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Chap 3 Bf-final

Apr 14, 2018

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Junaid Malik
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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.