Chapter 8: The Time Value of Money

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The Time Value of Money

• Chapter 8

Mar 9, 2009
2
Learning Objectives

• The time value of money and its importance to
  business. (Axiom 2)
• The future value and present value of a lump sum,
  single amount.
• The future value and present value of a series of
  payments/receipts, an annuity.
• How to use a financial calculator Buy it Now!
• Applications of time value of money concepts,
  i.e., business decisions, which alternative to
  choose, loan payments, lottery winnings, etc.

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The Time Value of Money

• Money grows over time when it earns interest.
  This is called compounding.
• Money that is to be received at some time in the
  future is worth less than the same dollar amount
  to be received today, due to inflation.
• Similarly, a debt of a given amount to be paid in
  the future is less burdensome than that debt to
  be paid now

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A few examples

• Indians/Manhattan Island
• 16 year old, summer jobs
• 30 year mortgage payments,
• 100,000 500, 27 years ago
• 1,000,000 5,000 today
• I was still paying 500 after 27 years
• Today I am paying nothing!
• Time value of money

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The Future Value of a Single Amount

• Suppose that you have 100 today and plan to put
  it in a bank account that earns 8 per year.
• How much will you have after 1 year? 3 years?
• 10 years?
• After one year
• 100 (.08 x 100) 100 8 108
• OR 100(1 .08) or 100 (1.08)1 108
• OR after three years
• 100(1.08)(1.08)(1.08) 100(1.08)3
• (1.08 x 1.08 x 1.08 1.2597)
• 100(1.2597) 125.97

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The Future Value of a Single Amount

• Suppose that you have 100 today and plan to put
  it in a bank account that earns 8 per year. How
  much will you have

• After five years
• 100 (1.08)5 100 (1.4693) 146.93
• Equation

• Table Method

• Equation

FV PV (FVIF k, n)
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Future Value of a Single Amount

• Table Method (see table I, p. 485 A1)
• Formula FV PV (FVIF k, n)
• What is the Future Value of 100, invested for 3
  years (n) at 5 (k) interest?
• FV 100 (FVIF, 5, 3 years)
• FVIF (1.05)3 1.1576
• 100 (1.1576) 115.76
• Invested for 5 years at 8 Interest?
• 100 (1.4693) 146.93

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Financial Calculator Solution - FV
Example You invest 100 at 8. How much is it
worth after 5 years?
Using Formula
FV 100 (1.08)5 146.93
Calculator Enter N 5 I/YR 8 PV
-100 PMT 0 CPT FV ?
146.93
5 8 -100 0 ?
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Present Value of a Single Amount

• Value today of an amount to be received or paid
  in the future.

Example You need to have 100 in 8 years in
order to buy and IPOD. If you can invest your
money at 10, how much would you have to invest
today? Example, zero coupon bond
100
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Present Value of a Single Amount

• Table method (see table II, p. 486 A-2)
• Formula PV FV (PVIF k, n)
• What amount must be invested today at 5 to
  accumulate to 115.76 in 3 years?
• PV 115.76 (PVIF 5, 3 years)
• 115.76 (0.8638) 100.00
• What is the present value of 100 to be received
  in 8 years at a discount rate of 10
• 100 (PVIF 10, 8) 100 (0.4665)
  46.65 (Savings Bond)

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Financial Calculator Solution - PV
Previous Example Expect to receive 100 in EIGHT
years. If can invest at 10, what is it worth
today?
Calculator Enter N 8 I/YR 10 FV
100 PMT 0 CPT PV ?
- 46.65
0
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Annuities

• An annuity is a series of equal cash flows spaced
  evenly over time.
• For example, you pay your landlord an annuity
  since your rent is the same amount, paid on the
  same day of the month for the entire year (or
  your car payment)

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Future Value of an Annuity
You deposit 100 each year (end of year) into a
savings account for your retirement. How much
would this account have in it at the end of 3
years if interest were earned at a rate of 8
annually?
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Future Value of an Annuity
0 1 2 3
0
100
100
100
How much would this account have in it at the end
of 3 years if interest were earned at a rate of
8 annually?
100(3.2464) 324.64
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Future Value of an Annuity

• Table method formula (see table III, p. 487 or
  p. A-3)
• FVA PMT (FVIFA k, n)
• What is the future value of an annuity of 100
  for 3 years at 8 interest?, or
• FVA 100 (FVIFA 8, 3)
• FVa 100 (3.2464) 324.64

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Future Value of an Annuity Calculator Solution
Enter N 3 I/YR 8 PMT -100 PV
0 CPT FV ?
324.64
3 8 0 -100 ?
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Present Value of an Annuity

• How much would the following cash flows be worth
  to you today if you could earn 8 on your
  deposits?

• How much would I be willing to pay for an income
  producing asset that would give me 100 per year
  return for three years if I wish to earn say 8
  on my investments?

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Present Value of an Annuity

• How much would the following cash flows be worth
  to you today if you could earn 8 on your
  deposits?

100(2.5771) 257.71
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Present Value of an Annuity

• Table Method (see table IV, p. 488 or A - 4)
• Formula PVA PMT (PVIFA k, n)
• What is present value of an annuity of 100 that
  will pay 8 interest for 3 years
• PVA 100 (PVIFA 8, 3) 100 (2.5771)
  257.71
• Investment examples
• Machine cost 250. Buy?
• Machine cost 260. Buy?

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Present Value of an Annuity Calculator Solution
PV?
Enter N 3 I/YR 8 PMT 100 FV
0 CPT PV ?
-257.71
3 8 ? 100 0
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Annuity Due

• An annuity is a series of equal cash payments
  spaced evenly over time. (draw timeline)
• Ordinary Annuity The cash payments occur at the
  END of each time period.
• Annuity Due The cash payments occur at the
  BEGINNING of each time period.
• Lottery is an annuity due see example, p 210
  (183)
• (How many of you buy lottery tickets?)

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Future Value of an Annuity Due
You deposit 100 each year (beginning of year)
into a savings account. How much would this
account have in it at the end of 3 years if
interest were earned at a rate of 8 annually?
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Future Value of an Annuity Due
How much would this account have in it at the end
of 3 years if interest were earned at a rate of
8 annually?
100(3.2464)(1.08)350.61
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Future Value of an Annuity Due

• Table Method
• FVad PMT(FVIFA k,n) (1 k)
• How much would this account have in it at the end
  of 3 years if interest were earned at the rate of
  8 annually
• FVad 100 (3.2464)(1 .08)
• FVad 324.64 (1.08) 350.61

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Present Value of an Annuity Due

• How much would the following cash flows be worth
  to you today if you could earn 8 on your
  deposits?

100 / (1.08)2
100/(1.08)1
100.00
92.60
85.73
278.33
100(2.5771)(1.08) 278.33
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Present Value of an Annuity Due

• Table method (see table IV, A - 4)
• Formula PMT PMT (PVIFA k, n-1)
• 100 for 3 years at 8
• PVad 100 100(PVIFA 8, 2 years)
• PVad 100 100(1.7833)
• 100 178.33 278.33,
• or
• Formula PVad PMT(PVIF k,n)(1 k)
• PVad 100(2.5771) x (1.08) 278.33

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Annuity Due Calculator Solution

• Same as regular annuity, except
• Multiply the answer by (1 K)

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Amortized Loans

• A loan that is paid off in equal amounts that
  include principal as well as interest.
• Solving for loan payments.

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Amortized Loans

• You borrow 5,000 from your parents to purchase a
  used car. You agree to make payments at the end
  of each year for the next 5 years. If the
  interest rate on this loan is 6, how much is
  your annual payment?

ENTER N 5 I/YR 6 PV 5,000 FV
0 CPT PMT ?
1,186.98
5 6 5,000 ? 0
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Compounding more than Once per Year

• If m number of compounds, then
• N n x m and K k / m

Compounding Frequency
Annual i.e. N 4 K 12 Semi-annual N
4 x 2 8 K 12 / 2 6
Quarterly N 4 x 4 16 K 12 / 4
3 Monthly N 4 x 12 48 K 12 / 12
1
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Amortized Loans

• You borrow 20,000 from the bank to purchase a
  used car. You agree to make payments at the end
  of each month for the next 4 years. If the annual
  interest rate on this loan is 9, how much is
  your monthly payment?

20,000 PMT(40.184782)
PMT 497.70
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Amortized Loans

• You borrow 20,000 from the bank to purchase a
  used car. You agree to make payments at the end
  of each month for the next 4 years. If the annual
  interest rate on this loan is 9, how much is
  your monthly payment?

ENTER N 48 I/Mo. .75 PV 20,000 FV
0 CPT PMT ?
497.70
Note N 4 12 48 I/YR 9/12 .75
48 .75 20,000 ? 0
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Amortized Loans

• Say you know how much you can afford to pay
  monthly. How much can you borrow to buy that new
  car?
• If you can afford 500 per month, and interest
  rates are 6, for 5 years, then
• N 5 x 12 60 i/y 6 / 12 0.5
• Loan 500( PVIFA 0.5, 60)
• Loan 25,862.78 Prius!

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Perpetuities

• A perpetuity is a series of equal payments at
  equal time intervals (an annuity) that will be
  received into infinity.

• Example A share of preferred stock pays a
  constant dividend of 5 per year. What is the
  present value if k 8?

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Perpetuities

• A perpetuity is a series of equal payments at
  equal time intervals (an annuity) that will be
  received into infinity. (i.e., a defined benefit
  retirement plan)

Example A share of preferred stock pays a
constant dividend of 5 per year. What is the
present value if k 8?

• If k 8 PVP 5/.08 62.50

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Solving for k, in any of the formulas

• If you know three of the four elements (n, i/y,
  pv, fv) you can solve for the fourth
• Example, if FV 230, PV 200, and n 2,
• Using FV PV (FVIF k,n), and solving for k,
• 230 200 (FVIF k, 2), or
• FVIF k, 2 230 / 200 1.15
• Looking at n 2 for 1.15, it falls between 7
  (1.1449) and 8 (1.1664). Interpolating, we get
• K 7.24

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Solving for k - Calculator Solution
Example A 200 investment has grown to 230
over two years. What is the ANNUAL return on this
investment?
Enter known values N 2 I/YR ? PV
-200 PMT 0 FV 230 Solve for I/YR
?
7.24
0
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Solving for n, in any of the formulas

• Same as solving for k
• If FV 230, PV 200, and k 7.24,
• Using FV PV(FVIF k,n), and solving for n
• 230 200(FVIF 7.24, n), or
• FVIF 7.24, n 230/200 1.15
• Looking between 7 an 8 for 1.15, you find it at
  n 2
• Calculator solution

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Solving for n - Calculator Solution
Example A 200 investment has grown to 230 at
a 7.24 rate of return. How long did it take?
Enter known values N ? I/YR 7.24 PV
-200 PMT 0 FV 230 Solve for n ?
2.00
0
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REVIEW

• Review self test questions (p. 226 227)
• Which formula are you using
• What are you solving for
• Equations, page 225-226