Chapter 3 – Time value of Money




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Chapter 3 – Time value of Money

This chapter discusses how to calculate the present value, future value, internal rate of return, and modified internal rate of return of a cash flow stream. Understanding how (and when) to use these formulas is essential to your success as a financial manager! Formulas and examples are included with these notes.


Numbers are rounded to 4 decimal places in tables and formula. However, the actual (non-rounded) numbers are used in the calculations.
Time Value of Money Concepts


  1. Time-Line Conventions




  1. $1 received today (cash inflow)

  2. $1 paid in five years (cash outflow)

  3. $1 received at the end of the third year

  4. $1 received at the beginning of the third year

  5. Four-year annuity of $1 per year, first cash flow received at t = 1 (ordinary annuity)

  6. Four-year annuity of $1 per year, first cash flow received at t = 0 (annuity due)

  7. Four-year annuity of $1 per year, first cash flow received at t = 2 (deferred annuity)







0

1

2

3

4

5

A.

$1
















B.
















-$1

C.










$1







D.







$1










E.




$1

$1

$1

$1




F.

$1

$1

$1

$1







G.







$1

$1

$1

$1




  1. Notation

C0 = cash flow at time 0

C = cash flow (used when all cash flows are the same)

r = discount rate or interest rate

t = time period (e.g., t = 4), or number of years (e.g., t years in the future)

m = number of compounding periods per year (e.g., with monthly compounding, m = 12)

g = growth rate in cash flow


  1. Annual Compounding, Single Payments




  1. Future value of $1 as of time 1. Interest rate = 5%.




0

1

2

3

4

5

$1

$0

$0

$0

$0

$0

Formula: C0 (1 + r)t = $1(1.05)1 = $1.0500

Financial Calculator: N = 1, I/Y = 5, PV = -1, PMT = 0, FV = Answer
Note on financial calculators – The calculator inputs described above are for a Texas Instruments BAII Plus calculator. (Many other financial calculators require similar inputs.)
Notice that you enter a -1 as the PV and the solution is +1.05. Here is the intuition: deposit $1 in the bank (negative cash flow), withdraw $1.05 in one year (positive cash flow). If you had entered +1 as the PV, the solution would be –1.05.


  1. Future value of $1, as of time 5. Interest rate = 5%.




0

1

2

3

4

5

$1

$0

$0

$0

$0

$0

Formula: C0 (1 + r)t = $1(1.05)5 = $1.2763

Financial Calculator: N = 5, I/Y = 5, PV = -1, PMT = 0, FV = Answer


  1. Present value of $1, received at time 1. Discount rate = 5%.




0

1

2

3

4

5

$0

$1

$0

$0

$0

$0

Formula: C1 / (1 + r)t = $1/(1.05)1 = $0.9524

Financial Calculator: N = 1, I/Y = 5, PV = Answer, PMT = 0, FV = -1


  1. Present value of $1, received at time 5. Discount rate = 5%.




0

1

2

3

4

5

$0

$0

$0

$0

$0

$1

Formula: C5 / (1 + r)t = $1/(1.05)5 = $0.7835

Financial Calculator: N = 5, I/Y = 5, PV = Answer, PMT = 0, FV = -1


  1. Compounding periods less than one year




  1. Future value of $1, as of time 5. Interest rate = 5%, compounded “m” times per year.




0

1

2

3

4

5

$1

$0

$0

$0

$0

$0

General (non-continuous) formula: C0 (1 + r/m)tm

Continuous compounding formula: C0 ert

Note: “e” = 2.718281828


Semi-annual compounding: $1(1 + (0.05/2))(5)(2) = $1.280085

Monthly compounding: $1(1 + (0.05/12))(5)(12) = $1.283359

Daily compounding: $1(1 + (0.05/365))(5)(365) = $1.284003

Continuous compounding: $1 e(5)(0.05) = $1.284025


Note: Some may use 360 days as the length of one year, other may take into account leap years (366 days every four years). The effects of these changes (from a 365-day year) are extremely small.
Financial Calculator (for semi-annual): N = 10, I/Y = 5/2, PV = -1, PMT = 0, FV = Answer

Financial Calculator (for monthly): N = 60, I/Y = 5/12, PV = -1, PMT = 0, FV = Answer

Financial Calculator (for daily): N = 1825, I/Y = 5/365, PV = -1, PMT = 0, FV = Answer


  1. Present value of $1, received at time 5. Discount rate = 5%, compounded m times per year.




0

1

2

3

4

5

$0

$0

$0

$0

$0

$1

General (non-continuous) formula: C5 / (1 + r/m)tm

Continuous compounding formula: C5 / ert
Semi-annual compounding: $1 / [1 + (0.05/2)](5)(2) = $0.781198

Monthly compounding: $1 / [1 + (0.05/12)](5)(12) = $0.779205

Daily compounding: $1 / [1 + (0.05/365)](5)(365) = $0.778814

Continuous compounding: $1 / e(5)(0.05) = $0.778801


Financial Calculator (for semi-annual): N = 10, I/Y = 5/2, PV = Answer, PMT = 0, FV = -1

Financial Calculator (for monthly): N = 60, I/Y = 5/12, PV = Answer, PMT = 0, FV = -1

Financial Calculator (for daily): N = 1825, I/Y = 5/365, PV = Answer, PMT = 0, FV = -1


  1. Constant Finite Annuities




  1. Four-year annuity of $1 per year, first cash flow received at t = 1. Interest and discount rate = 5%.




0

1

2

3

4

5

$0

$1

$1

$1

$1

$0

Standard formula for the future value of a finite annuity = C [(1 + r)t – 1] / r


This standard formula for the future value of a finite annuity gives a value as of the last period of the annuity (time 4 in this example). The 1.05 is raised to the fourth power because there are 4 payments in the annuity.


Value as of time 4 = $1 [(1.054 – 1) / 0.05] =

$4.3101

You can calculate the value of the cash flows at other points in time by multiplying or dividing by 1+r, where r (the interest and discount rate) is 5% in this example.


For instance, assume you want to know the value of the above cash flow stream at t = 6. Time 6 is two years after time 4. To calculate, use the standard formula to determine the value at t = 4, then multiply by 1.052 to determine the value at t = 6. (Use the second power because you are calculating the value two years after time 4.) The solution is:


Value as of time 6 = $1 [(1.054 – 1) / 0.05] 1.052 =

$4.7519

As a second example, assume that you want to know the value of the above cash flow stream at t = 1. Time 1 is three years before time 4. To calculate, use the standard formula to determine the value at t = 4, then divide by 1.053 to determine the value at t = 1. (Use the third power because you are calculating the value three years before time 4.) The solution is:




Value as of time 1 = $1 [(1.054 – 1) / 0.05] / 1.053 =

$3.7232

Therefore, “multiply” when you want to determine the value at a later date, “divide” when you want to determine the value at an earlier date.


Some more examples:


Value as of time 0 = $1 [(1.054 – 1) / 0.05] / 1.054 =

$3.5460

Value as of time 1 = $1 [(1.054 – 1) / 0.05] / 1.053 =

$3.7232

Value as of time 2 = $1 [(1.054 – 1) / 0.05] / 1.052 =

$3.9094

Value as of time 3 = $1 [(1.054 – 1) / 0.05] / 1.051 =

$4.1049

Value as of time 4 = $1 [(1.054 – 1) / 0.05] =

$4.3101

Value as of time 5 = $1 [(1.054 – 1) / 0.05] 1.051 =

$4.5256

Value as of time 6 = $1 [(1.054 – 1) / 0.05] 1.052 =

$4.7519

Value as of time 7 = $1 [(1.054 – 1) / 0.05] 1.053 =

$4.9895

Financial Calculator (time 0): N = 4, I/Y = 5, PV = 0, PMT = -1, FV = Answer. Divide answer by 1.054.

Financial Calculator (time 1): N = 4, I/Y = 5, PV = 0, PMT = -1, FV = Answer. Divide answer by 1.053.

Financial Calculator (time 2): N = 4, I/Y = 5, PV = 0, PMT = -1, FV = Answer. Divide answer by 1.052.

Financial Calculator (time 3): N = 4, I/Y = 5, PV = 0, PMT = -1, FV = Answer. Divide answer by 1.051.

Financial Calculator (time 4): N = 4, I/Y = 5, PV = 0, PMT = -1, FV = Answer.

Financial Calculator (time 5): N = 4, I/Y = 5, PV = 0, PMT = -1, FV = Answer. Multiply answer by 1.051.

Financial Calculator (time 6): N = 4, I/Y = 5, PV = 0, PMT = -1, FV = Answer. Multiply answer by 1.052

Financial Calculator (time 7): N = 4, I/Y = 5, PV = 0, PMT = -1, FV = Answer. Multiply answer by 1.053
You can also use the formula for the present value of a finite annuity to calculate the value of a cash flow stream at different points in time.
Standard formula for the present value of a finite annuity = C { [1 – (1 / (1 + r))t] / r}
The standard formula gives a value one period before the first payment of the annuity (time 0 in this example). The 1.05 is raised to the fourth power because there are 4 payments in the annuity.


Value as of time 0 = $1 { [1 – (1/1.05)4] / (0.05) } =

$3.5460

As before, you can calculate the value at other points in time by multiplying or dividing by 1+r, (1.05 in this example).


Two examples:


Value as of time 1 = $1 { [1 – (1/1.05)4] / (0.05) } 1.051 =

$3.7232




Value as of time 6 = $1 { [1 – (1/1.05)4] / (0.05) } 1.056 =

$4.7519

Financial Calculator (time 0): N = 4, I/Y = 5, PV = Answer, PMT = -1, FV = 0

Financial Calculator (time 1): N = 4, I/Y = 5, PV = Answer, PMT = -1, FV = 0. Multiply answer by 1.051.

Financial Calculator (time 6): N = 4, I/Y = 5, PV = Answer, PMT = -1, FV = 0. Multiply answer by 1.056.




  1. Three-year annuity of $1 per year, first cash flow received at t = 0. Interest and discount rate = 5%.




0

1

2

3

4

5

$1

$1

$1

$0

$0

$0

The standard future value annuity formula gives a value as of the last year of the annuity (year 2 in this example). This is a three-year annuity. Therefore, 1.05 is raised to the third power in the formula.




Value as of time 2 = $1 [(1.053 – 1) / 0.05] =

$3.1525

The value at other points in time can be calculated by multiplying or dividing by 1.05, raised to the appropriate power.




Value as of time 0 = $1 [(1.053 – 1) / 0.05] / 1.052 =

$2.8594

Value as of time 4 = $1 [(1.053 – 1) / 0.05] 1.052 =

$3.4756

The standard present value annuity formula gives a value one period before the first payment of the annuity. Therefore, the formula will give you a value at t = -1. You need to multiply by 1 + r to get the value by t = 0.




Value as of time 0 = $1 { [1 – (1/1.05)3] / (0.05) } 1.051 =

$2.8594

The values at time 2 and 4:




Value as of time 2 = $1 { [1 – (1/1.05)3] / (0.05) } 1.053 =

$3.1525

Value as of time 4 = $1 { [1 – (1/1.05)3] / (0.05) } 1.055 =

$3.4756




  1. Five-year annuity of $1 per year, first cash flow received at t = 3. Interest and discount rate = 5%.




0

1

2

3

4

5

6

7

8

$0

$0

$0

$1

$1

$1

$1

$1

$0

The standard future value annuity formula gives the value as of the last year of the annuity (t = 7 in this example). This is a five-year annuity. Therefore, 1.05 is raised to the fifth power.




Value as of time 7 = $1 [(1.055 – 1) / 0.05] =

$5.5256


Values at different points in time using the future value annuity formula. A couple of examples


Value as of time 0 = $1 [(1.055 – 1) / 0.05] / 1.057 =

$3.9270

Value as of time 4 = $1 [(1.055 – 1) / 0.05] / 1.053 =

$4.7732

Value as of time 8 = $1 [(1.055 – 1) / 0.05] 1.051 =

$5.8019

The standard present value annuity formula gives the value as of the year before the first payment of the annuity (t = 2 in this example).




Value as of time 2 = $1 { [1 – (1/1.05)5] / (0.05) } =

$4.3295


Values at different points in time using the present value annuity formula. A couple of examples:


Value as of time 0 = $1 { [1 – (1/1.05)5] / (0.05) } / 1.052 =

$3.9270

Value as of time 4 = $1 { [1 – (1/1.05)5] / (0.05) } 1.052 =

$4.7732

Value as of time 8 = $1 { [1 – (1/1.05)5] / (0.05) } 1.056 =

$5.8019




  1. Growing Finite Annuities




  1. Four-year growing annuity, growing at 10% per year. First cash flow (equal to $1) received at t = 1. Interest and discount rate = 5%.




0

1

2

3

4

5

$0

$1

$1.1

$1.21

$1.331

$0

Standard formula for the present value of a finite growing annuity (for when r is not equal to g) =

Cfirst [1 – [(1 + g) / (1 + r)]t ] / (r – g). This formula gives the value one period before the first payment (t = 0 in this example).
Cfirst is the first cash flow of the annuity. In this above example, Cfirst = C1 = $1.


Value as of time 0 = $1 { [1 – (1.10/1.05)4] / (0.05 – 0.10) }

$4.0904


Values at different points in time using the present value growing annuity formula. Multiply or divide by 1+r (raised to the appropriate power) to determine the value at other points in time.


Value as of time 2 = $1 { [1 – (1.10/1.05)4] / (0.05 – 0.10) } 1.052

$4.5096

Value as of time 4 = $1 { [1 – (1.10/1.05)4] / (0.05 – 0.10) } 1.054

$4.9719

Value as of time 5 = $1 { [1 – (1.10/1.05)4] / (0.05 – 0.10) } 1.055

$5.2205




  1. Four-year growing annuity, growing at 10% per year. First cash flow (equal to $1) received at t = 3. Interest and discount rate = 5%.




0

1

2

3

4

5

6

7

$0

$0

$0

$1

$1.1

$1.21

$1.331

$0

The standard formula for the present value of a growing annuity gives you a value at time 2 (one period before the first payment). This is a 4-year annuity. Therefore, 1.05 is raised to the 4th power.




Value as of time 2 = $1 { [1 – (1.10/1.05)4] / (0.05 – 0.10) }

$4.0904


Values at different points in time using the present value growing annuity formula. A few examples:


Value as of time 0 = $1 { [1 – (1.10/1.05)4] / (0.05 – 0.10) } /1.052

$3.7101

Value as of time 5 = $1 { [1 – (1.10/1.05)4] / (0.05 – 0.10) } 1.053

$4.7351




  1. Perpetual constant annuities




  1. Perpetual constant annuity of $1 (cash flows start at time 1, interest and discount rate = 5%)




0

1

2

3

4

5



$0

$1

$1

$1

$1

$1

$1

The standard formula for the present value of a perpetual constant annuity is C / r. The formula gives you the value one period before the first payment.




Value as of time 0 = $1 / 0.05

$20.0000


Values at different points in time using the present value perpetual constant annuity formula


Value as of time 1 = ($1 / 0.05) 1.051

$21.0000

Value as of time 4 = ($1 / 0.05) 1.054

$24.3101




  1. Perpetual constant annuity of $1 (cash flows start at time 0, interest and discount rate = 5%)




0

1

2

3

4

5



$1

$1

$1

$1

$1

$1

$1

The standard formula for the present value of a perpetual constant annuity gives you the value one period before the first payment (t = -1 in this example). Therefore, you need to multiply by 1.05 to get the value at t = 0.




Value as of time 0 = ($1 / 0.05) 1.05

$21.0000


Values at different points in time using the present value perpetual annuity formula. Two examples:


Value as of time 1 = ($1 / 0.05) 1.052

$22.0500

Value as of time 4 = ($1 / 0.05) 1.055

$25.5256




  1. Perpetual constant annuity of $1 (cash flows start at time 5, interest and discount rate = 5%)




0

1

2

3

4

5



$0

$0

$0

$0

$0

$1

$1

The standard formula for the present value of a perpetual constant annuity gives you the value one period before the first payment (t = 4 in this example).




Value as of time 4 = ($1 / 0.05)

$20.0000


Values at different points in time using the present value perpetual annuity formula. Some examples:


Value as of time 0 = ($1 / 0.05) / 1.054

$16.4540

Value as of time 5 = ($1 / 0.05) 1.051

$21.0000




  1. Perpetual growing annuity, growing at 3% per year (first cash flow, received at time 1, equals $1, interest and discount rate = 5%)




0

1

2

3



$0

$1

$1.03

$1.0609

3% more

The standard formula for the present value of a perpetual growing annuity is Cfirst / (r – g). The formula gives you the value one period before the first payment.




Value as of time 0 = $1 / (0.05 – 0.03)

$50.0000


Values at different points in time using the present value perpetual growing annuity formula. Some examples:


Value as of time 1 = [$1 / (0.05 – 0.03)] 1.051

$52.5000

Value as of time 4 = [$1 / (0.05 – 0.03)] 1.054

$60.7753




  1. Perpetual growing annuity, growing at 3% per year (first cash flow, received at time 11, equals $1, interest discount rate = 5%)




10

11

12

13



$0

$1

$1.03

$1.0609

3% more

The standard formula for the present value of a perpetual growing annuity gives you the value one period before the first payment (t = 10 in this example).




Value as of time 10 = [$1 / (0.05 – 0.03)]

$50.0000


Values at different points in time using the present value perpetual growing annuity formula. A few examples:


Value as of time 0 = [$1 / (0.05 – 0.03)] / 1.0510

$30.6957

Value as of time 9 = [$1 / (0.05 – 0.03)] / 1.051

$47.6190

Value as of time 11 = [$1 / (0.05 – 0.03)] 1.051

$52.5000




  1. Two growth-rate example: $1 at time 1, 10% growth rate until time 4, 3% growth rate after time 4 (in perpetuity). Use a 5% discount rate




0

1

2

3

4

5



$0

$1

$1.1

$1.21

$1.331

$1.3709

3% more

Value the first four payments using the finite growing annuity formula. Value the payments starting at time 5 using the perpetual growing annuity formula


The first four payments


Value as of time 0 = $1 { [1 – (1.10/1.05)4] / (0.05 – 0.10) }

$4.0904

The payments starting at time 5




Value as of time 0 = {[($1) (1.13) (1.03)] / [(0.05 – 0.03)]} / 1.054

$56.3934

Solution = $4.0904 + $56.3934 = $60.4837


Note: [($1) (1.13) (1.03)] = $1.3709 = the payment at t = 5


  1. Two growth-rate example: $1 at time 0, 10% growth rate until time 12, 3% growth rate after time 12 (in perpetuity). Use a 5% discount rate




0

1

2

. . .

11

12

13



$1

$1.1

$1.21



$2.8531

$3.1384

$3.2326

3% more

Value the first thirteen payments (t = 0 to t = 12) using the finite growing annuity formula. Value the payments starting at time 13 using the perpetual growing annuity formula


The first thirteen payments


Value as of time 0 = $1 { [1 – (1.10/1.05)13] / (0.05 – 0.10) } 1.051

$17.4471

The payments starting at time 13




Value as of time 0 = {[($1) (1.112) (1.03)] / [(0.05 – 0.03)]} / 1.0512

$90.0011

Solution = $17.4471 + $90.0011 = $107.4482


Note: [($1) (1.112) (1.03)] = $3.2326 = the payment at t = 13


  1. Two growth-rate example: $1 at time 3, 10% growth rate until time 20, 3% growth rate after time 20 (in perpetuity). Use a 5% discount rate




0

1

2

3

4

. . .

20

21



$0

$0

$0

$1

$1.1



$5.0545

$5.2061

3% more

Value the first eighteen payments (t = 3 to t = 20) using the finite growing annuity formula. Value the payments starting at time 21 using the perpetual growing annuity formula


The first eighteen payments


Value as of time 0 = $1 { [1 – (1.10/1.05)18] / (0.05 – 0.10) } / 1.052

$23.7689

The payments starting at time 21




Value as of time 0 = {[($1) (1.117) (1.03)] / [(0.05 – 0.03)]} / 1.0520

$98.1063

Solution = $23.7689 + $98.1063 = $121.8752


Note: [($1) (1.117) (1.03)] = $5.2061 = the payment at t = 21


  1. Internal rate of return (IRR)

Definition: the IRR = the discount rate that causes the sum of the present values of all cash flows to equal zero.


IRR calculation examples


  1. Two cash flows




0

1

2

3

4

5

-$1

$0

$0

$0

$0

$2

-$1 + $2 / (1 + r)5 = $0

r = ($2 / $1)(1/5) – 1 = 14.8698% = IRR

Financial Calculator: N = 5, I/Y = Answer, PV = -1, PMT = 0, FV = 2




0

1

2

3

4

5

$1

$0

$0

$0

$0

-$2

$1 + -$2 / (1 + r)5 = $0

r = ($2 / $1)(1/5) – 1 = 14.8698% = IRR

Financial Calculator: N = 5, I/Y = Answer, PV = 1, PMT = 0, FV = -2




  1. Perpetual constant annuities




0

1

2

3

4



-$10

$1

$1

$1

$1

$1

-$10 + $1 / r = $0

r = $1 / $10 = 10% = IRR


  1. Perpetual growing annuities




0

1

2

3

4



-$10

$1

$1.03

$1.0609

$1.0927

3% more

-$10 + $1 / (r – 3%) = $0

r = ($1 / $10) + 3% = 13% = IRR


  1. Other cash flow patterns – solve by your calculator or computer. Example – finite annuity




0

1

2

3

4

5

-$3

$1

$1

$1

$1

$1

-$3 + $1 { [1 – (1/(1+r))5] / r } = $0

r = 19.8577% = IRR

Financial Calculator: N = 5, I/Y = Answer, PV = -3, PMT = 1, FV = 0




  1. Modified Internal Rate of Return (MIRR)




  • Step one: Using the discount rate, take a PV (to time zero) of the negative cash flows

  • Step two: Using the interest rate, take a FV (to time t, where t is the time of the last cash flow) of the positive cash flows

  • Step three: Calculate the IRR of the two cash flows calculated in the first two steps



  1. Using an interest and discount rate = 5%, what is the MIRR of the following cash flow stream?




0

1

2

3

4

5

-$3

$1

$1

$1

$1

$1


Step 1: PV of negative cash flows (at time 0) = -$3

Step 2: FV of positive cash flows (at time 5) = $5.5256

Step 3: IRR = ($5.5256 / $3)(1/5) – 1 = 12.9932% = MIRR


  1. Using an interest and discount rate = 5%, what is the MIRR of the following cash flow stream?




0

1

2

3

4

5

+$3

-$1

-$1

-$1

-$1

-$1


Step 1: PV of negative cash flows (at time 0) = -$4.3295

Step 2: FV of positive cash flows (at time 5) = $3.8288

Step 3: IRR = ($3.8288 / $4.3295)(1/5) – 1 = -2.4277% = MIRR


  1. Using an interest and discount rate = 5%, what is the MIRR of the following cash flow stream?




0

1

2

3

4

5

+$3

$0

$0

-$1

-$1

$1


Step 1: PV of negative cash flows (at time 0) = -$1.6865

Step 2: FV of positive cash flows (at time 5) = $4.8288

Step 3: IRR = ($4.8288 / $1.6865)(1/5) – 1 = 23.4154% = MIRR


  1. Application – loan amortization schedules

A 30-year home loan has an annual interest rate of 8%. Interest is compounded monthly. What is the monthly payment on a fully amortizing, level payment loan for $100,000? $733.7646


Use this payment to filling in the following loan amortization table for the home loan described above.


Month

Beginning Balance

Total Payment

Interest Payment

Principal Payment

Ending Balance

0













$100,000.000

1

$100,000.000

$733.7646

$666.6667

$67.0979

$99,932.9021

2

$99,932.9021

$733.7646

$666.2193

$67.5452

$99,865.3569

3

$99,865.3569

$733.7646

$665.7690

$67.9955

$99,797.3613

4

$99,797.3613

$733.7646

$665.3157

$68.4488

$99,728.9125

5

$99,728.9125

$733.7646

$664.8594

$68.9052

$99,660.0073

The loan balance will be $0 after the 360th payment.









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