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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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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

slide10
FV1=PV (1+i)
  • =$100(1+0.06)
  • =$100(1.06)
  • =$106
slide16

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

slide22

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

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

slide29

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.
slide31

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
slide33

Ordinary annuity

FVIFAk,n = [(1/k) ( (1+ k)n – 1)]

slide35

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

slide37

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

slide39
PMT

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

$5,000 = PMT (3.993)

$1,252.19=PMT

slide41

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

slide43

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

slide44

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

slide46

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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