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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 .
- 6. Determine the present value of a perpetuity.

Power of time value of money

- Money Angles: by Andrew Tobias.

1. Chessboard with the King

2. Manhattan

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

- 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

FV1=PV (1+i)

- =$100(1+0.06)
- =$100(1.06)
- =$106

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

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

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

- Annuity: equal annual cash flows.
- Ordinary annuity: at the end of each period.
- Annuity due: at the beginning of each eriod.

- 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

FVIFAk,n = [(1/k) ( (1+ k)n – 1)]

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

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

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

- 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

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 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

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

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 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

- 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

- 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.

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