SLIDE 1 Time Value of Money
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Chapter 6
SLIDE 2 Learning Objectives
1.
Distinguish between an ordinary annuity and an annuity due, and calculate the present and future values of each.
2.
Calculate the present value of a level perpetuity and a growing perpetuity.
3.
Calculate the present and future values of complex cash flow streams.
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SLIDE 3 Principals Applies in this Chapter
Principle 1: Money Has a Time Value Principle 3: Cash Flows Are the Source of Value.
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SLIDE 4
Ordinary Annuities
An annuity is a series of equal dollar payments that are made at the end of equidistant points in time, such as monthly, quarterly, or annually. If payments are made at the end of each period, the annuity is referred to as ordinary annuity. If payments are made at the beginning of the period, the annuity is referred to as an annuity due.
SLIDE 5 Ordinary Annuities (cont.)
Example How much money will you accumulate by the end
- f year 5 if you deposit $5,000 each year for the next ten
years in a savings account that earns 6% per year?
Determine the answer by using the equation for computing
the FV of an ordinary annuity.
SLIDE 6
Figure 6.1 Future Value of a Five-Year Annuity
SLIDE 7
The Future Value of an Ordinary Annuity
SLIDE 8
The Future Value of an Ordinary Annuity
Using equation 6-1c, FV = $5000 {[ (1+.06)5 - 1] ÷ (.06)}
= $5,000 { [0.3382] ÷ (.06) } = $3,000 {5.6371} = $28,185.46
SLIDE 9
The Future Value of an Ordinary Annuity
Using a Financial Calculator
N=5 I/Y = 6.0 PV = 0 PMT = -5000 FV = $28,185.4648
SLIDE 10
Solving for the PMT in an Ordinary Annuity
SLIDE 11 CHECKPOINT 6. CHECKPOINT 6.1: 1: CHECK Y CHECK YOURSELF URSELF
Solving for PMT If you can earn 12 percent on your investments, and you would like to accumulate $100,000 for your newborn child’s education at the end
- f 18 years, how much must you invest annually to reach your goal?
SLIDE 12 Step 1: Picture the Problem
You would like to save $100,000 for your child’s education
i=12% Years
Cash flow PMT PMT PMT
1 2 … 18
The FV of annuity
for 18 years At 12% = $100, 000 $100,000
We are solving for PMT
SLIDE 13
Step 2: Decide on a Solution Strategy
SLIDE 14
Step 3: Solution (cont.)
Using a Financial
Calculator
N=18 1/y = 12.0 PV = 0 FV = 100000 PMT = -1,793.73
SLIDE 15 Step 4: Analyze
If we contribute $1,793.73 every year for 18 years, we
should be able to reach our goal of accumulating $100,000 if we earn a 12% return on our investments.
Note the last payment of $1,793.73 occurs at the end of year
- 18. In effect, the final payment does not have a chance to
earn any interest.
SLIDE 16
Solving for the Interest Rate in an Ordinary Annuity
You can also solve for “interest rate” you must earn on your
investment that will allow your savings to grow to a certain amount of money by a future date.
In this case, we know the values of T, PMT, and FVT in
equation 6-1c and we need to determine the value of i.
SLIDE 17
Solving for the Interest Rate in an Ordinary Annuity (cont.)
Example: In 20 years, you are hoping to have saved $100,000
towards your child’s college education.
If you are able to save $2,500 at the end of each year for the
next 20 years, what rate of return must you earn on your investments in order to achieve your goal?
SLIDE 18
Solving for the Interest Rate in an Ordinary Annuity (cont.)
Using a Financial
Calculator
N = 20 PMT = -$2,500 FV = $100,000 PV = $0 i = 6.77
SLIDE 19
Solving for the Number of Periods in an Ordinary Annuity
It is easier to solve for number of periods using financial
calculator or Excel spreadsheet, rather than mathematical formula.
SLIDE 20
Solving for the Number of Periods in an Ordinary Annuity (cont.)
Using a Financial
Calculator
1/y = 5.0 PV = 0 PMT = -6,000 FV = 50,000 N = 7.14
Using an Excel Spreadsheet
= NPER(rate, pmt, pv, fv) = NPER(5%,-6000,0,50000) = 7.14 years
SLIDE 21
The Present Value of an Ordinary Annuity
The Present Value (PV) of an ordinary annuity measures the
value today of a stream of cash flows occurring in the future.
Figure 6.2 shows the PV of ordinary annuity of receiving
$500 every year for the next 5 years at an interest rate of 6%?
SLIDE 22
Figure 6.2 Timeline of a Five-Year, $500 Annuity Discounted Back to the Present at 6 Percent
SLIDE 23
The Present Value of an Ordinary Annuity (cont.)
SLIDE 24 Step 1: Picture the Problem
What is the present value of a 10 year ordinary annuity of
$10,000 per year given a 10 percent discount rate?
i=10% Years
Cash flow $10,000 $10,000 $10,000
1 2 … 10
Sum up the present Value of all the cash flows to find the PV of the annuity
SLIDE 25
Step 2: Decide on a Solution Strategy
SLIDE 26
Step 3: Solution (cont.)
Using a Financial
Calculator
N = 10 1/y = 10.0 PMT = -10,000 FV = 0 PV = 61,445.67
SLIDE 27
Annuities Due
Annuity due is an annuity in which all the cash flows occur at the beginning of each period. For example, rent payments on apartments are typically annuities due because the payment for the month’s rent occurs at the beginning of the month.
Most consumer loans are annuities due
SLIDE 28
Annuities Due
Computation of future value of an annuity due requires compounding the cash flows for one additional period, beyond an ordinary annuity. FVADT = (1+i)FVAT Computation of present value of an annuity due requires compounding the cash flows for one additional period, beyond an ordinary annuity. PVADT = (1+i)PVAT
SLIDE 29 Perpetuities
A perpetuity is an annuity that continues forever or has no
- maturity. For example, a dividend stream on a share of
preferred stock. There are two basic types of perpetuities:
Growing perpetuity in which cash flows grow at a constant
rate from period to period over time.
Level perpetuity in which the payments are constant over
time.
SLIDE 30
Calculating the Present Value of a Level Perpetuity
PV = the present value of a level perpetuity PMT = the constant dollar amount provided by the perpetuity i = the interest (or discount) rate per period
SLIDE 31
Present Value of a Growing Perpetuity
In growing perpetuities, the periodic cash flows grow at a constant rate each period.
SLIDE 32
Complex Cash Flow Streams
The cash flows streams in the business world may not always involve one type of cash flows. The cash flows may have a mixed pattern of cash inflows and outflows, single and annuity cash flows.
SLIDE 33 Complex Cash Flows
Suppose you are going to invest $1,000 in a project that will
generate cash flows of $100, 200, 300, 400 and 500 over the next 5 years. The discount rate is 15%.
What it the present value of the cash flows?
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SLIDE 34 Complex Cash Flows
Step 1: Picture the problem
|_____|_____|_____|____|____|
-1000 100 200 300 400 500 Step 2: Decide on a solution strategy
We will need to compute the PV of each cash flow and add
them up
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SLIDE 35 Complex Cash Flows
Step 3: Solution
Year 0: -1000 Year 1: 100/(1.15) = 86.9565 Year 2: 200/(1.15)2 = 151.2287 Year 3: 300/(1.15)3 = 197.2549 Year 4: 400/(1.15)4 = 228.7013 Year 5: 500/(1.15)5 = 248.5884 Sum = -1,000 + 912.7298 = -$87.2702
Step 4: Analyze:
The expenditure is greater than the PV of the cash inflows This is not a good project
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SLIDE 36 Complex Cash Flows
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SLIDE 37
Figure 6-4 Present Value of Single Cash Flows and an Annuity ($ value in millions)
SLIDE 38 Complex Cash Flows
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