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CH 28 - 1CH 28: TIME VALUE OF MONEYI. Basics

[1] Definitions of variablest = 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 valueFVA = Future value of an annuity.PV = Present value.PVA = Present value of an annuity.

[2] Compounding and discounting of interestInterest can be compounded or discounted yearly, quarterly, monthly, weekly, daily, or evencontinuously.

EX) Annual interest rate = 8%. Horizon = 2 years.i n

1) yearly compounding 8% 2 periods2) semiannual compounding 8% / 2 = 4% 4 periods3) quarterly compounding 8% / 4 = 2% 8 periods4) monthly compounding 8% / 12 = (2/3)% 24 periods5) daily compounding 8% / 365 = (8/365)% 730 periods

[3] Cash flowsCash inflow = + valueCash outflow = - value

(1) Single cash flow: C

(2) Different streams: CtPMT PMT PMT PMT

(3) Ordinary annuity(end of each period) 0 1 2 n-1 n

PMT PMT PMT PMT(4) Annuity due

(beginning of each period) 0 1 2 n-1 n

[4] Time line

Cash flow

Discounting(Present value) Compounding

(Future value)

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CH 28 - 2

II. Future Value: FVIF = ( + i)i nn

, 1

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

1,000

{C0s value at t=1} = 1,000(1+0.03)1

= 1,030{C0s value at t=2} = 1,000(1+0.03)(1+0.03) = 1,000(1+0.03)2= 1,060.90{C0s 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

{C1s value at t=3} + {C2s value at t=3} + {C3s 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

[3] Annuity: FVIFA = ( + i)i

ii,n

n t

t

n n

1 1 1

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

{C1s value at t=3} + {C2s value at t=3} + {C3s 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) (1+ 0.03) (1+ 0.03)

(1+ 0.03) (1+ 0.03)

2 1 0

t 3-t

+ +

=

=

= = +

=

= = 1 000 1 000

0

2

1

3

, ,t t

1,000 FVIFA(0.03,3) 1,000(1 0.03) 1

0.033,090.90

3

[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

1,000 1,000 1,000


= 1,000(1+0.03)3+ 1,000(1+0.03)2+ 1,000(1+0.03)1 = 3,183.627

= = +

+ =1,000 FVIFA(0.03,3) FVIF(0.03,1) 1,000(1 0.03) 1

0.03(1 0.03) 3,183.627

31

Note that 1,000FVIFA(0.03,3) is the value at t=2

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CH 28 - 3

III. Present Value: PVIF =( + i)

i n n,

1

1

[1] Single cash flow: C3=1,000; i=0.03; {Value at t=0}?0 1 2 3

1,000

{C3s value at t=2} = 1,000 +

1

1 0 03970 8737864

( . ).

{C3s value at t=1} = 1,000 +

+

1

1 0 03

1

1 0 039425959091

( . ) ( . ).

{C3s value at t=0} = 1,000 +

+

+

1

1 0 03

1

1 0 03

1

1 0 039151416594

( . ) ( . ) ( . ).

{Value at t=0} = 1,000 PVIF(0.03,3) 1,0001

(1 0.03)915.1416594

3 =

+

[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

{C1s value at t=0} + {C2s value at t=0} + {C3s value at t=0}=1,000 PVIF(0.03,1)+1,300 PVIF(0.03,2)+900 PVIF(0.03,3)

1,0001

(1 0.03)1,300

1

(1 0.03)900

1

(1 0.03)3,019.875961

1 2 3

= +

+ +

+ +

[3] Annuity: PVIFA =( + i)

i

ii n t

t

n n

,

( )1

1

1 1

1

1= =

+

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


=1,000 1

(1+ 0.03)1,000

(1+ 0.03)1,000

(1+ 0.03)

=1,000 1

(1+ 0.03) (1+ 0.03) (1+ 0.03)

1

(1+ 0.03)

1 2 3

1 2 3 t

+ +

+ +

=

= = +

=

1 1

1 11000

1

3

,t

1,000 PVIFA(0.03,3) 1,000

1 -1

(1 0.03)

0.032,828.611355

3

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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


= 1,000 + 1,000 1

(1+ 0.03)1,000

(1+ 0.03)

=1,000 1

(1+ 0.03) (1+ 0.03)

=1,000 1

(1+ 0.03) (1+ 0.03) (1+ 0.03)

1

(1+ 0.03)

1 2

1 2

1 2 3

t

+

+ +

+ +

+

=

+

= = +

+

=

1

1 1

1 11 0 03

1 000 1 0 031

3

( . )

, ( . )t

1,000 PVIFA(0.03,3) FVIF(0.03,1) 1,000

1 - 1

(1 0.03)

0.03(1 0.03) 2,913.469696

3

Note that 1,000PVIFA(0.03,3) is the value at t=-1

IV. Present value of perpetuity = = +

PMT PVIFA(i, ) PMT

1-1

(1 i)

iPMT

1

i

Exercise: i=0.05 {Value of the following cash flows at time=4}?0 1 2 3 4 29 30

1,000 2,000 2,000 2,000 2,000 2,000 1,5001 000 0 05 4 1 500 0 05 26 2 000 0 05 29 0 05 4

1 000 1 0 05 1 500 1

1 0 052 000

1 1

1 0 05

0 051 0 0 5 38 445 506474

26

294

, ( . , ) , ( . , ) , ( . , ) ( . , )

, ( . ) ,( . )

, ( . )

.( . ) , .

+ +

= + + +

+

+

+

FVIF PVIF PVIFA FVIF

Exercise: 30-year mortgage loan; interest 7.75%; Loan =$200,000; Monthly payment?

200 000 00775

12

360

1 1

1 00775

12

0077512

200 000

1 1

1 00775

12

00775

12

1 432 824493 1 43282

360

360

, ( .

, )

.

.

,

.

.

, . , .

= =

+

=

+

=

PMT PVIFA PMT

PMT

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CH 28 - 5V. Summary[1] Interest factors

(1) FVIFi,n= ( )1 1+ i n Future value interest factor

(future value of $1 at the end of n periods).0 1 2 n-1 n

C

(2) FVIFAi,n= ( ) ( )

1 1 1

1

+ +

= i =

i

i n.

n t

t

n n

Future value interest factor for an annuity

(Sum of annuity of $1 per period for n periods).0 1 2 n-1 n

PMT PMT PMT PMT

(3) PVIFi,n=1

11

( )+

in

Present value interest factor

(Present value of $1 due n period in the future).0 1 2 n-1 n

C

(4) PVIFAi,n= ( + i)

i

i nt

t

n n1

1

1 1

1

1= =

+

( )

Present value interest factor for an annuity

(present value of an annuity of $1 per period for nperiod).

0 1 2 n-1 n

PMT PMT PMT PMT

(5) PV of perpetuity =PMT PVIFA = PMT

i=

PMT

ii,

1

[2] Basic equations(1) FV = C x FVIFi,n: future value of a single cash flow.(2) FVA = PMT x FVIFAi,n: future value of an annuity. (At the same point of time with the

last receipt/payment)

(3) PV = C x PVIFi,n: present value of a single cash flow.(4) PVA = PMT x PVIFAi,n: present value of an annuity. (One period prior to the first

receipt/payment)

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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 afteoperating 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?

6M PVIFA(0.10,5) FVIF(0.10,1) 6M

11

(1 0.10)

0.10(1 0.10) 2.68

51

=

+

+ =$25,019,19

b. What is the present value of the cash inflows?

4M PVIFA(0.10,15) PVIF(0.10,4) 4M

11

(1 0.10)

0.10 (1 0.10)8.58

15

=

+

+

=1

4 $20,780,21

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 berejected.

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 securitycosts $900. The second investment involves simply putting the same amount of money in a bank savings accounthat pays an 8 percent nominal (quoted) interest rate, but with quarterly compounding. Your grandmother regardthe two investments as being equally safe and liquid, so the required effective annual rate of return on the securityis the same as that on the bank deposit. She has asked you to calculate the value of the security to help her decidewhether 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= 1050PV = 50 PVIF(0.02,4) 50 PVIF(0.02,8) 50 PVIF(0.02,12) 1,050 PVIF(0.02,16)

= 501

(1 + 0.02)50

1

(1 + 0.02)50

1

(1 + 0.02)1,050

(1 + 0.02)4 8 12 16

+ + +

+ + + =

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