# Lesson 1. Net Present Value. Prof. Beatriz de Blas

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1 Lesson 1. Net Present Value Prof. Beatriz de Blas April 2006

2 1. Net Present Value 1 1. Introduction When deciding to invest or not, a rm or an individual has to decide what to do with the money today. So we need to compare money today with money in the future. What s the relationship between \$1 today and \$1 tomorrow? Is it worth the same \$1 today as \$1 tomorrow/yesterday? time t time t + 1 \$1! \$1? more? less? This is what is called the time-value-of-money concept.

3 1. Net Present Value 2 2. The one-period case Three equivalent concepts: future value, present value and net present value. Example (page 61 in book): a nancial analyst at a leading real estate rm is thinking about recommending that Kaufman & Broad invest in a piece of land that costs \$85,000. She is certain that next year the land will be worth \$91,000, a sure \$6,000 gain. Given that the guaranteed interest rate in the bank is 10%, should Kaufman & Broad undertake the investment in land? Cash inflow \$91,000 Time 0 1 Cash outflow \$85,000

4 1. Net Present Value Future value (or compound value): the value of a sum after investing over one or more periods. If the money is invested in the bank, next year they would have since then \$85; 000 (1 + 0:1) = \$93; 500 future value \$93; 500 > \$91; 000 Invest everything in the bank. The general formula is where F V = C 0 (1 + r) (1) C 0 is cash ow today (at time 0), and r is the appropriate interest rate.

5 1. Net Present Value Present value: the amount of money that should be put in the alternative investment (the bank) in order to obtain the expected amount next year. In our example, present value is, solving for P V P V (1 + 0:1) = \$91; 000 P V = \$91; 000 1:1 = \$82; 727:27 since present value \$82; 727:27 < \$85; 000 then Do not to buy the land.

6 1. Net Present Value 5 The general formula is where P V = C r (2) C 1 is cash ow at date 1; and r is the appropriate interest rate (the one required to buy the land) also called discount rate.

7 1. Net Present Value Net present value: the present value of future cash ows minus the present value of the cost of the investment. The formula is NP V = P V Cost: (3) In our case, we would have NP V = \$91; 000 1:1 \$85; 000 = \$2; 273 Because NP V < 0 Purchase of the land should not be recommended. NOTE: all three methods reach the same conclusions.

8 1. Net Present Value 7 3. The multiperiod case 3.1 Future value and compounding General formula where F V = C 0 (1 + r) T (4) C 0 is cash ow at date 0; r is the appropriate interest rate, and T is the number of periods over which the cash is invested. Example: \$1:10 of dividend, expected to grow at a 40% per year over the next 2 years, then in 2 years the dividend will be F V = C 0 (1 + r) T = \$1:10 (1:40) 2 = \$2:156

9 1. Net Present Value 8 Notice that (1 + 0:40) 2 = 1 + 0: :40 (5) Compounding: the process of leaving money in the capital market and lending it for another year. Notice that \$2:156 > \$1: [\$1:10 0:40] = \$1:98 Simple interest: interest payments are not reinvested, i.e. T r: In our example, 2 0:40 = 0:80 Compound interest: each interest payment is reinvested, i.e. r T : In our example, then 0:40 2 = 0:16 \$2:156 = \$1:10 (1 + 0:16 + 0:80)

10 1. Net Present Value 9 What rate is enough? Alternatively, we may not know the interest rate. Example: Assume the total cost of a college education will be \$50; 000 when your child enters college in 12 years. You have \$5; 000 to invest today. What rate of interest must you earn on your investment to cover the cost of your child s education? F V = C 0 (1 + r) T \$50; 000 = \$5; 000(1 + r) 12 \$50; 000 \$5; 000 = (1 + r)12 r = 50; 000 5; 000! = 0:2115

11 1. Net Present Value Present value and compounding Discounting: the process of calculating the present value of a future cash ow. It is the opposite of compounding. Example: how much would an investor have to set aside today in order to have \$20; 000 ve years from now if the current rate is 15%? P V (1 + r) T = C T! P V = C T (1 + r) T P V (1 + 0:15) 5 = \$20; 000 P V = \$20; 000 = \$9; 943:53 (1:15) 5 Present value factor: the factor used to calculate the present value of a future cash ow. 1 (1 + r) T :

12 1. Net Present Value Multiperiod net present value NPV of a T period project can be written as NP V = C 0 + C r + C 2 (1 + r) 2 + ::: + C T (1 + r) T NP V = C 0 + X T i=1 C i (1 + r) i: (6)

13 1. Net Present Value Compounding periods Compounding an investment m times a year for T years provides for future value of wealth where F V = C r m mt (7) C 0 is the initial investment, r is the stated annual interest rate (also called annual percentage rate), m is how many times a year the investment is compounded, and T is the total number of periods of the investment.

14 1. Net Present Value 13 Example: if you invest \$50 for 3 years at 12% compounded semiannually, your investment will grow to F V = \$ : = \$50(1+0:06) 6 = \$70:93 Example: what is the end-of-year wealth of \$1 invested for one year at a stated annual interest rate of 24% compounded monthly? F V = \$ : = \$1(1+0:02) 12 = \$1:2682 Example: what is the wealth at the end of 5 years of investing \$5; 000 at a stated annual interest rate of 12% per year, compounded quarterly? F V = \$5; : = \$5; 000(1 + 0:03) 20 = \$9; 030:50

15 1. Net Present Value 14 E ective annual interest rates: the annual rate that would give us the same end-of-investment wealth after T years. In the example above, \$50(1 + EAR) 3 = \$70: :93 3 EAR = 1 = 0:1236 = 12:36%: 50 So, investing at 12:36% compounded annually is the same as investing at 12% compounded seminannually. That is, (1 + r EAR ) = 1 + r m m (8) r EAR = 1 + r m m 1: (9)

16 1. Net Present Value 15 Continous Compounding The general formula for the future value of an investment compounded continuously over many periods can be written as where F V = C 0 e rt = C 0 exp(rt ) (10) C 0 is cash ow at date 0; r is the stated annual interest rate, T is the number of periods over which the cash is invested.

17 1. Net Present Value Simpli cations Perpetuity: forever. a constant stream of cash ows that lasts C C C P V = C (1 + r) + C (1 + r) 2 + C (1 + r) 3 + ::: = C r : (11) Example: the British bonds called consols. The present value of a consol is the present value of all of its future coupons.

18 1. Net Present Value 17 Example: consider a perpetuity paying \$100 a year. If the relevant interest rate is 8%; what is the value of the consol? P V = \$100 = \$1; 250 0:08 What is the value of the consol if the interest rate goes down to 6%? P V = \$100 0:06 = \$1; 666:67 Note: the value of the perpetuity rises with a drop in the interest rate, and vice versa.

19 1. Net Present Value 18 Annuity: a stream of constant cash ows that lasts for a xed number of periods. C C C C T P V = C (1 + r) + C (1 + r) 2 + C (1 + r) 3 + ::: C (1 + r) T P V = C r " 1 # 1 (1 + r) T (12) (13) Intuition: an annuity is valued as the di erence between two perpetuities: one that starts at time 1 minus another one that starts at time T + 1:

20 1. Net Present Value 19 Example: If you can a ord a \$400 monthly car payment, how much car can you a ord if interest rates are 7% on 36 month loans? P V = \$400 0: : = \$12; 954:59 Example: What is the present value of a 4 year annuity of \$1000 per year that makes its rst payment 2 years from today if the discount rate is 9%? First, let s bring all the value to the starting time of the annuity P V = \$ \$1000 1:09 + \$1000 1: \$1000 1:09 3 = \$3531:29 this amount today is P V = \$3531:29 1:09 2 = \$2972:22

21 1. Net Present Value Net present value: rst principles of - nance Making consumption choices over time: the intertemporal budget constraint So, given P V (consumption) P V (total wealth) Initial assets A 0 Annual wage w t at any period t Real interest rate r Living from t = 0 to t = T

22 1. Net Present Value 21 and individual s consumption should be such that c 0 + c r +:::+ c T (1 + r) T A 0+w 0 + w r +:::+ w T (1 + r) T Let s consider a person who lives for two periods. This person s budget constraint can be explained with the net present value. Assumptions: no initial assets, income w 0 and w 1 ; r is the real interest rate in the nancial market.

23 1. Net Present Value 22 This can be represented c1 w1 Slope: (1+r) w0 c0 c 0 + c r w 0 + w r if all is consumed today (i.e. c 1 = 0), then c 0 = w 0 + w r but if all is saved and consumed tomorrow (i.e. c 0 = 0), then c 1 = w 0 (1 + r) + w 1 ; which are the intercepts to plot the budget constraint.

24 1. Net Present Value 23 Changes in the interest rate change the slope of the budget constratint. Changes in income shift the budget constraint. Making consumption choices over time: consumer preferences Individual s preferences can be represented by utility functions. Indi erence curves (IC): combinations of goods that give the consumer the same level of happiness or satisfaction. Some properties: higher IC are preferred to lower IC IC are downward sloping (the slope is the marginal rate of substitution)

25 1. Net Present Value 24 IC do not cross each other IC are bowed inward (convex) Making consumption choices over time: optimal consumption choice Individuals choose consumption intertemporally by maximizing their utility subject to the budget constraint. For our example, the problem can be written as follows: subject to maxu(c c 0 ;c 0 ; c 1 ) 1 c 0 + c r w 0 + w r :

26 1. Net Present Value 25 Illustrating the investment decision Consider an investor who has an initial endowment of income, w 0 ; of \$40; 000 this year and w 1 = \$55; 000 next year. Suppose that he faces a 10% interest rate and is o ered the following investment: \$30,000 \$25,000 r = 0.1 First of all, should the individual undertake the project? Compute NPV \$30; 000 NP V = \$25; :1 = \$2; 272:72 > 0! YES!

27 1. Net Present Value 26 Budget constraint without investment that is c 1 + c 0 (1 + r) = w 0 (1 + r) + w 1 ; c 1 + 1:1c 0 = \$99; 000 Budget constraint with investment invests at t = 0 he will have left Notice that if he w 0 \$25; 000 c 0 : At time t = 1 he will get w 1 + \$30; 000: Then the budget constraint becomes c 1 + c 0 (1:1) = = (\$40; 000 \$25; 000)(1:1) + \$55; \$30; 000 that is c 1 + 1:1c 0 = \$101; 500

28 1. Net Present Value 27 If we plot both budget constratints we obtain C 1 101,500 Slope= (1+0.1) 99,000 90,000 92, C 0 That is, the consumption possibilities frontier is expanded if we undertake this investment project, which has a NPV>0.

29 1. Net Present Value 28 Where will the consumer stay? We need some assumption on the preferences. Assume u(c 0 ; c 1 ) = c 0 c 1 : Then the problem to solve is subject to max c 0 ;c 1 c 0 c 1 c 1 + 1:1c 0 = \$99; 000 in the case of no investment. Optimal solution (A) is: (c 0 ; c 1 ) = (\$45; 000; \$49; 500): If we consider the investment opportunity, the problem to solve is subject to max c 0 ;c 1 c 0 c 1 c 1 + 1:1c 0 = \$101; 500

30 1. Net Present Value 29 and the optimal solution (B) is: (c 0 ; c 1 ) = (\$46; 136:36; \$50; 749:99): If we plot it, we have C 1 101,500 Slope= (1+0.1) 99,000 B A 90,000 92, C 0 The investment decision allows the individual to reach a higher indi erence curve, and therefore, he is better o.

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