Chapter 2 Time value of money

Transcription

1 Chapter 2 Time value of money Interest: the cost of money Economic equivalence Interest formulas single cash flows Equal-payment series Dealing with gradient series Composite cash flows. Power-Ball Lottery

2 Decision dilemma take a lump sum or annual installments A suburban Chicago couple won the Power-ball. They had to choose between a single lump sum $104 million, or $198 million paid out over 25 years (or $7.92 million per year). The winning couple opted for the lump sum. Did they make the right choice? What basis do we make such an economic comparison?

3 Option A (lump sum) Option B (installment plan) $104 M $7.92 M $7.92 M $7.92 M 25 $7.92 M

4 What do we need to know? To make such comparisons (the lottery decision problem), we must be able to compare the value of money at different point in time. To do this, we need to develop a method for reducing a sequence of benefits and costs to a single point in time. Then, we will make our comparisons on that basis.

5 End-of-period convention 0 1 Beginning of Interest period End of interest period 0 1

6 Methods of calculating interest Simple interest: rarely used; but the point is that there are different types of interest Compound interest: what people generally mean when they say interest

7 Investment variables P present value F future value I total interest i interest rate N number of interest periods n identifier of an interest period B n balance at the end of interest period n

8 Compounding process $1,080 $1, $1, $1,000 $1,080 2 $1,

9 $1, $1,000 F = $1,000( ) = $1,

10 Compound interest formula n = 0: P n = 1: F = P(1 + i) 1 n = 2: F = F(1 + i) = P(1 + i) 2 1 M n = N : F = P(1 + i) N The fundamental law of engineering economy 2 (1 + i) N compound amount factor

11 Practice problem: Warren Buffett s Went public in 1965: $18 per share Worth today (August 22, 2003): $76,200 Annual compound growth: 24.58% Current market value: $ Billion If he lives till 100 (current age: 73 years as of 2003), his company s total market value will be? Berkshire Hathaway

12 Market value Assume that the company s stock will continue to appreciate at an annual rate of 24.58% for the next 27 years. F = P(1 + i) N = $100.36( ) 27 = $37,902 $ trillion

13 Practice problem If you deposit $100 now (n = 0) and $200 two years from now (n = 2) in an account that pays 10% interest, how much would you have at the end of year 10?

14 Solution F $100 $200 $100(1+0.10) 10 = $100(2.59) = $259 $200(1+0.10) 8 = $200(2.149) = $429 F = $259 + $429 = $688

15 Practice problem Consider the following sequence of deposits & withdrawals over a period of 4 years. If you earn 10% interest, what would be the balance at the end of 4 years? 0 1 $1,000 $1,000 $1, $1,500 4?

16 $1,210? $1,000 $1,000 $1,100 $1,000 $2,100 $2,310 -$1,210 $1,100 $1,500 $1,210 + $1,500 $2,710 $2,981

17 Solution End of Period Beginning balance Deposit made Withdraw Ending balance n = 0 0 $1,000 0 $1,000 n = 1 $1,000( ) =$1,100 n = 2 $2,100( ) =$2,310 n = 3 $1,100( ) =$1,210 n = 4 $2,710( ) =$2,981 $1,000 0 $2,100 0 $1,210 $1,100 $1,500 0 $2, $2,981

18 Economic equivalence What do we mean by economic equivalence? Why do we need to establish an economic equivalence? How do we establish an economic equivalence?

19 Economic equivalence Economic equivalence exists between cash flows that have the same economic effect and could therefore be traded for one another. Even though the amounts and timing of the cash flows may differ, the appropriate interest rate makes them equal.

20 Equivalence in personal finance F If you deposit P dollars today for N periods at i, you will have F dollars at the end of period N. 0 F = P( 1+ i) N N P F P

21 Alternate way of defining equivalence P F dollars at the end of period N is equal to a single sum P dollars now, if your earning power is measured in terms of interest rate i. 0 N F P = F(1 + i) N (1 + i) -N present worth factor 0 N

22 Practice problem At 8% interest, what is the equivalent worth of $2,042 now, 5 years from now? $2,042 If you deposit $2,042 today in an account that pays 8% interest annually how much would you have at the end of 5 years? F

23 Solution F = $2,042( ) 5 = $3,000 Using interest tables: (F/P,8%,5) = $2,042 X = $3,000

24 At what interest rate would these two amounts be equivalent? $2,042 i =? $3,

25 Equivalence between two cash flows Step 1: Determine the base period, say, year 5. Step 2: Identify the interest rate to use. Step 3: Calculate equivalence value. $2,042 0 $3,000 5 i i i 5 = 6%, F = $2, 042( ) = $2, = 8%, F = $2, 042( ) = $3, = 10%, F = $2, 042( ) = $3,289

26 Example - equivalence Various dollar amounts that will be economically equivalent to $3,000 in 5 years, given an interest rate of 8%. $3,000 P = = $2,042 5 ( ) P F $2,042 $2,205 $2,382 $2,572 $2,778 $3,

27 Example $200 V $100 $80 $120 $150 $ Compute the equivalent lump-sum amount at n = 3 at 10% annual interest.

28 Approach V $200 $100 $80 $120 $150 $

29 V V 3 = $ = $ $200 $200( ) -1 + $100( ) -2 = $ $100 $80 $120 $150 $ $100( ) 3 + $80( ) 2 + $120( ) + $150 = $511.90

30 Practice problem How many years would it take an investment to double at 10% annual interest? log 2 F = 2 P= P( ) 2= 1.1 N = = = N N log1.1 log 2 log years N 0 P 2P N =?

31 Rule of 72 Approximately how long it will take for a sum of money to double N = 72 interest rate (%) = 7.2 years

32 Practice problem You just purchased 100 shares of stock at $60 per share. You will sell when the market price has doubled. If you expect the stock price to increase 20% per year, how long do you expect to wait until selling?

33 Practice problem Given: i = 10%, Find: C that makes the two cash flow streams to be indifferent A B $1,000 $ C C

34 Approach Step 1: Select the base period to use, say n = 2. Step 2: Find the equivalent lump sum value at n = 2 for both A and B. Step 3: Equate both equivalent values and solve for unknown C. A B $1,000 $ C C

35 Solution A V 2 = $500( ) 2 + $1,000(+0.0) -1 = $1, A $500 $1, B V 2 = C( ) + C = 2.1C C C To find C: B 2.1C = $1, C = $

36 Practice problem At what interest rate would you be indifferent between the two cash flows? A B $1,000 $ $502 $502 $

37 Approach Step 1: Select the base period to compute the equivalent value (say, n = 3) Step 2: Find the net worth of each at n = 3. A $ $1,000 B $502 $502 $

38 Establish equivalence at n = 3 3 Option A : F3 = $500(1 + i) + $1,000 2 Option B : F3 = $502(1 + i) + $502(1 + i) + $502 Find the solution by trial and error, say i = 8% 3 Option A : F3 = $500(1.08) + $1,000 = $1,630 2 Option B : F3 = $502(1.08) + $502(1.08) + $502 = $1,630