Some examples of NPV and IRR

Transcription

1 Economics 623 Human Capital: Lectue 2 Sping 212 Some examples of NPV and IRR The numbe of examples ae legion, especially when one ealizes that loans have the opposite sign values of the cash flows than investment pojects. In a loan, we eceive money in peiod (today) and make payments in the futue. I am in the maket to buy anothe ca. My cuent ca is woth $3, and the ca I wish to puchase is $25,. I can put down $5, and have to boow the est at ate 6.8% fo thity six months. Daw pictue of cash flows Positive inflow at time of $17, and monthly outflows of payments (pmt). To detemine the size of the monthly payments,x, solve fo x: 17, 3 12 i=1 x ( )i = The tem in the denominato eflects that inteest is compounded monthly (applied to the outstanding balance of the loan) while the inteest ate is quoted as an annual ate. My fancy calculato tells me the monthly payments will be 523 dollas pe month (and some change). If you don t have a fancy calculato we can solve the equation diectly. Let n denote the length of the loan in months (3 x 12) and q = /m to be the effective inteest ate pe month. Let P V equal the loan amount (P V eminds us that the loan comes in today and payments ae in the futue). Wite the equation as: P V n i=1 x (1 + q) i = Notice that 1 1+q 1 as q. So the tem 1/(1 + q)j declines as j inceases, which eflects that payments made fathe into the futue ae smalle in today s dollas. Because the monthly payment is constant it can be moved outside the summation sign. Let u = 1/(1 + q), and the equation becomes [ n ] P V = x u j j=1 = x B n (u), B n (u) = x = P V/B n (u) n j=1 u j 1

2 So to find x, we need to only find an expession fo B n (u). Multiply B n (u) by u and substact it fom B n (u) to get Substitute fo u = 1/(1 + q) to yield Substituting in B n (u) = u + u 2 + u u n u B n (u) = u 2 + u u n+1 B n (u) u B n (u) = u u n+1 = u(1 u n ) (1 u)b n (u) = u(1 u n ) B n (u) = u(1 un ) 1 u x = q P V 1 (1 + q) n Continuous Time Flows x = , = = = Recall that inteest compounded n times pe yea such as monthly o quately fo t yeas accumulates as a(t). a(t) = ( 1 + ) n t n Continuous compounding is then the compounding peiod is infinitely small, thus n. ( a(t) = lim n 1 + ) n t = e t n So, an initial amount, A compounded continuously at inteest ate fo t peiods (whee the time units on matches that of t) is A(t) = A a(t) = A e t. 2

3 High School vesus College Conside a peson who just gaduated fom high school. They have an offe of employment that pays them a salay of Y hs (t) (pe unit of time t. They need not make any investment and will wok at this establishment until etiement. (I ealize assuming no job change is unealistic, but notice that to pemit tunove and job change equies that we state when it will occu and its payoff. So fo now, let s abstact fom such complications.) Quantities ae in eal tems, and the assumption is that the high school job pays Y hs (t) = Y hs until age T (say 65). The maket inteest ate is, assumed constant and the Pesent value is: P V hs = T Y hs e t d t T = Y hs e t d t = Y T hs = Y hs [ e T 1 ] = Y hs [ ] 1 e T. But the powe of discounting is such that fo easonable sized values of T and fo, e T. So the pesent value of holding the job foeve is Y hs / but this is just the pice one should be willing to pay fo an asset that pays Y h s pe yea in pepetuity. Now conside the cost of going to college. The (full) flow cost of college is C t = C fo s peiods (e.g., 4 yeas o 5 yeas o...). Again let the eal inteest ate be denoted as and the eanings of a college gaduate be Y c (t) = Y c. The diect benefit is P V c = = C s Ce t d t + = (Y c + C)e s [ 1 e s ] + Y c e s C s Y c e t d t Substituting the PV of High School as the oppotunity cost of going to college, the NPV of obtaining a college degee is 3

4 NP V c = (Y c + C)e s C Y hs Can now investigate compaative statics of the effect of diffeential eanings and out of pocket costs and financial aid. Should wite NP V c as NP V c (s, Y c, C, Y hs, ), but to keep things simple do not. The NP V c inceasing in Y c, and deceasing in C and Y hs, and in s (the longe it takes to go though college the lowe the net pesent value of a college education. What is the inceased cost of taking s + 1 yeas vesus s yeas? At the stat of the college education, NP V c (s) NP V c (s + 1) = 1 (Y c + C)(e s e (s+1) ) which is appoximately (using calculus) d NP V c (s) d s (Y c + C)e s So the inceased cost is the additional yea of cost plus the fogone income as a college gaduate. But the discount facto discounts back to time the stat of the investment pocess. If you decide in the sping of what was to be the last semeste, in contempoay dollas, postponing only one yea, so the appopiate discount facto is e not e s. So, if diect costs ae $15, and eanings as a college gaduate ae $3,, and the inteest ate is 3%, the incemental cost of an additional yea is 45,.97 = 43, 67. Enjoy. What is the effect of an incease in the eal inteest ate? This is the basic model fo undestanding enollments in college. Pedictions follow fom the compaative statics mentioned above. 1. College enollment will decline if costs of college ise (holding othe factos constant) 2. College enollment will incease if the gap between eanings of college gaduates and high school gaduates widens. 3. Most college students will be young. Why? The cost of college inceases with the oppotunity of going to school. Thus, most students will be young befoe thei value in the maket inceases and the oppotunity cost is too lage to make the investment. Thee is a second eason fo why students ae likely to be young, and that is that the investment etun inceases with the length of the payoff peiod. I assumed the payoff peiod was sufficiently long e T so that we could conside the flows inconsequential ( ). Steams ove elatively shot duation of 5 o 1 yeas ae not close to zeo. Thus, have the incentive to invest in college education when 18 because the oppotunity cost of you time is low, and you have the longest peiod to eap the benefits. 4

5 Human capital theoy is a common way to undestand the distibution of eanings. Eanings acoss a wide vaiety of societies have thee common featues: 1. Aveage eanings of full-time wokes ise with the level of education. 2. The most apid incease in eanings occus ealy in one s woking life, thus giving a concave shape to the age eanings pofile of both men and women; 3. Age eanings pofiles tend to fan out, so that education elated eanings diffeences late in wokes lives ae geate than those ealy on; Daw standad concave age eanings pofile. The fist one is obvious and indeed necessay: people with moe education foegone consumption while in school, and must be paid moe late to ecove thei investment. Indeed, we could imagine an economy in which people ae identical, same pefeences, same poductive abilities. Thee can be two jobs, one that equies no schooling and anothe that equies some schooling. Since people ae identical they ae indiffeent which job they do, so thei utility has to be the same. So, induce some people to accept the job that equie some education, eanings in that job has to be highe than in the job without education. Wokes in the job with education ae not bette off, just paid moe when olde and eceive less when young. Stong eanings gowth occus when young? Think of a poduction function, in which human capital depends on schooling and expeience. (Human capital is the acquitision of knowledge and abilities, lean some things in school othes on the job.) In most poduction function, the maginal etuns ae lagest when input levels ae low. Why do eanings fan out? Ask. People who ae moe able o can lean moe quickly will accumulate knowledge faste. And the fast leanes ae likely to be the ones that acquied the most schooling. And it is tue that moe educated wokes ae less likely to be unemployed. If we think of leaning by doing investment pocess while woking, unemployed wokes accumulate less human capital. Pivate Vesus Social Retuns Inceased eanings due to education (and the investment in human capital) o inceased eanings because of moving fom a poo to good labo maket ae examples of pivate etuns. 5