CH 28 - 1 CH 28: TIME VALUE OF MONEY I. Basics [1] Definitions of variables t= time index. Ct = cash flow at time t. PMT = the periodic payment = annuity payment (same amount of cash flow). n= number of periods. i= interest rate/ discount rate/ required rate of return/ cost of capital/ opportunity cost per period. FV = Future value FVA = Future value of an annuity. PV = Present value. PVA = Present value of an annuity. [2] Compounding and discounting of interest Interest can be compounded or discounted yearly, quarterly, monthly, weekly, daily, or even continuously. EX) Annual interest rate = 8%. Horizon = 2 years. n i 1) yearly compounding 8% 2 periods 2) semiannual compounding 8% / 2 = 4% 4 periods 3) quarterly compounding 8% / 4 = 2% 8 periods 4) monthly compounding 8% / 12 = (2/3)% 24 periods 5) daily compounding 8% / 365 = (8/365)% 730 periods [3] Cash flows Cash inflow = + value Cash outflow = - value (1) Single cash flow: C (2) Different streams: Ct (3) Ordinary annuity (end of each period) (4) Annuity due (beginning of each period)

PMT

PMT

PMT

PMT

0

1

2

n-1

n

PMT

PMT

PMT

PMT

0

1

2

n-1

[4] Time line Cash flow Discounting (Present value)

Compounding (Future value)

n

CH 28 - 2 II. Future Value : FVIFi ,n= ( 1 + i)

n

[1] Single cash flow: C0=1,000; i=0.03; {Value at t=3}? 0 1 2

3

1,000 {C0’s value at t=1} = 1,000×(1+0.03)1 = 1,030 {C0’s value at t=2} = 1,000×(1+0.03)×(1+0.03) = 1,000×(1+0.03)2 = 1,060.90 {C0’s value at t=3} = 1,000×(1+0.03)×(1+0.03)×(1+0.03) = 1,000×(1+0.03)3 = 1,092.727 {Value at t=3} = 1,000 × FVIF(0.03,3) = 1,000×(1+0.03)3 = 1,092.727 [2] Different streams: C1=1,000; C2=1,300; C3=900; i=0.03; {Value at t=3}? 0 1 2 3 1,000

1,300

900

{C1’s value at t=3} + {C2’s value at t=3} + {C3’s value at t=3} = 1,000× FVIF(0.03,2) + 1,300× FVIF(0.03,1)+ 900 = 1,000×(1+0.03)2 + 1,300×(1+0.03)1 + 900 = 3,299.90 n (1 + i) n − 1 n −t FVIFA = ( 1 + i) = [3] Annuity: ∑ i i,n t =1 PMT=1,000; n=3; i=0.03; {FV of the annuity (value at the end of the annuity)}? 0 1 2 3 1,000

1,000

1,000

{C1’s value at t=3} + {C2’s value at t=3} + {C3’s value at t=3} = 1,000×(1+0.03)2 + 1,000×(1+0.03)1 + 1,000×(1+0.03)0 = 3,090.90 = 1,000 × (1+ 0.03) 2 + (1+ 0.03)1 + (1+ 0.03)0

[

]

⎡ ⎤ ⎡ ⎤ = 1,000 × ⎢∑ (1+ 0.03) t ⎥ = 1,000 × ⎢∑ (1+ 0.03) 3-t ⎥ ⎣ t =0 ⎦ ⎣ t =1 ⎦ (1 + 0.03)3 − 1 = 1,000 × FVIFA(0.03,3) = 1,000 × = 3,090.90 0.03 [4] Annuity due: PMT=1,000; n=3; i=0.03; {FV of the annuity due (value at the end of the annuity due)}? 0 1 2 3 2

3

1,000

1,000

1,000

{C0’s value at t=3} + {C1’s value at t=3} + {C2’s value at t=3} = 1,000×(1+0.03)3 + 1,000×(1+0.03)2 + 1,000×(1+0.03)1 = 3,183.627 (1 + 0.03)3 − 1 = 1,000 × FVIFA(0.03,3) × FVIF(0.03,1) = 1,000 × × (1 + 0.03)1 = 3,183.627 0.03 • Note that 1,000×FVIFA(0.03,3) is the value at t=2

CH 28 - 3 1 ( 1 + i)n [1] Single cash flow: C3=1,000; i=0.03; {Value at t=0}? 0 1 2 III. Present Value : PVIFi ,n=

3 1,000

1 ≈ 970.8737864 (1 + 0.03) 1 1 {C3’s value at t=1} = 1,000 × × ≈ 942.5959091 (1 + 0.03) (1 + 0.03) 1 1 1 . {C3’s value at t=0} = 1,000 × × × ≈ 9151416594 (1 + 0.03) (1 + 0.03) (1 + 0.03) 1 {Value at t=0} = 1,000 × PVIF(0.03,3) = 1,000 × ≈ 915.1416594 (1 + 0.03)3 {C3’s value at t=2} = 1,000 ×

[2] Different streams: C1=1,000; C2=1,300; C3=900; i=0.03; {Value at t=0}? 0 1 2 3 1,000

1,300

900

{C1’s value at t=0} + {C2’s value at t=0} + {C3’s value at t=0} =1,000 × PVIF(0.03,1) +1,300 × PVIF(0.03,2) +900 × PVIF(0.03,3) 1 1 1 = 1,000 × + 1,300 × + 900 × ≈ 3,019.875961 1 2 (1 + 0.03) (1 + 0.03) (1 + 0.03) 3 1 1− n 1 (1 + i ) n PVIFA = = [3] Annuity: ∑ i ,n t i t =1 ( 1 + i) PMT=1,000; n=3; i=0.03; {PV of the annuity (value at the beginning of the annuity)}? 0 1 2 3 1,000

1,000

1,000

{C1’s value at t=0} + {C2’s value at t=0} + {C3’s value at t=0} = 1,000 ×

1 1 1 + 1,000 × + 1,000 × 1 2 (1+ 0.03) (1+ 0.03) (1+ 0.03) 3

⎡ ⎤ ⎡ 3 ⎤ 1 1 1 1 1 000 = 1,000 × ⎢ + + = × , ∑ ⎢ 1 2 3⎥ t⎥ (1+ 0.03) ⎦ ⎣ (1+ 0.03) (1+ 0.03) ⎣ t =1 (1+ 0.03) ⎦ 1 1(1 + 0.03) 3 = 1,000 × PVIFA(0.03,3) = 1,000 × ≈ 2,828.611355 0.03

CH 28 - 4 [4] Annuity due: PMT=1,000; n=3; i=0.03; {PV of the annuity due (value at the beginning of the annuity due)}? 0 1 2 3 1,000

1,000

1,000

{C0’s value at t=0} + {C1’s value at t=0} + {C2’s value at t=0} 1 1 = 1,000 + 1,000 × + 1,000 × 1 (1+ 0.03) (1+ 0.03) 2

⎡ ⎤ 1 1 = 1,000 × ⎢1 + + 1 2⎥ ⎣ (1+ 0.03) (1+ 0.03) ⎦ ⎡ ⎤ 1 1 1 = 1,000 × ⎢ + + × (1 + 0.03) 1 2 3⎥ (1+ 0.03) ⎦ ⎣ (1+ 0.03) (1+ 0.03) ⎡ 3 ⎤ 1 = 1,000 × ⎢∑ × (1 + 0.03) t⎥ ⎣ t =1 (1+ 0.03) ⎦ 1 (1 + 0.03)3 = 1,000 × PVIFA(0.03,3) × FVIF(0.03,1) = 1,000 × × (1 + 0.03) ≈ 2,913.469696 0.03 • Note that 1,000×PVIFA(0.03,3) is the value at t=-1 1 11 (1 + i)∞ IV. Present value of perpetuity = PMT × PVIFA(i, ∞) = PMT × ≈ PMT × i i Exercise: i=0.05 {Value of the following cash flows at time=4}? 0 1 2 3 4 29 30 1-

1,000 2,000 2,000 2,000 2,000 2,000 1,500 1,000 × FVIF (0.05,4) + 1,500 × PVIF (0.05,26) + 2,000 × PVIFA(0.05,29) × FVIF (0.05,4) 1 1 (1 + 0.05) 29 = 1,000 × (1 + 0.05) 4 + 1,500 × + × × (1 + 0.05) 4 ≈ 38,44550647 2 000 , . (1 + 0.05) 26 0.05 Exercise: 30-year mortgage loan; interest 7.75%; Loan =$200,000; Monthly payment? 1 1− 360 ⎛ 0.0775⎞ 1 + ⎜ ⎟ ⎝ 0.0775 12 ⎠ 200,000 = PMT × PVIFA( ,360) = PMT × 12 ⎛ 0.0775⎞ ⎜ ⎟ ⎝ 12 ⎠ 200,000 PMT = = 1,432.824493 ≈ 1,432.82 1 ⎡ ⎤ 360 ⎥ ⎢1 − ⎢ ⎛⎜1 + 0.0775⎞⎟ ⎥ 12 ⎠ ⎥ ⎢ ⎝ ⎢ ⎥ ⎛ 0.0775⎞ ⎜ ⎟ ⎢ ⎥ ⎝ 12 ⎠ ⎢ ⎥ ⎢⎣ ⎥⎦ 1−

CH 28 - 5 V. Summary [1] Interest factors (1) FVIFi,n = (1 + i) n ≥ 1 0

Future value interest factor (future value of $1 at the end of n periods). 2 n-1

1

n

C (1 + i ) n − 1 ≥ n. i

n

(2) FVIFAi,n = ∑ (1 + i ) n−t = t =1

0

1

(3) PVIFi,n =

PMT

1 ≤1 (1 + i ) n

0

(Sum of annuity of $1 per period for n periods). n-1 n

2

PMT

Future value interest factor for an annuity

PMT

PMT

Present value interest factor (Present value of $1 due n period in the future). 2 n-1 n

1

C

(4) PVIFAi,n =

n

1

∑ ( 1 + i) t =1

0

t

1− =

1 PMT

1 (1 + i) n ≤n i

Present value interest factor for an annuity (present value of an annuity of $1 per period for n period). n-1 n

2 PMT

(5) PV of perpetuity = PMT × PVIFAi,∞ = PMT × [2] Basic equations (1) FV = C x FVIFi,n : (2) FVA = PMT x FVIFAi,n : (3) PV = C x PVIFi,n : (4) PVA = PMT x PVIFAi,n :

PMT

PMT

PMT 1 = i i

future value of a single cash flow. future value of an annuity. (At the same point of time with the last receipt/payment) present value of a single cash flow. present value of an annuity. (One period prior to the first receipt/payment)

CH 28 - 6 [Exercise problem] A project costs $6M per year for 5 years, starting immediately. You reckon that it will produce an cash inflow after operating costs of $4M a year for 15 years, starting 5 years from now. The opportunity cost of capital is 10 percent. 0

1

2

3

4

5

6

7

8

17

18

19

-6

-6

-6

-6

-6

+4

+4

+4

+4

+4

+4

+4

a. What is the present value of costs? 1− 6M × PVIFA(0.10,5) × FVIF(0.10,1) = 6M ×

1 (1 + 0.10)

5 1

× (1 + 0.10) = $25,019,192.68

0.10

b. What is the present value of the cash inflows? 1− 4M × PVIFA(0.10,15) × PVIF(0.10,4) = 4M ×

1 15

(1 + 0.10) 0.10

×

1 = $20,780,218.58 (1 + 0.10) 4

c. Based on these cost and cash inflow estimates, what is your recommendation? Because the present value of the costs is greater than the present value of cash inflows, the project should be rejected. 28-4

Your grandmother has asked you to evaluate two alternative investments for her. The first is a security that pays $50 at the end of each of the next 3 years, with a final payment of $1,050 at the end of Year 4. This security costs $900. The second investment involves simply putting the same amount of money in a bank savings account that pays an 8 percent nominal (quoted) interest rate, but with quarterly compounding. Your grandmother regards the two investments as being equally safe and liquid, so the required effective annual rate of return on the security is the same as that on the bank deposit. She has asked you to calculate the value of the security to help her decide whether it is a good investment. What is its value relative to the bank deposit? One period = 1 quarter; i = 0.08/4 = 0.02; C4 = 50, C8 = 50, C12 = 50, C16 = 1050 PV = 50 × PVIF(0.02,4) + 50 × PVIF(0.02,8) + 50 × PVIF(0.02,12) + 1,050 × PVIF(0.02,16) = 50 ×

1

28-5

+ 50 ×

1

8

+ 50 ×

1

12

+ 1,050 ×

1

= $893.1595531 < $900 16 (1 + 0.02) (1 + 0.02) (1 + 0.02) (1 + 0.02) Because the present value is smaller than the price of the security, the security is not a good investment. 4

Assume that your father is now 55 years old, that he plans to retire in 12 years, and that he expects to live for 20 years after he retires, that is, until he is 87. He wants a fixed retirement income that has the same purchasing power at the time he retires as $60,000 has today (he realizes that the real value of his retirement income will decline year by year after he retires, but he wants level payments during retirement anyway). His retirement income will begin the day he retires, 12 years from today, and he will receive 20 annual payments. Inflation is expected to be 5 percent per year from today forward. He currently has $100,000 in savings, and he expects to earn a return on his savings of 8 percent per year, annual compounding. To the nearest dollar, how much must he save during each of the next 12 years (with deposits being made at the end of each year) to meet his retirement goal? 55

56

57

66

67

+D

+D

+D

+D

68

86

+100,000 -60,000(1+0.05)12 -60,000(1+0.05)12 -60,000(1+0.05)12

87

CH 28 - 7 a.

The value of the 20 retirement incomes at age of 67: 60,000 × (1 + 0.05)12 × PVIFA(0.08,20) × FVIF(0.08,1) 1 1− (1 + 0.08) 20 = 60,000 × (1 + 0.05)12 × × (1 + 0.08)1 = 1,142,552.44 0.08 b. Value of 100,000 at 67: 100,000 × FVIF(0.08,12) = 100,000 × (1+0.08)12 = 251,817.0117

c.

(Value of 12 deposits at 67) + (Value of 100,000 at 67) = (Value of 20 retirement incomes at 67) (Value of 12 deposits at 67) = (Value of 20 retirement incomes at 67) - (Value of 100,000 at 67) (Value of 12 deposits at 67) = 1,142,552.44 – 251,817.0117 = 890,735.428 Deposit × FVIFA(0.08,12) = 890,735.428 890,735.428 890,735.428 = Deposit = = 46,937.31846 FVIFA(0.08,12) ⎡ (1 + 0.08)12 − 1 ⎤ ⎢ ⎥ 0.08 ⎣ ⎦

Finance 602: Review Problems 1. Calculate the following values. a. The present value of $100 received in year 10 at a 11 percent discount rate. b. The present value of $100 received in year 15 at a 17 percent discount rate. c. The present value of $100 received in each of year 1 through year 20 at a 22 percent discount rate. d. The present value of $100 received in each of year 3 through year 12 at a 13 percent discount rate. 2. A project costs $200,000 per year for 5 years, starting immediately. You reckon that it will produce an cash inflow after operating costs of $170,000 a year for 10 years, starting 3 years from now. The opportunity cost of capital is 17 percent. a. What is the present value of costs? b. What is the present value of the cash inflows? c. Based on these cost and cash inflow estimates, what is your recommendation? 3. As winner of a breakfast cereal competition, you can choose one of the following prizes: (a) $100,000 now (b) $180,000 at the end of 5 years (c) $11,400 a year forever (d) $19,000 for each of 10 years If the interest rate is 13 percent, which is the most valuable prizes? [Hint: calculate the present values] 4. Siegfried Basset is 65 years of age and has a life expectancy of 12 years. He wishes to invest $20,000 in an annuity that will make a level payment at the end of each year until his death. If interest rate is 8.5 percent, what income can Mr. Basset expect to receive each year? 5. James and Helen Turnip are saving to buy a boat at the end of 5 years. If the boat will cost $20,000 and they can earn 11 percent a year on their savings, how much do they need to put aside at the end of years 1 through 5? 6. Kangaroo Autos is offering free credit on a new $10,000 car. You pay $1000 down and then $300 a month for the next 30 months. Turtle Motors next door does not offer free credit but will give you $1000 off the list price. If the rate of interest is 11 percent a year, which company is offering the better deal? [Hint: calculate the present value of payments] 7. How long will it take your investment to triple its value at 13 percent interest? 8. For an investment of $1000 today, the Tiburon Finance Company is offering to pay you $1600 at the end of 8 years. What is the annually compounded rate of interest?