Sigle-Paymet Factors (P/F, F/P) Example: Ivest $1000 for 3 years at 5% iterest. F =? i =.05 $1000 F 1 = 1000 + (1000)(.05) = 1000(1+.05) F 2 = F 1 + F 1 i = F 1 (1+ = 1000(1+.05)(1+.05) = 1000(1+.05) 2 F 3 = 1000(1+.05) 3 EGR 312-04 1 Sigle-Paymet Factors (P/F, F/P) Fudametal questio: What is the future value, F, if a sigle preset worth, P, is ivested for periods at a ROR of i% assumig compoud iterest? $? Geeral Solutio: $P EGR 312-04 2 Sigle-Paymet Factors (P/F, F/P) Fudametal questios: 1. What is the future value, F, if a sigle preset worth, P, is ivested for periods at a ROR of i% assumig compoud iterest? I geeral, F = P(1+ 2. What is the preset value, P, if a future value, F, is desired, assumig P is ivested for periods at i% compoud iterest? P = F/(1+ EGR 312-04 3 1
Sigle-Paymet Factors (P/F, F/P) Stadard Notatio: If watig to kow F give some P is ivested for periods at i% iterest use (F/P,i,) Example: if i = 5% per moth, = 6 moths, If watig to kow P give some F if P is to be ivested for periods at i% iterest use (P/F,i,) Example: if i = 7.5% per year, = 4 years, EGR 312-04 4 Sigle-Paymet Factors (P/F, F/P) Stadard Notatio Equatio: To fid the value of F give some P is ivested for periods at i% iterest use the equatio F = P(F/P,i,) To fid the value of P give some F if P is to be ivested for periods at i% iterest use P = F(P/F,i,) The compoud iterest factor tables o pages 581-609 provide factors for various combiatios of i ad. EGR 312-04 5 Sigle-Paymet Factors (P/F, F/P) Example: If you were to ivest $2000 today i a CD payig 8% per year, how much would the CD be worth at the ed of year four? F = $2000(F/P,8%,4) F = $2000( ) from pg. 593 F = $2721 or, F = $2000(1.08) 4 F = $2000(1.3605) F = $2721 EGR 312-04 6 2
Sigle-Paymet Factors (P/F, F/P) Example: How much would you eed to ivest today i a CD payig 5% if you eeded $2000 four years from today? P = $2000(P/F,5%,4) P = $2000( ) from pg. 590 P = $1645.40 or, P = $2000/(1.05) 4 P = $2000/(1.2155) P = $1645.40 EGR 312-04 7 Uiform Series Preset Worth (P/A, A/P) To aswer the questio: what is P give equal paymets (istallmets) of value A are made for periods at i% compouded iterest? P =? i Note: the first paymet occurs at the ed of period 1. Examples? Reverse mortgages Preset worth of your remaiig car paymets A EGR 312-04 8 Uiform Series Preset Worth (P/A, A/P) To aswer the questio: what is P give equal paymets (istallmets) of value A are made for periods at i% compouded iterest? Stadard Notatio: (P/A,i,) (1 1 P A i(1 EGR 312-04 9 3
Uiform Series Preset Worth (P/A, A/P) To aswer the related questio: what is A give P if equal istallmets of A are made for periods at i% compouded iterest? Stadard Notatio: (A/P,i,) i(1 A P (1 1 Examples? Estimatig your mortgage paymet EGR 312-04 10 Uiform Series Preset Worth (P/A, A/P) Example: What is your mortgage paymet o a $90K loa if you are quoted 6.25% iterest for a 30 year loa. (Remember to first covert to moths.) P = $90,000 i = = A = A = EGR 312-04 11 Uiform Series Future Worth (F/A, A/F) To aswer the questio: What is the future value at the ed of year if equal istallmets of $A are paid out begiig at the ed of year 1 through the ed of year at i% compouded iterest? i F =? A EGR 312-04 12 4
Uiform Series Future Worth (F/A, A/F) Kowig: P = F/(1+ i(1 A P (1 1 The: F i(1 A (1 (1 1 ad, i A F (1 1 (1 1 F A i EGR 312-04 13 Uiform Series Future Worth (F/A, A/F) Example: If you ivest i a college savigs pla by makig equal ad cosecutive paymets of $2000 o your child s birthdays, startig with the first, how much will the accout be worth whe your child turs 18, assumig a iterest rate of 6%? A = $2000, i = 6%, = 18, fid F. F = 2000(F/A,6%,18) F = $2000(30.9057) F = $61,811.40 or, EGR 312-04 14 For example: No-Uiform Cash Flows You ad several classmates have developed a keychai ote-takig device that you believe will be a huge hit with college studets ad decide to go ito busiess producig ad sellig it. 1) Sales are expected to start small, the icrease steadily for several years. 2) Cost to produce expected to be large i first year (due to learig curve, small lot sizes, etc.) the decrease rapidly over the ext several years. EGR 312-04 15 5
Arithmetic Gradiet Factors (P/G, A/G) Cash flows that icrease or decrease by a costat amout are cosidered arithmetic gradiet cash flows. The amout of icrease (or decrease) is called the gradiet. $175 $150 $125 $100 $2000 $1500 $1000 $500 G = $25 Base = $100 G = -$500 Base = $2000 EGR 312-04 16 Arithmetic Gradiet Factors (P/G, A/G) Equivalet cash flows: $175 $150 $125 $100 $100 => + $75 $50 $25 G = $25 Base = $100 Note: the gradiet series by covetio starts i year 2. EGR 312-04 17 Arithmetic Gradiet Factors (P/G, A/G) To fid P for a gradiet cash flow that starts at the ed of year 2 ad ed at year : $G $2G $G 0 1 2 3 G (1 1 P i i(1 (1 $P where (P/G,i,) = or P = G(P/G,i,) 1 (1 1 i i(1 (1 EGR 312-04 18 6
Arithmetic Gradiet Factors (P/G, A/G) To fid P for the arithmetic gradiet cash flow: $175 P =? $150 $125 $100 P 1 =? $100 + P 2 =? $75 $50 $25 i = 6% P 1 = P 2 = P = Base(P/A, i, ) + G(P/G, i, ) = EGR 312-04 19 Arithmetic Gradiet Factors (P/G, A/G) To fid P for the decliig arithmetic gradiet cash flow: $2000 $1500 $1000 $500 P 1 =? $2000 - P 2 =? $1500 $1000 $500 i = 10% P 1 = P 2 = P = Base(P/A, i, ) - G(P/G, i, ) = EGR 312-04 20 Arithmetic Gradiet Factors (P/G, A/G) To fid the uiform aual series, A, for a arithmetic gradiet cash flow G: $G $2G $G 0 1 2 3 0 1 2 3 A = G(P/G,i,) (A/P,i,4) = G(A/G,i,) $A Where (A/G,i,) = 1 i (1 1 EGR 312-04 21 7
Geometric Gradiet Factors (P g /A) A Geometric gradiet is whe the periodic paymet is icreasig (decreasig) by a costat percetage: $133 $121 $100 $110 A 1 = $100, g = 0.1 A 2 = $100(1+g) A 3 = $100(1+g) 2 A = $100(1+g) -1 EGR 312-04 22 Geometric Gradiet Factors (P g /A) To fid the Preset Worth, P g, for a geometric gradiet cash flow G: $133 $121 $100 $110 1 g 1 i Pg A 1 1 i g g i $P g Pg A1 i 1 g i EGR 312-04 23 Determiig Ukow Iterest Rate To fid a ukow iterest rate from a sigle-paymet cash flow or uiform-series cash flow, the followig methods ca be used: 1) Use of Egieerig Eco Formulas 2) Use of factor tables 3) Spreadsheet (Excel) a) =IRR(first cell: last cell) b) =RATE(,A,P,F) EGR 312-04 24 8
Determiig Ukow Iterest Rate Example: The list price for a vehicle is stated as $25,000. You are quoted a mothly paymet of $658.25 per moth for 4 years. What is the mothly iterest rate? What iterest rate would be quoted (yearly iterest rate)? Usig factor table: $25000 = $658.25(P/A,i,48) (P/A,i,48) = i = (HINT: start with table 1, pg. 581) 0r aually EGR 312-04 25 Determiig Ukow Iterest Rate Example (cot d) Usig formula: 48 (1 1 $25000 $658.25 48 i(1 48 (1 1 37.9795 48 i(1 Use calculator solver or Excel trial ad error method to fid i. EGR 312-04 26 Determiig Ukow Number of Periods () To fid a ukow umber of periods for a siglepaymet cash flow or uiform-series cash flow, the followig methods ca be used: 1)Use of Egieerig Eco. Formulas. 2)Use of factor tables 3)Spreadsheet (Excel) a) =NPER(i%,A,P,F) EGR 312-04 27 9
Determiig Ukow Number of Periods () Example: Fid the umber of periods required such that a ivest of $1000 at 5% has a future worth of $5000. P = F(P/F,5%,) $1000 = $5000(P/F,5%,) (P/F,5%,) = = EGR 312-04 28 10