Fiacial Aalysis Lecture 4 (4/12/217)
Fiacial Aalysis Evaluates maagemet alteratives based o fiacial profitability; Evaluates the opportuity costs of alteratives; Cash flows of costs ad reveues; The timig of paymets is importat. Why?
What is discoutig? A process that accouts for time prefereces Coverts future values to preset values Compoudig Preset Value Future Value A value expressed i dollars received immediately Discoutig A value expressed i dollars received at some future time
Defiitio of Discoutig The process of covertig values expressed i dollars received at oe poit i time to a equivalet value expressed i dollars received at a earlier poit i time Compoudig is the reverse process) FUTURE NOW
The iterest rate Time preferece: = huma ature + potetial ivestmets Moey ca make moey over time Corollary: usig moey costs moey The iterest rate determies the relatioship betwee preset ad future values
Iterest rate as a trade-off (the ecoomy of Robiso Crusoe, Buogioro & Gilles 23) Amout ext year (C 1 ) B C 1 * T E 2 C * I* A dc 1 dc r E E * 1 Preset cosumptio (C ) (1 ) Source: Buogioro ad Gilles 23, p. 374
The iterest rate Also: the iterest rate is the percetage of the amout ivested or borrowed that is paid i iterest after oe uit of time V V iv pricipal iterest 1 V 1 V (1 i) Future Value : V V (1 i) Preset Value : V V (1 i) V (1 i)
Discoutig Multiple Paymets/Costs
The Net Preset Value (NPV) The NPV is the preset value of reveues mius the preset value of costs: NPV R1 R2 R... 1 2 (1 i) (1 i) (1 i) C1 C2 C... 1 2 (1 i) (1 i) (1 i)
Cash flows 8 6 4 Chuck's Christmas Tree Farm 4246 7246 Cash flows ($) 2-2 -4-6 -164-11342 Shearig, sprayig, weedig, etc. -8-1 -12 Lad, machiery, seedligs, platig, etc. 1 2 3 4 5 6 7 8 Time (years)
Derivatio of the ifiite aual series formula R R R V... 2 3 1 i 1i 1i 1. Leave $1 i a bak accout forever at a iterest rate of 5%. How much ca you withdraw each year? 2. Aswer: $1*.5=$5/yr 3. I other words: Vi R V R/ i
Ifiite aual series Preset value: The paymet: V R i R iv The iterest: i R V NOW R R R R R R R R
Ifiite series of periodic paymets A a s p e d s t a d 2 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 Reveues ($/ac) 1 5 1 5 4 8 1 2 1 6 2 2 4 2 8 3 2 T im e ( y e a r s ) Let s use the ifiite aual paymet formula, but substitute the aual iterest rate with a 4 year compoud iterest rate: V R i 4 4 R R(1 i) R R R 4 (1 i) 1
Ifiite periodic series Preset value: V R t (1 i) 1 The paymet: R V (1 i) 1 t The iterest: i t R/ V 11
Ifiite series of periodic paymets (the aspe example) So, how much is the preset value of the reveues geerated by the aspe stad at a 6% iterest rate? Solutio: V R $18 (1 i) 1 (1.6) 1 t 4 $19.38
Fiite series of aual paymets Examples: Calculatig regular, aual paymets o a loa for a fix period of time; Calculatig aual ret/tax paymets or maagemet costs for a fix period of time; Or, calculatig mothly paymets. Calculatig mothly iterest rates: 12 1/12 i [ i1] 1 ( i1) 1 i/12 m
Fiite series of aual paymets Derivatio of the formula: R Year V R/ i R (1 i) i(1 i) V / R i V R R R[(1 i) 1] i i(1 i) i(1 i)
Fiite series of aual paymets Pr eset Value: V R[(1 i) 1] i(1 i) Future Value (i year ): V Paymet to achieve a give Preset ad Future Value: Vi (1 i) Vi R (1 i) 1 (1 i) 1 R[(1 i) 1] i
Example You wat to buy a house i Seattle for $5,. You have $1, to put dow, so you get a loa with a 5.% iterest. How much would you have to pay each moth to pay the loa off withi 3 years?
Solutio procedure cot. 1. Covert the aual iterest rate of 5% to a mothly iterest rate im 12 12 [ i1] 1 [.51] 1.474.474% 2. Plug i the mothly iterest rate i the fiite aual paymets formula: R m Vi (1 i ) $4,.474 (1.474) 36 m m 36 (1 im ) 1 (1.474) 1 $743.2517 3.32194 $2,12.22
Fiite series of periodic paymets There is a fixed amout (R) that you receive or pay every t years for years (where is a iteger multiple of t); Example: A itesively maaged black locust stad (Robiia pseudoacacia) is coppiced three times at 2-year itervals. After the third coppice (at age 6), the stad has to be replated. At ages 2, 4 ad 6 yrs the stad produces $1, per acre. Usig a 5% discout rate, what would the preset value of these harvests be?
Solutio procedure What do we kow? 1. R 2 =$1, 2. =6 yrs, t=2 yrs 3. i=5%=.5 What do we eed to kow? Preset Value (V ) What formula to use? Use the fiite aual paymet formula with a 2-year compoud iterest rate.
Solutio procedure cot. First let s calculate the 2 year compoud iterest rate: i (1.5) 2 1 165.3298% 2 Plug i the 2-yr iterest rate ito the fiite aual series formula: V R[(1 i) 1] 3 $1, [(1 1.6533) 1] 3 i(1 i) 1.6533(1 1.6533) $17, 679.23436 $572.47 3.88238
Fiite periodic paymets formula I geeral: V R[(1 i) 1] t [(1 i) 1](1 i) The paymet to achieve a give preset value: R V [(1 i) 1](1 i) t [(1 i) 1]
Discoutig with Iflatio
Defiitio Iflatio: a icrease i average price level, reducig the purchasig power of a uit currecy (deflatio is the reverse process) Iflatio rate: average aual rate of icrease i the price of goods
Measurig Iflatio Cosumer Price Idex (CPI)*: measures the average icrease i the cost of a stadard collectio of cosumer goods (market basket) Producer Price Idex (PPI): measures the average icrease i the cost of a stadard collectio of productio iputs *CPI: the Cosumer Price Idex for All Urba Cosumers (CPI-U) for the U.S. City Average for All Items, 1982-84=1.
The Average Aual Iflatio Rate k k CPI 2 ( t2t1) 1 CPI Example: Calculate the average aual iflatio rate for the last 3 years (1985-215) Solutio: Use the website at http://stats.bls.gov to get CPIs: CPI (215 1985) 215 1 27.8 3 1 CPI1985 15.5 t t 1.2285 2.285%
Compoets of the Iterest Rate The omial rate: icludes both the cost of capital ad iflatio; The real rate: is the rate eared o a ivestmet after accoutig for iflatio. This is the real retur for ivestig oe s moey. the omial rate the iflatio rate + the real rate i k + r
Combiig Iterest Rates Let i = the omial rate; r = the real rate; ad k = the iflatio rate. R(1 r) (1 k) R i (1 r)(1 k) 1 R rkrk rk (1 i) (1 i) i rkrk; r 1; k 1 (1 k) (1 r)
Combiig Iterest Rates Example: You bought a house i 1985 for $12,. I 215 it was appraised at $45,. How much was your real rate of retur o this house if the average aual iflatio rate betwee 1985 ad 215 was 2.285%?
Solutio: Which formula to use? How do we calculate i? r (1 i) (1 k) 1 i (2151985) V V 215 1985 1 $45, 11 1.4544 $12, Calculate r: r 1.4544 1.217 2.2% 1.2285
Deflatig ad Iflatig Deflatig: The process of covertig a value expressed i the currecy of a give poit i time ito a value expressed i the currecy of a earlier time with the same purchasig power ; Iflatig: is the reverse process. Note: Historical iflatio rates are available to iflate past values to the preset.
Compoudig Adjusts for time preferece Preset Value Future Value A value expressed i dollars received immediately Discoutig Iflatig A value expressed i dollars received at some future time Adjusts for the chage i purchasig power A value expressed i dollars with the same purchasig power as dollars today Real Future Value Deflatig Nomial Future Value A value expressed i dollars that have the purchasig power of dollars i the year whe the value occurs
V V * Deflatig ad Iflatig omial value occurig i year, real value occurig i year, ad referece year. Real value: (1 ) ( ) * V k V Nomial value: * (1 ) ( ) V k V Note: Deflatig/iflatig is mathematically same as discoutig/compoudig but coceptually very differet.
Example 1 How much would a salary of $69, i 22 be worth i curret (214) dollars if the forecasted average aual iflatio rate is 4%? Solutio: * 1. What do we kow? V22 $69,, 214 2. What do we eed to kow? V22? 3. Which formula to use? (22214) * V (1.4) V 22 22 6 1.4 $69, $54,531.7
Example 2 How much would a salary of $15, i 1976 be worth i curret dollars (214)? Solutio: 1. What do we kow? V $15,, 214 * 1976 2. What do we eed to kow? V? 3. Which formula to use? V (1 k) V, (1976214) * 1976 1976 where k 1976 2141976 CPI CPI 214 1976 1
Example 2 V 1 CPI 1 214 * 1976 38 1976 CPI 1976 38 V 38 CPI CPI 214 1976 38 V * 1976 CPI CPI 214 1976 V * 1976 236.3 $15, $62,214.41 56.9
Rules of discoutig with iflatio Discout omial future values with a omial rate ad discout real future values with a real rate; Whe a preset value is compouded by a real rate, the the result is a real future value; Whe a preset value is compouded by a omial rate, the the result is a omial future value.
Discoutig with iflatio Other terms: costat, relative or deflated $ Real Future value (V ) deflatig k Other terms: curret, actual or iflated $ Nomial Future Value (V *) r iflatig i=r+k+rk i Preset value (V ) Note: It is ofte hard to tell if a future value is real or omial
A hybrid poplar platatio The platatio ca be established for $6/ac o a lad that ca be reted for $1/ac/year. You expect the lad ret to go up at about the same rate as the iflatio rate (=4%/year). After 7 years, the platatio will produce 2 tos of chips per acre. The curret price for chips is $1/to ad you expect this price to go up at the rate of the iflatio. What is the preset value of the poplar project at a 8% real iterest rate?
The cash flows Cash Flow ($) 2 15 1 5-5 -1 Real Cash Flows of the Poplar Platatio -7-1 -1-1 -1-1 -1 1 2 3 4 5 6 7 k = 4% r = 8% Time (years) 2 Year Future Values Preset Real Nomial Values (costat k (curret ($) values) values) i -7-7. -7. 1-1 -14. -92.59 r 2-1 -18.16-85.63 3-1 -112.49-79.38 4-1 -116.99-73.5 5-1 -121.67-68.6 6-1 -126.53-63.2 7 2 2631.86 1,166.98 NPV N/A N/A 4.69 i rkrk.8.4.8.4.1232 12.32%