Chapter 15: Time Value of Money

Chapter 5: Time Value of Money I. Introduction A. Why does finance worry about time value of money? 1. Most financial decisions involve costs and benefits that are spread out over time = => Need to adjust cash flows for time value of money 2. Firms have to decide between projects that generate cash flows in different periods of times. Time value of money allows comparison of these cash flows. B. Main ideas 1. Future value (FV) of a cash flow a. Shows compounding or growth over time b. Shows what periodic payments have to be made at given interest rate so that desired sum of money can be obtained at specified future date c. Notation:

i. Let PV0 be present value or the beginning amount at time 0 (measured in dollars)

ii. Let FVn = future value at the end of “n” periods (measured in dollars) iii. Let k be interest rate iv. Let n be number of periods interest is compounded.

d. Formula to determine FVn

n FVn =PV 0 (1+k)

Financial tables list FVIFk,N = future value interest for one dollar compounded annually at k percent for n

N periods. FVIFk,N = (1+k) 2. Present value (PV) of cash flow a. Present value is current dollar value of a future amount of money b. Main idea: a dollar today is worth more than a $1 tomorrow c. It is amount today that must be invested at given interest rate to reach future amount d. Known as discounting = Reverse of compounding. Future value is discounted back to present value e. Interest rate = discount rate = opportunity cost = required return = cost of capital f. Formula to determine present value

FV PV = n 0 (1+k)n

Financial tables list PVIFk,N = present values interest factors for one dollar discounted at k percent for N

N periods. PVIFk,N = 1/(1+k) 3. Example: Cash flow received over 5 years. k = 5%

Year 0 1 2 3 4 5 Cash Flow $50 $100 $150 $200 $250 $300

Cash Flow Present Value Cash Flow Future Value Year 0 $50/(1.05)0= $50.00 Year 0 $50(1.05)5= $63.81 Year 1 $100/(1.05)1= $95.24 Year 1 $100(1.05)4= $121.55 Year 2 $150/(1.05)2= $136.05 Year 2 $150(1.05)3= $173.64 Year 3 $200/(1.05)3= $172.77 Year 3 $200(1.05)2= $220.50 Year 4 $250/(1.05)4 $238.10 Year 4 $250(1.05)1= $262.50 Year 5 $300/(1.05)5= $235.06 Year 5 $300(1.05)0= $300.00

PV0= $927.21 FV5= $1,142.01

II. Annuities, perpetuities, and mixed stream cash flows A. Annuities 1. Comments a. Annuities are equally-spaced cash flows of equal size

b. Annuities can be either cash inflows or cash outflows c. An ordinary (deferred) annuity has cash flows that occur at the end of each period d. An annuity due has cash flows that occur at the beginning of each period 2. Ordinary annuities a. Notation: i. Let C be the equal-sized cash inflow ii. Let N be the number of periods iii. Let k be the interest rate b. Present value of ordinary annuity

Year 0 1 2 3 . . . N Cash flow 0 C C C C

i. Math:

C C C C Key Equation: PV = + + + ⋯ + 0 (1+k)1 (1+k) 2 (1+k) 3 (1+k) N

N1 N PV =C =C (1+k)-i 0 i  i=1(1+k) i=1

Now some mathematical manipulation of above equation: First factor out C to get:

1 1 1 1 PV = C[ + + + ⋯ + ] 0 (1+k)1 (1+k) 2 (1+k) 3 (1+k) N

1 Now factor out to obtain: (1+k)

1 1 1 1 PV =C [1 + + + ⋯ + ] 0 (1+k) (1+k)1 (1+k) 2 (1+k) N-1

1 Next set d = to get: (1+k)

2 N-1 PV0 =Cd [1 + d + d + ⋯ + d ] (†)

Note 0 < d < 1.

Now a digression about infinite geometric series and truncated geometric series: 1. First, infinite geometric series. Let d be a fraction such that 0 < d < 1.

1 Theorem: 1 + d + d2 + d 3 d 4  ⋯ = 1 - d

Proof: X = 1 + d + d2 + d 3 d 4  ⋯ (*)

Multiply both sides of equation (*) by d to get:

dX = d + d2 + d 3 + d 4 d 5  ⋯ (**)

Subtract left-hand side of (**) from left-hand side of (*), subtract right-hand side of (**) from right-hand side of (*) to obtain:

X - dX = 1

Dividing both sides by 1 – d obtains:

1 X = 1+d+d2 +d 3 +d 4 +⋯ = QED! 1-d 2. Next truncated geometric series.

1-dN Theorem: Let 0 < d < 1, then 1+d+d2 + +d N-1 = 1-d

Proof:

1 =1+d+d2 +d 3 +⋯ +d N-1 +d N +d N+1 + ⋯ (*) 1-d

Multiply both sides of equation (*) by dN to obtain:

dN =dN +d N+1 +d N+2 +d N+3 +⋯ +d 2N-1 +d 2N +d 2N+1 + ⋯ (**) 1-d

Again subtract left-hand side of (**) from left-hand side of (*) and right-hand side of (**) from right-hand side of (*) to get:

1-dN =1+d+d2 +d 3 +⋯ +d N-1 1-d

Equation (†) on page 4 and the above theorem on truncated geometric series implies that:

N N 骣 1 骣 1 1-琪 1-琪 2 N-1 1 桫(1+k) 桫1+k PV =Cd [1 + d + d + ⋯ + d ]=C =C 0 1 (1+k)1- k (1+k)

ii. Financial tables: On page 118 of the text the author writes that PVAN = PMT x (PVIFAk,N) where PVAN is the present value of the annuity, PMT = C or the equal-sized cash flow, and PVIFAk,N is the present value interest factor for a one-dollar annuity discounted at k percent for N years. Note:

N 1  N 1-  -i 1+k  PVIFAk,N = (1+k) = i=1 k

For example: PVIFA5%,10 = 7.722

10 1  1-  1.05   =7.721735 .05

iii. Using Microsoft Excel Function: =PV(rate, Nper, pmt, FV, type) where rate = interest rate (10% = 0.10) Nper = N pmt = C = equal sized cash flow FV = future value (no value given in this case) type = 1 if annuity due. = 0 or omitted if ordinary annuity Example: PV(0.05, 10, -1, , )=$7.72

c. Example: What is the present value of an ordinary annuity paying $2,000 each of three years using an annual interest rate of 10%?

Year 0 1 2 3 Cash flow $2,000 $2,000 $2,000

i. Math:

$2000 $2000 $2000 PVA = + + 3 (1.10) (1.10)2 (1.10) 3

PVA3 =$1818.18+1652.89+1502.63=$4,973.70 ii. Financial table

PVA3=pmt x PVIFA10%,3 = (2000)x(2.487)=$4,974 iii. Microsoft Excel function: =PV(0.10,3,-2000, , )=$4,973.70 d. Future value of annuity i. Math

Year 0 1 2 3 . . . N-1 N Cash flow C C C C C

N-1 N-2 N-3 FVAN =C(1+k) +C(1+k) +C(1+k) +⋯ +C(1+k)+C

N N-t FVAN =C (1+k) t=1

ii. Financial tables: See page 116 of text

FVAN = PMT x FVIFAi,N PMT = C = equal-sized annuity payment

FVIFAk,N = future value interest factor for an annuity.

N N-t Note : FVIFAk,N = (1+k) t=1 Can show that:

1 FVIFA = [(1+k)N -1] k,N k

iii. Microsoft Excel Function =FV(rate, Nper, pmt, PV, type) where rate = interest rate in decimal form Nper = N pmt = C PV = present value, omitted in this case Type = 1 if annuity due, 0 or omitted if not e. Example of future value: Assume an annuity pays $2,000 at the end of every year for a three year period. Using a 10% interest rate, what is the future value of this cash flow at the end of the third year?

Year 0 1 2 3 Cash flow $2,000 $2,000 $2,000

i. Math:

2 1 FVA3 =2000(1.10) +2000(1.10) +2000

FVA3 =2420.00+2200+2000=$6,620.00

ii. Financial tables:

FVA3=pmt x FVIFA10%,3 = (2000) (3.310)=$6,620.00 iii. Excel function: = FV(0.10,3,-2000, , ) =$6,620.00 3. Annuity Due vs. Ordinary Annuity Compare present values and future values of both a annuity due and an ordinary annuity with periodic payments of $1,000, annual interest rate of 7%, and 5 year maturity.

0 1 2 3 4 5 Cash flow: $0 $1,000 $1,000 $1,000 $1,000 $1,000 Ordinary annuity Cash flow: $1,000 $1,000 $1,000 $1,000 $1,000 $0 Annuity due

Ordinary annuity Annuity due Present value at year 0 $4,100.20 $4,387.21 Future value at end of year 5 $5,750.74 $6,153.29

For the ordinary annuity: $1000 $1000 $1000 $1000 $1000 PVA = + + + + =$4,100.20 5 (1.07)1 (1.07) 2 (1.07) 3 (1.07) 4 (1.07) 5

4 3 2 1 FVA5 =$1000(1.07) +$1000(1.07) +$1000(1.07) +$1000(1.07) +$1000=$5,750.74

For the annuity due: $1000 $1000 $1000 $1000 PVA =$1000+ + + + =$4,387.21 5 (1.07)1 (1.07) 2 (1.07) 3 (1.07) 4

5 4 3 2 1 FVA5 =$1000(1.07) +$1000(1.07) +$1000(1.07) +$1000(1.07) +$1000(1.07) =$6,153.29

C. Perpetuities 1. Perpetuity is special kind of annuity that never matures. With a perpetuity, the periodic cash flow continues forever. 2. Calculate the present value of a perpetuity paying C dollars at the end of every year

C C C C C PV= + + + + +⋯ (1+k) (1+k)2 (1+k) 3 (1+k) 4 (1+k) 5

C Factor out from right-hand side of above equation to obtain: 1+k

C 1 1 1 1  ⋯ PV= 1+1 + 2 + 3 + 4 +  (1+k) (1+k) (1+k) (1+k) (1+k) 

Term in brackets on right-hand side of above equation is infinite geometric

1 1 1+k = = series that simplifies to 1 1+k-1 . Plugging this into bracket of 1- k 1+k 1+k

above equation obtains: C (1+k) C PV= = (1+k) k k

3. Key result: Present value of perpetuity equals periodic

PV=C payment divided by interest rate or k .

4. Ex: How much would I have to deposit today in order to withdraw $1,000 each year forever if I earn an annual interest rate of 8% on my deposit? Answer: PV = $1,000/0.08=$12,500.00 D. Present value of uneven cash flows or a mixed stream 1. A mixed stream is a series of cash flows that exhibits no particular pattern.

2. Let FVi be future cash flow received at the end of year i, i = 1, . . . , N. 3. Given interest rate k, present value (PV) of mixed stream cash flow is equal to:

FV FV FV FV PV=1 + 2 + 3 +⋯ + N (1+k)1 (1+k) 2 (1+k) 3 (1+k) N

4. Example: What is present value of following mixed stream assuming interest rate was 5%? 8%?

Year 0 1 2 3 4 5 Cash flow $0 $400 $800 $500 $400 $300

Present value of mixed stream cash flow @ 5%:

$400 $800 $500 $400 $300 PV= + + + + =$2,102.63 (1.05)1 (1.05) 2 (1.05) 3 (1.05) 4 (1.05) 5

Present value of mixed stream cash flow @ 8%:

$400 $800 $500 $400 $300 PV= + + + + =$1,951.34 (1.08)1 (1.08) 2 (1.08) 3 (1.08) 4 (1.08) 5

III. Compounding more frequently than annually A. General formula for more frequent compounding 1. i = annual interest rate m = number of times per year interest is compounded

i 2. FV =PV×(1+ )m×n n m

i 3. Semiannual compounding: FV =PV×(1+ )2×n n 2

i 4. Quarterly compounding: FV =PV×(1+ )4×n n 4

i 5. Monthly compounding: FV =PV×(1+ )12×n n 12 6. Example: 12% interest rate compounded either annually, semiannually, quarterly, or monthly. Initial $1,000 deposit and end of month balance over two year period

Annual Semiannual Quarterly Monthly interest factor 1.12 1.06 1.03 1.01 Month 0 1,000.00 1,000.00 1,000.00 1,000.00 1 1,010.00 2 1,020.10 3 1,030.00 1,030.30 4 1,040.60 5 1,051.01 6 1,060.00 1,060.90 1,061.52 7 1,072.14 8 1,082.86 9 1,092.73 1,093.69 10 1,104.62 11 1,115.67 12 1,120.00 1,123.60 1,125.51 1,126.83 13 1,138.09 14 1,149.47 15 1,159.27 1,160.97 16 1,172.58 17 1,184.30 18 1,191.02 1,194.05 1,196.15 19 1,208.11 20 1,220.19 21 1,229.87 1,232.39 22 1,244.72 23 1,257.16

24 1,254.40 1,262.48 1,266.77 1,269.73

B. Nominal and effective interest rates 1. Nominal interest rate (k) = stated or contractual rate of interest charged by lender or promised by borrower 2. effective (annual) interest rate (EAR) = rate actually paid or earned 3. EAR = (1+k/m)m – 1 4. In general: effective rate is greater than nominal rate whenever compounding occurs more than once per year 5. Example: What is effective rate of interest on credit card if its nominal rate is 18% per year, compounded monthly? Answer: EAR = (1+0.18/12)12 – 1 = 19.56% C. Continuous compounding

i x n 1. With continuous compounding, then FVn = PV x e , where e is the exponential function or e approximately equals = 2.7183. 2. Example: What is the future value of an initial $200 deposit (PV = $200) after two years (n = 2) in an account paying 8% interest (i = 0.08), compounded continuously?

2x0.08 0.16 Answer: FV2 = $200e =$200e =$200(1.1735)=$234.70 IV. Loan amortization A. Suppose a person wanted to borrow $6,000. He or she will repay the loan in equal annual end of the year payments over a 4 year period at an annual interest rate of 10%. Let X = loan payment.

X X X X $6,000= + + + 1.10 1.102 1.10 3 1.10 4 轾1 1 1 1 $6,000=X + + + 臌犏1.10 1.102 1.10 3 1.10 4

轾1 1 1 1 $6,000=X + + + =X(3.169865) 臌犏1.10 1.102 1.10 3 1.10 4

X=$6,000/3.169865=$1,892.83 B. Loan amortization schedule: end beginning end-of-year end of year loan interest principal principal of balance payment 0.10x(1) (2)-(3) (1)-(4) year (1) (2) (3) (4) (5) 1 $6,000.00 $1,892.83 $600.00 $1,292.83 $4,707.17 2 4,707.17 1,892.83 470.72 1,422.11 3,285.06 3 3,285.06 1,892.83 328.51 1,564.32 1,720.73 4 1,720.73 1,892.83 172.07 1,720.76 -0.02