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 of value of money

• History of Interest Rates

$1000 ( 1 + .08)400 = ?

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

Future value

1.Simple compounding

2.Complex compounding

nm

m

kPVFV )1(

n

t

iFV )1(100

Future value

• FV1=PV (1+i)

• =$100(1+0.06)

• =$100(1.06)

• =$106

09.106)2

06.01(100 2 FV

Compound twice a year

14.106)4

06.01(100 4 FV

Compound four times a year

17.106)12

06.01(100 12 FV

Compound 12 times a year

18.106)360

06.01(100 360 FV

Compound 360 times a year

1836.106100 )106.0( ne

Continuous compounding

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

%8

108.1

1)08.01(

1)1

08.01(

1

1

effK

%16.8

10816.1

1)04.01(

1)2

08.01(

2

2

effK

%24.8

10824.1

1)02.01(

1)4

08.01(

4

4

effK

89.628,1$

)62889.1(000,1$

)05.01(000,1$

)1(10

nn iPVFV

Future value and future value interest factor

FVn=PV(FVIFi,n)

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

n

n

nn ipvFV

)09.01(58.2

)09.01(300$774$

)1(

PV=$300, Vn=$774; i=9 % N= ？

10

10

)1(791.1

)1(100$10.179$

)1(

i

i

iPVFV nn

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

ninFVPV)1(

1

PRESENT VALUE

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

279$

)558.0(500$

)(500$

500$

791.11

)06.01(1

10

PV

(PVIF i, n)

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

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

Present value

• FV10 =$1,500

• N= 10 years

• discount rate= 8 %

• PV=$1500(0.463)

=$694.50

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

• Annuity: equal annual cash flows.

• Ordinary annuity: at the end of each period.

• Annuity due: at the beginning of each eriod.

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

 

50.818,2$

00.500$00.530$00.562$50.595$00.631$

500$

)060.1(500$)124.1(500$)191.1(500$)262.1(500$

500$)06.01(500$

)06.01(500$)06.01(500$)06.01(500$ 2345

FV

1

0

)1(n

t

tn iPMTFV

FVIFAk,n = [(1/k) ( (1+ k)n – 1)]

Ordinary annuity

106,2$

)747.0(500$)792.0(500$)840.0(500$)890.0(500$)943.0(500$

500$500$

500$500$500$

54

32

)06.01(1

)06.01(1

)06.01(1

)06.01(1

)06.01(1

PV

Present value of an Annuity

n

tiPMTPV

1)1(

1

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

n

tiPMTPV

1)1(

1

PVIFAK,n = (1/k) [( 1 – 1/(1+k)n]

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

)(000,1$000,1$ 10%,5)05.01(1

yrPVIFAPV t

PV= $1,000(7.722)

= $7,722

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

PMT

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

$5,000 = PMT (3.993)

$1,252.19=PMT

PMT

PMT

PVIFAPMT

PMT

yr

tt

58.101,2$

)855.2(000,6$

)(000,6$

000,6$

4%,15

4

1)15.01(

1

AMORTIZED LOANS

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

2 2,101.58 719.76 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

 

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 Value of $100 Compounded at Various Intervals FOR 1 YEAR AT i PERCE

NT

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

1. 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

Warm up Quiz.

Terms:

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

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

1 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).

2 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

3. 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).

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

PERPETUITIES

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

PV = $500/0.08 = $6,250

Power of time of value of money

• History of Interest Rates

$1000 ( 1 + .08)400 = ?

Power of time value of money

• Money Angles: by Andrew Tobias.

Chessboard with the King

Manhattan