The time value of money
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The Time Value of Money. Chapter 5. LEARNING OBJECTIVES. 1.   Explain the mechanics of compounding when invested. 2.  Present value and future value. 3.  Ordinary annuity and its future value. 4.  An ordinary annuity and an annuity due. 5.  Non-annual future or present value of a sum .

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The Time Value of Money

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The time value of money

The Time Value of Money

Chapter 5


Learning objectives

LEARNING OBJECTIVES

  • 1.  Explain the mechanics of compounding when invested.

  • 2.  Present value and future value.

  • 3.  Ordinary annuity and its future value.

  • 4.  An ordinary annuity and an annuity due.

  • 5.  Non-annual future or present value of a sum .

  • 6. Determine the present value of a perpetuity.


Power of time of value of money

Power of time of value of money

  • History of Interest Rates

    $1000 ( 1 + .08)400 = ?


Power of time value of money

Power of time value of money

  • Money Angles: by Andrew Tobias.

    1.Chessboard with the King

    2.Manhattan


Terms

Terms

  • Compound Interest

  • Future value and Present Value

  • Annuities

  • Annuities Due

  • Amortized Loans

  • Compound Interest with Non-annual Periods

  • Present Value of an Uneven Stream·

  • Perpetuities


Compound interest

COMPOUND INTEREST

  • FV1=PV (1+i) (5-1)

  • Where FV1=the future value of the investment at the end of one year

  • i=the annual interest (or discount) rate

  • PV=the present value, or original amount invested at the beginning of the first year


Future value

Future value

1.Simple compounding

2.Complex compounding


The time value of money

Future value


The time value of money

  • FV1=PV (1+i)

  • =$100(1+0.06)

  • =$100(1.06)

  • =$106


The time value of money

Compound twice a year


The time value of money

Compound four times a year


The time value of money

Compound 12 times a year


The time value of money

Compound 360 times a year


The time value of money

Continuous compounding


The time value of money

Illustration of Compound Interest Calculations

Year Beginning Value Interest Earned Ending Value

1 $100.00 $6.00 $106.00

2 106.00 6.36 112.36

3 112.36 6.74 119.10

4 119.10 7.15 126.25

5 126.25 7.57 133.82

6 133.82 8.03 141.85

7 141.85 8.51 150.36

8 150.36 9.02 159.38

9 159.38 9.57 168.95

10 168.95 10.13 179.08


The time value of money

Future value and future value interest factor


Fvn pv fvifi n

FVn=PV(FVIFi,n)


The time value of money

Table 5-2

FVIFi,n or the Compound Sum of $1

N 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

1 1.010 1.020 1.030 1.040 1.050 1.060 1.070 1.080 1.090 1.100

2 1.020 1.040 1.061 1.082 1.102 1.124 1.145 1.166 1.188 1.210

3 1.030 1.061 1.093 1.125 1.158 1.191 1.225 1.260 1.295 1.331

4 1.041 1.082 1.126 1.170 1.216 1.262 1.311 1.360 1.412 1.464

5 1.051 1.104 1.159 1.217 1.276 1.338 1.403 1.469 1.539 1.611

6 1.062 1.126 1.194 1.265 1.340 1.419 1.501 1.587 1.677 1.772

7 1.072 1.149 1.230 1.316 1.407 1.504 1.606 1.714 1.828 1.949

8 1.083 1.172 1.267 1.369 1.477 1.594 1.718 1.815 1.993 2.144

9 1.094 1.195 1.305 1.423 1.551 1.689 1.838 1.999 2.172 2.358

10 1.105 1.219 1.344 1.480 1.629 1.791 1.967 2.159 2.367 2.594

11 1.116 1.243 1.384 1.539 1.710 1.898 2.105 2.332 2.580 2.853

12 1.127 1.268 1.426 1.601 1.796 2.012 2.252 2.518 2.813 3.138

13 1.138 1.294 1.469 1.665 1.886 2.133 2.410 2.720 3.066 3.452

14 1.149 1.319 1.513 1.732 1.980 2.261 2.579 2.937 3.342 3.797

15 1.161 1.346 1.558 1.801 2.079 2.397 2.759 3.172 3.642 4.177


The time value of money

PV=$300, Vn=$774; i=9 % N= ?


The time value of money

PV=$100; FVn=$179.10; n=10 years. I= ?


The time value of money

PRESENT VALUE


The time value of money

FV10=$500, n=10, i=6 % PV = ?


Pvif i n

(PVIF i, n)

  • present-value interest factor for I and n (PVIF i, n),

    (PVIF i, n) = 1/(1+i)


Present value

Present value

  • FV10 =$1,500

  • N= 10 years

  • discount rate= 8 %

  • PV=$1500(0.463)

    =$694.50


The time value of money

Table 5-3

PVIFi,n or the Present Value of $1

N 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

1 0.990 0.980 0.971 0.962 0.952 0.943 0.935 0.926 0.917 0.909

2 0.980 0.961 0.943 0.925 0.907 0.890 0.873 0.857 0.842 0.826

3 0.971 0.942 0.915 0.889 0.864 0.840 0.816 0.794 0.772 0.751

4 0.961 0.924 0.888 0.855 0.823 0.792 0.763 0.735 0.708 0.683

5 0.951 0.906 0.863 0.822 0.784 0.747 0.713 0.681 0.650 0.621

6 0.942 0.888 0.837 0.790 0.746 0.705 0.666 0.630 0.596 0.564

7 0.933 0.871 0.813 0.760 0.711 0.655 0.623 0.583 0.547 0.513

8 0.914 0.837 0.766 0.703 0.645 0.592 0.544 0.500 0.460 0.424

9 0.905 0.820 0.744 0.676 0.614 0.558 0.508 0.463 0.422 0.386


Annuities

ANNUITIES

  • Annuity: equal annual cash flows.

  • Ordinary annuity: at the end of each period.

  • Annuity due: at the beginning of each eriod.


The time value of money

  • Table 5-4

  • Illustration of a Five-Year $500 Annuity Compounded at 6 percent

  • YEAR 0 1 2 3 4 5

  • DOLLAR DEPOSITS AT END OF YEAR 500 500 500 500 500

  • $500.00

    • 530.00

    • 562.00

    • 595.50

    • 631.00

  • Future value of the annuity$2,818.50


  • The time value of money

    Ordinary annuity

    FVIFAk,n = [(1/k) ( (1+ k)n – 1)]


    The time value of money

    Present value of an Annuity


    The time value of money

    Table 5-6

    Illustration of a Five-Year $500 Annuity Discounted to the Present at 6 percent

    YEAR 0 1 2 3 4 5

    Dollars received at the 500 500 500 500 500

    the end of year $471.50

    445.00

    420.00

    396.00

    373.50

    PV annuity $2,106.00


    The time value of money

    PVIFAK,n = (1/k) [( 1 – 1/(1+k)n]


    The time value of money

    Table 5-7

    PVIFi,n or the Present Value of an Annuity of $1

    N 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

    1 0.990 0.980 0.971 0.962 0.952 0.943 0.935 0.926 0.917 0.909

    2 1.970 1.942 1.913 1.886 1.859 1.833 1.808 1.783 1.759 1.736

    3 2.941 2.884 2.829 2.775 2.723 2.673 2.642 2.577 2.531 2.487

    4 3.902 3.808 3.717 3.630 3.546 3.465 3.387 3.312 3.240 3.170

    5 4.853 4.713 4.580 4.452 4.329 4.212 4.100 3.003 3.890 3.791

    6 5.795 5.601 5.417 5.242 5.076 4.917 4.767 4.623 4.486 4.355

    7 6.728 6.472 6.230 6.002 5.786 5.582 5.389 5.206 5.033 4.868

    8 7.652 7.326 7.020 6.733 6.463 6.210 5.971 5.747 5.535 5.335

    9 8.566 8.162 7.786 7.435 7.108 6.802 6.515 6.247 5.995 5.759

    10 9.471 8.983 8.530 8.111 7.722 7.360 7.024 6.710 6.418 6.145


    The time value of money

    n=10 years, I=5 percent, and current PMT=$1,000

    PV= $1,000(7.722)

    = $7,722


    The time value of money

    PMT

    Annuity: $5,000, n =5 years, i=8 percent, PMT:?

    $5,000 = PMT (3.993)

    $1,252.19=PMT


    The time value of money

    AMORTIZED LOANS


    The time value of money

    Loan Amortization Schedule Involving a $6,000 Loan at 15 Percent to Be Repaid in Four Years

    Year Annuity Interest Portion Repayment of Outstanding

    Of The Annuity1 The Principal Loan Balance

    Portion Of The After The An-

    Annuity2 nuity Payment

    1 $2,101.58 $900.00 $1,201.58 $4,798.42

    22,101.58 719.76 1,381.82 3,416.60

    3 2,101.58 512.49 1,589.09 1,827.51

    4 2,101.58 274.07 1,827.51


    Annuities due

    ANNUITIES DUE

    • FVn (annuity due)=PMT(FVIFA I,n)(1+I) (5-10)

      FV5=$500(FVIFA5%,5)(1+0.06)

      =$500(5.637)(1.06)

      =$2,987.61

       from $2,106 to $2,232.36,

      PV=$500(PVIFA6%,5)(1+0.06)

      =$500(4.212)(1.06)

      =$2,232.36


    The time value of money

    End year

    Loan payment

    (1)

    Beginning principal

    (2)

    payments

    End of year principal(5)

    [(2)-(4)]

    Interest(3)

    [0.1 × (2)]

    Principal(4)

    [(1) - (3)]

    1

    $1892.74

    $6000.00

    $600.00

    $1292.74

    $4707.26

    2

    $1892.74

    $4707.26

    $470.73

    $1422.01

    $3285.25

    3

    $1892.74

    $3285.25

    $328.53

    $1564.21

    $1721.04

    4

    $1892.74

    $1721.04

    $172.10

    $1720.64


    The time value of money

    The Value of $100 Compounded at Various IntervalsFOR 1 YEAR AT i PERCENT

    I = 2% 5% 10% 15%

    Compounded annually $102.00 $105.00 $110.00 $115.00

    Compounded semiannually 102.01 105.06 110.25 115.56

    Compounded quarterly 102.02 105.09 110.38 115.87

    Compounded monthly 102.02 105.12 110.47 116.08

    Compounded weekly (52) 102.02 105.12 110.51 116.16

    Compounded daily (365) 102.02 105.13 110.52 116.18


    Present value of an uneven stream

    PRESENT VALUE OF AN UNEVEN STREAM

    YEAR CASH FLOW YEAR CASH FLOW

    1 $500 6 500

    2 200 7 500

    3 -400 8 500

    4 500 9 500

    5 500 10 500


    The time value of money

    • Present value of $500 received at the end of one year

    • = $500(0.943) = $471.50

    • 2. Present value of $200 received at the end of tree years

    • = $200(0.890) = 178.00

    • 3. Present value of a $400 outflow at the end of three years

    • = -400(0.840) = -336.00

    • 4. (a) Value at the end of year 3 and a $500 annuity, years 4 through 10

    • = $500 (5.582) = $2,791.00

    • (b) Present value of $2,791.00 received at the end of year 3

    • = 2,791(0.840) = 2,344.44

    • 5. Total present value = $2,657.94


    Quiz 1

    Quiz 1

    Warm up Quiz.

    Terms:

    : n = 5, m = 4, I =12 percent, and PV =$100 solve for fv


    Quiz 2

    Quiz 2

    What is the present value of an investment involving $200 received at the end of years 1 through 4, a $300 cash outflow at the end of year 5 to 8, and $500 received at the end of years 9 through 10, given a 5 percent discount rate?


    Quiz 3

    Quiz 3

    • A 25 year-old graduate has his $50,000 salary a year. How much will he get when he reaches to 60 (35 years later)year-old with a value rate of 8%(annual compounding).

    • The graduate will have his $80,000 salary at age of 30. How much will he get when he reaches to his age of 60(30 years later) with the value rate of 8%(semi-annual compounding).


    Quiz 4

    Quiz 4

    • The graduate will have his $100,000 salary at age of 40. How much will he get when he reaches to his age of 60(20 years later) with the value rate of 12%(quarterly-annual compounding).

    • Compute the future value from 25-30/30-40/40-60 year old with the same rate and the compounding rate.


    Perpetuities

    PERPETUITIES

    $500 perpetuity discounted back to the present at 8 percent?

    PV = $500/0.08 = $6,250


    Power of time of value of money1

    Power of time of value of money

    • History of Interest Rates

      $1000 ( 1 + .08)400 = ?


    Power of time value of money1

    Power of time value of money

    • Money Angles: by Andrew Tobias.

      Chessboard with the King

      Manhattan


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