X FVIF,7,10% (1.9487) = 5000 X = Annuity (FV) = 3000 X (FVIFA,7,10% -1) = 3000 X = Total Value = = nd F BGN/END, /- PV, 7 N, 10 I/Y, COMP FV, M+, clear calculator 2ndF CA, then: /- PMT, 6 N,10 I/Y, COMP FV, M+, RCL, M+, the answer = nd F BGN/END 2.54 (a) 1000 X FVIFA,10,12% (17.548) X FVIFA,5,12% (6.3528) less 200 X FVIF,6,12% (1.9738) (b) 1000 X (FVIFA,11,12% - 1)(19.654) X (FVIFA,6,12% -1) (7.1152) less 200 X FVIF,7,12% (2.2107) Note - this question could be done using a financial calculator but was intended to use the tables only to calculate the result. With a calculator, each line would be similar to /- PV, 12 I/Y, 5 N, COMP FV, M+, and RCL, M+, on the last line. 26

20 2.55 (a) PV = 5000 X PVIFA,5,14% (3.4331) = /- PMT, 14 I/Y, 5 N, COMP PV, the answer = (b) PV = 5000 X (PVIFA,4,14% + 1) (3.9137) = then: 2 nd F BGN/END, COMP PV, the answer = , 2 nd F BGN/END 2.56 (a) PV = 600 X (PVIFA,7,10% + 1) (5.8684) = nd F BGN/END, 600 +/- PMT, 8 N, 10 I/Y, COMP PV, the answer = (b) PV = 600 X (PVIFA,7,15% +1) (5.1604) = then: 15 I/Y, COMP PV, the answer = (c) PV = 600 X (PVIFA,7,18% +1) (4.8115) = then: 18 I/Y COMP PV, the answer = nd F BGN/END 2.57 (a) Using a financial calculator 2 nd F BGN/END, 24 N, /- PMT, 9 12 = I/Y, COMP PV, the answer =

21 (b) clear calculator 2ndF CA, then: FV, 24 N, 9 12 = I/Y, COMP PV the answer = nd F BGN/END Using a financial calculator only (a) 2 (x,y), 10 2 nd F EFF, the answer = 10.25% (b) 4 (x,y), 8 2 nd F EFF, the answer = 8.24% (c) 6 (x,y), 12 2 nd F EFF, the answer = 12.62% (d) 12 (x,y), 18 2 nd F EFF, the answer = 19.56% This is the last question in this chapter. 28