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Chapter 3
Time Value ofMoney
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After study ing Chapter 3,
you should be able to:
1. Understand what is meant by "the time value of money."
2. Understand the relationship between present and future value.
3. Describe how the interest rate can be used to adjust the value ofcash flows both forward and backward to a single point in
time.4. Calculate both the future and present value of: (a) an amount
invested today; (b) a stream of equal cash flows (an annuity);and (c) a stream of mixed cash flows.
5. Distinguish between an ordinary annuity and an annuity due.
6. Use interest factor tables and understand how they provide ashortcut to calculating present and future values.
7. Build an amortization schedule for an installment-style loan.
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Obviously, $10,000 today.
You already recognize that there isTIME VALUE TO MONEY!!
The In teres t Rate
Which would you prefer -- $10,000today or$10,000 in 5 years?
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TIME provides us with the oppor tun i ty
to put our money to work and earnINTEREST.
Why TIME?
Why is TIME such an important
element in your decision?
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The Interest Rate
Interest is the money paid (earned) for the use ofmoney.
Which should you prefer--- $1,000 today or $2,000ten years from today? To answer this question, itis necessary to position time adjusted cash flowsat a single point in time.
The rate of interest can be used to adjust thevalue of cash flows to a single point in time sothat a fair comparison can be made
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Types o f In teres t
Compound Interest
Interest paid (earned) on any previousinterest earned, as well as on theprincipal borrowed (lent).
Simple Interest
Interest paid (earned) on only the originalamount, or principal, borrowed (lent).
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Simp le In terest Formu la
The dollar amount of SI is a function of three variables:principal; interest rate; and the number of time periods forwhich principal has been borrowed.
Formula SI = P0(i)(n)
SI: Simple Interest
P0: Principal, or original amount borrowedor lent (t=0)
i: Interest Rate per Period
n: Number of Time Periods
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SI = P0
(i)(n)= $1,000(.07)(2)= $140
Simp le In teres t Example
Assume that you deposit $1,000 in anaccount earning 7% simple interest for
2 years. What is the accumulatedinterestat the end o f the 2nd year?
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FV =P0+SI
=$1,000+$140
=$1,140
FVn= P0+ SI = P0+P0(i)(n ) =P0[1 + (i)(n)]
Future Value is the value at some future timeof a present amount of money, or a series of
payments, evaluated at a given interest rate.
Simp le In teres t (FV)
What is the Future Value (FV) of thedeposit?
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ThePresent Valueis s imply the
$1,000you or ig inal ly deposi ted.
That is the value today!
Present Value is the current value of a futureamount of money, or a series of payments,evaluated at a given interest rate.
PV0 = P0 = FVn/ [1 + (i)(n)]
Simp le In teres t (PV)
What is the Present Value (PV) of theprevious problem?
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Assume that you deposit $1,000 ata compound interest rate of7% for
2 years.
Futu re Value
Sing le Depos i t (Graph ic)
0 1 2
$1,000
FV2
7%
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FV1 = P0 (1+i)1 = $1,000(1.07)
= $1,070
Compound Interest
You earned $70 interest on your $1,000
deposit over the first year.This is the same amount of interest you
would earn under simple interest.
Futu re Value
Sing le Depos i t (Formu la)
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FV1 = P0(1+i)1 = $1,000 (1.07)
= $1,070
FV2 = FV1 (1+i)1
= P0 (1+i)(1+i) = $1,000(1.07)(1.07)
= P0(1+i)2
= $1,000(1.07)2
= $1,144.90
You earned an EXTRA $4.90in Year 2 with
compound over simple interest.
Future Value
Sing le Depos i t (Formu la)
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FV1 = P0(1+i)1
FV2 = P0(1+i)2
General Future Value Formula:
FVn = P0 (1+i)n
or FVn = P0 (FVIFi,n) -- See Table I
General Futu re
Value Formu la
etc.
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FVIFi,nis found on Table Iat the end of the book.
Valuat ion Using Tab le I
Period 6% 7% 8%
1 1.060 1.070 1.080
2 1.124 1.145 1.1663 1.191 1.225 1.260
4 1.262 1.311 1.360
5 1.338 1.403 1.469
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FV2 = $1,000 (FVIF7%,2)= $1,000 (1.145)
= $1,145 [Due to Rounding]
Using Futu re Value Tab les
Period 6% 7% 8%
1 1.060 1.070 1.080
2 1.124 1.145 1.1663 1.191 1.225 1.260
4 1.262 1.311 1.360
5 1.338 1.403 1.469
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Julie Miller wants to know how large her deposit
of$10,000 today will become at a compound
annual interest rate of10% for5 years.
Story Prob lem Examp le
0 1 2 3 4 5
$10,000
FV5
10%
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Calculation based on Table I:
FV5 = $10,000(FVIF10%, 5)= $10,000(1.611)= $16,110 [Due to Rounding]
Story Prob lem So lut ion
Calculation based on general formula: FVn = P0 (1+i)
n
FV5 = $10,000 (1+ 0.10)5= $16,105.10
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We will use the Rule-of-72.
Doub le Your Money!!!
Quick! How long does it take todouble $5,000 at a compound rate
of 12% per year (approx.)?
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Approx . Yearsto Double= 72/ i%
72 / 12% = 6 Years
[Actual Time is 6.12 Years]
The Rule-of-72
Quick! How long does it take todouble $5,000 at a compound rate
of 12% per year (approx.)?
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Assume that you need $1,000in 2 years.Lets examine the process to determinehow much you need to deposit today at a
discount rate of7% compounded annually.
0 1 2
$1,000
7%
PV1PV0
Presen t Value
Sing le Depos i t (Graph ic)
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PV0 = FV2 / (1+i)2 = $1,000/ (1.07)2
= FV2
/ (1+i)2 = $873.44
Presen t Value
Sing le Depos i t (Formu la)
0 1 2
$1,000
7%
PV0
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PV0= FV1 / (1+i)1
PV0 = FV2 / (1+i)2
General Present Value Formula:
PV0 = FVn / (1+i)n
or PV0 = FVn (PVIFi,n) -- See Table II
General Present
Value Formu la
etc.
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PVIFi,nis found on Table IIat the end of the book.
Valuation Using Tab le II
Period 6% 7% 8%
1 .943 .935 .926
2 .890 .873 .857
3 .840 .816 .794
4 .792 .763 .735
5 .747 .713 .681
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PV2 = $1,000 (PVIF7%,2)= $1,000 (.873)
= $873 [Due to Rounding]
Using Present Value Tab les
Period 6% 7% 8%
1 .943 .935 .926
2 .890 .873 .8573 .840 .816 .794
4 .792 .763 .735
5 .747 .713 .681
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Julie Miller wants to know how large of adeposit to make so that the money willgrow to $10,000 in 5 years at a discountrate of10%.
Story Prob lem Examp le
0 1 2 3 4 5
$10,000
PV0
10%
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Calculation based on general formula: PV0 = FVn / (1+i)
n
PV0 = $10,000/ (1+ 0.10)5= $6,209.21
Calculation based on Table I:
PV0 = $10,000 (PVIF10%, 5)= $10,000(.621)= $6,210.00 [Due to Rounding]
Story Prob lem So lut ion
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Types o f Annui t ies
Ordinary Annuity: Payments or receiptsoccur at the end of each period.
Annuity Due: Payments or receiptsoccur at the beginning of each period.
An Annu i tyrepresents a series of equalpayments (or receipts) occurring over a
specified number of equidistant periods.
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Examp les o f Annu i ties
Student Loan Payments
Car Loan Payments Insurance Premiums
Mortgage Payments Retirement Savings
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FVAn = R (FVIFAi%,n)FVA3 = $1,000 (FVIFA7%,3)
= $1,000 (3.215) = $3,215
Valuation Us ing Tab le III
Period 6% 7% 8%
1 1.000 1.000 1.000
2 2.060 2.070 2.080
3 3.184 3.215 3.246
4 4.375 4.440 4.506
5 5.637 5.751 5.867
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FVADn = R (FVIFAi%,n)(1+i)FVAD3 = $1,000 (FVIFA7%,3)(1.07)
= $1,000 (3.215)(1.07) = $3,440
Valuation Us ing Tab le III
Period 6% 7% 8%
1 1.000 1.000 1.000
2 2.060 2.070 2.080
3 3.184 3.215 3.246
4 4.375 4.440 4.506
5 5.637 5.751 5.867
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PVAn = R (PVIFAi%,n)PVA3 = $1,000 (PVIFA7%,3)
= $1,000 (2.624) = $2,624
Valuation Using Tab le IV
Period 6% 7% 8%
1 0.943 0.935 0.926
2 1.833 1.808 1.783
3 2.673 2.624 2.577
4 3.465 3.387 3.312
5 4.212 4.100 3.993
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PVADn = R (PVIFAi%,n)(1+i)PVAD3 = $1,000 (PVIFA7%,3)(1.07)
= $1,000 (2.624)(1.07) = $2,808
Valuation Using Tab le IV
Period 6% 7% 8%
1 0.943 0.935 0.926
2 1.833 1.808 1.783
3 2.673 2.624 2.577
4 3.465 3.387 3.312
5 4.212 4.100 3.993
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General Formula:
FVn
= PV0(1 + [i/m])mn
n: Number of Yearsm: Compounding Periods per Year
i: Annual Interest RateFVn,m: FV at the end of Year n
PV0: PV of the Cash Flow today
Frequency of
Compound ing
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Julie Miller has $1,000 to invest for2Years at an annual interest rate of
12%.
Annual FV2 = 1,000(1+ [.12/1])(1)(2)
= 1,254.40Semi FV2 = 1,000(1+ [.12/2])
(2)(2)
= 1,262.48
Impact o f Frequency
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Qrtly FV2 = 1,000(1+ [.12/4])(4)(2)
= 1,266.77
Monthly FV2 = 1,000(1+ [.12/12])(12)(2)
= 1,269.73
Daily FV2 = 1,000(1+[.12/365])(365)(2)= 1,271.20
Impact o f Frequency
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1. Calculate the payment per period.
2. Determine the interest in Period t.
(Loan Balanceat t-1) x (i% / m )3. Compute principal payment in Period t.
(Payment-Interestfrom Step 2)
4. Determine ending balance in Period t.(Balance-pr inc ipal paymentfrom Step 3)
5. Start again at Step 2 and repeat.
Steps to Amort izing a Loan
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Julie Miller is borrowing $10,000 at acompound annual interest rate of12%.
Amortize the loan ifannual payments are
made for5 years.
Step 1: Payment
PV0 = R (PVIFA i%,n)
$10,000 = R (PVIFA 12%,5)
$10,000 = R (3.605)
R = $10,000 / 3.605 = $2,774
Amort izing a Loan Example
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Amort izing a Loan Example
End ofYear
Pa ment Interest Princi al EndinBalance
0 --- --- --- $10 000
1 $2 774 $1 200 $1 574 8 426
2 2 774 1 011 1 763 6 663
3 2 774 800 1 974 4 689
4 2 774 563 2 211 2 478
5 2 775 297 2 478 0
$13 871 $3 871 $10 000
[Last Payment Slightly Higher Due to Rounding]
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Usefulness o f Amort izat ion
2. Calculate Debt Outstanding --The quantity of outstandingdebt may be used in financingthe day-to-day activities of the
1. Determine Interest Expense --Interest expenses may reduce
taxable income of the firm.