0 5 1 Assume interest rates are 43884 You have just won a lottery and must choose between the following two options Receive a cheque for 150000 today Receive 10000 a year for the next 25 years ID: 131970
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Slide1
Introduction to Valuation: The Time Value of Money
0
5Slide2
1
Assume interest rates are 4.3884%.
You have just won a lottery and must choose between the following two options:
Receive a
cheque for $150,000 today.Receive $10,000 a year for the next 25 years.
KEY QUESTIONS FOR YOU:
Which option gives you the biggest “winnings”
How should you tackle this kind of problem?
Slide3
Be able to compute the future value of an investment made today
Be able to compute the present value of cash to be received at some future date
Be able to compute the return on an investment
Be able to compute the number of periods that equates a present value and a future value given an interest rate
Be able to use a financial calculator and/or a spreadsheet to solve time value of money problems2
Key Concepts and SkillsSlide4
3
Consider the time line below:
PV is the Present Value, that is, the value
today
.
FV is the Future Value, or the value at a future date.
The number of time periods between the Present Value and the Future Value is represented by “t”.The rate of interest is called “r”. All time value questions involve the four values above: PV, FV, r, and t. Given three of them, it is always possible to calculate the fourth.
Time Value Terminology
0
1
2
3
t
PV
FV
……..Slide5
Present Value – earlier money on a time line
Future Value – later money on a time lineInterest rate – “exchange rate” between earlier money and later moneyDiscount rateCost of capital
Opportunity cost of capital
Required return
4Basic DefinitionsSlide6
Suppose you invest $1,000 for one year at 5% per year. What is the future value in one year?
Interest = 1,000(.05) = 50
Value in one year = principal + interest = 1,000 + 50 = 1,050
Future Value (FV) = 1,000(1 + .05) = 1,050
Suppose you leave the money in for another year. How much will you have two years from now?FV = 1,000(1.05)(1.05) = 1,000(1.05)2 = 1,102.505
1. Future ValuesSlide7
Simple interest
Compound interestConsider the previous exampleFV with simple interest = 1,000 + 50 + 50 = 1,100FV with compound interest = 1,102.50
The extra 2.50 comes from the interest of .05(50) = 2.50 earned on the first interest payment
6
Effects of CompoundingSlide8
7
Future Value of $100 at 10 Percent
Beginning Simple Compound Total
Total
Year Amount Interest
Interest
Interest
Amount
1 $100.00 $10.00 $ 0.00 $10.00 $110.00
2 110.00 10.00 1.00 11.00 121.00
3 121.00 10.00 2.10 12.10 133.10
4 133.10 10.00 3.31 13.31 146.41
5 146.41 10.00 4.64 14.64 161.05
Totals $50.00 $ 11.05 $ 61.05
Slide9
FV = PV(1 + r)t
FV = future valuePV = present valuer = period interest rate, expressed as a decimalT = number of periods
Future value interest factor = (1 + r)
t
8Future Values: General FormulaSlide10
Suppose you invest the $1,000 from the previous example for 5 years. How much would you have?
FV = 1,000(1.05)5 = 1,276.28The effect of compounding is small for a small number of periods, but increases as the number of periods increases. (Simple interest would have a future value of $1,250, for a difference of $26.28.)
9
Future Values – Example 2Slide11
Suppose you had a relative deposit $10 at 5.5% interest 200 years ago. How much would the investment be worth today?
FV = 10(1.055)200 = 447,189.84What is the effect of compounding?
Simple interest = 10 + 200(10)(.055) = 120.00
Compounding added $447,069.84 to the value of the investment
10Future Values – Example 3Slide12
Suppose your company expects to increase unit sales of widgets by 15% per year for the next 5 years. If you currently sell 3 million widgets in one year, how many widgets do you expect to sell in 5 years?
FV = 3,000,000(1.15)5 = 6,034,072
11
Future Value as a General Growth FormulaSlide13
What is the difference between simple interest and compound interest?
Suppose you have $500 to invest and you believe that you can earn 8% per year over the next 15 years.How much would you have at the end of 15 years using compound interest?How much would you have using simple interest?
12
Example 4Slide14
13
Example 5:
Q. You have just won a $1 million jackpot in the provincial lottery. You can
buy a ten year certificate of deposit which pays 6% compounded
annually. Alternatively, you can give the $1 million to your brother-in-law,
who promises to pay you 8%
simple interest
annually over the ten year
period. Which alternative will provide you with more money at the
end of ten years?Slide15
Deposit $5,000 today in an account paying 12%. How much will you have in 6 years? How much is simple interest? How much is compound interest?
14
Example 6:Slide16
How much do I have to invest today to have some amount in the future?
FV = PV(1 + r)tRearrange to solve for PV = FV / (1 + r)t
When we talk about discounting, we mean finding the present value of some future amount.
When we talk about the “value” of something, we are talking about the present value unless we specifically indicate that we want the future value.
152. Present ValuesSlide17
Suppose you need $10,000 in one year for the down payment on a new car. If you can earn 7% annually, how much do you need to invest today?
PV = 10,000 / (1.07)1 =
9,345.79
16
Present Value – One Period ExampleSlide18
You want to begin saving for your daughter’s college education and you estimate that she will need $150,000 in 17 years. If you feel confident that you can earn 8% per year, how much do you need to invest today?
PV = 150,000 / (1.08)17 = 40,540.34
17
Present Values – Example 2Slide19
Your parents set up a trust fund for you 10 years ago that is now worth $19,671.51. If the fund earned 7% per year, how much did your parents invest?
PV = 19,671.51 / (1.07)10 = 10,000
18
Present Values – Example 3Slide20
For a given interest rate – the longer the time period, the lower the present value
What is the present value of $500 to be received in 5 years? 10 years? The discount rate is 10%5 years: PV = 500 / (1.1)5
= 310.46
10 years: PV = 500 / (1.1)
10 = 192.7719Present Value – Important Relationship ISlide21
For a given time period – the higher the interest rate, the smaller the present value
What is the present value of $500 received in 5 years if the interest rate is 10%? 15%?Rate = 10%: PV = 500 / (1.1)5
= 310.46
Rate = 15%; PV = 500 / (1.15)
5 = 248.5920Present Value – Important Relationship IISlide22
What is the relationship between present value and future value?
Suppose you need $15,000 in 3 years. If you can earn 6% annually, how much do you need to invest today?If you could invest the money at 8%, would you have to invest more or less than at 6%? How much?
21
Example 4Slide23
Suppose you need $3,000 next year to buy a new computer. The interest rate is 8 percent per year. How much money should you set aside now in order to pay for the purchase?
What if you can wait 2 years till you purchase the computer, how much money should you set aside?
22
Example 5Slide24
23
Present Value vs Future Value
PV and FV are related!
Have you noticed that $620,921 becomes $1 million (and that $1 million requires $620,921) if you have a time period of 5 years and a discount rate of 10%?
$620,921
$1,000,000
FV at 10%
PV at 10%Slide25
You should understand that a dollar received today is worth more than a dollar received tomorrow.
Lesson: The value of cash flows received at different times can never be directly
compared. You must
first
discount all cash flows to a common date and then compare them. 24Key PrincipleSlide26
Kangaroo Autos is offering free credit on a $10,000 car. You pay $4,000 down and then the balance at the end of 2 years. Turtle Motors next door does not offer free credit buy will give you $500 off the list price. If the interest rate is 10 percent, which company is offering the better deal?
Suppose your friend wants to buy a car but does not like to be indebt to the car company. What would you recommend?
25
Example 6: To see if you really understand the concept of PV?Slide27
PV = FV / (1 + r)
tThere are four parts to this equationPV, FV, r and t
If we know any three, we can solve for the
fourth
26The Basic PV Equation - RefresherSlide28
Often we will want to know what the implied interest rate is in an investment
Rearrange the basic PV equation and solve for rFV = PV(1 + r)t
r = (FV / PV)
1/t
– 1273. Discount RateSlide29
You are looking at an investment that will pay $1,200 in 5 years if you invest $1,000 today. What is the implied rate of interest?
r = (1,200 / 1,000)1/5 – 1 = .03714 = 3.714
%
28
Discount Rate – Example 1Slide30
Suppose you are offered an investment that will allow you to double your money in 6 years. You have $10,000 to invest. What is the implied rate of interest?
r = (20,000 / 10,000)1/6 – 1 = .122462 = 12.25%
29
Discount Rate – Example 2Slide31
Suppose you have a 1-year old son and you want to provide $75,000 in 17 years towards his college education. You currently have $5,000 to invest. What interest rate must you earn to have the $75,000 when you need it?
r = (75,000 / 5,000)1/17
– 1 = .172688 = 17.27%
30
Discount Rate – Example 3Slide32
What are some situations in which you might want to know the implied interest rate?
You are offered the following investments:You can invest $500 today and receive $600 in 5 years. The investment is considered low risk.
You can invest the $500 in a bank account paying 4%.
What is the implied interest rate for the first choice and which investment should you choose?
31Example 4Slide33
32
Benjamin Franklin died on April 17, 1790. In his will, he gave 1,000 dollars to the city of Boston and $1000 to the city of Philadelphia. It was agreed that the money would be paid out 200 years after Franklin’s death in 1990. By that time, the Philadelphia bequest had grown to about $2 million; the Boston bequest had grown to $4.5 million. What was the rate of return for the two states earn?
Example 5Slide34
33
Suppose you deposit $5000 today in an account paying
r
percent per year. If you will get $10,000 in 10 years, what
rate of return are you being offered?Example 6Slide35
Start with basic equation and solve for t (remember your logs)
FV = PV(1 + r)tt = ln(FV / PV) /
ln
(1 + r
)344. Finding the Number of PeriodsSlide36
You want to purchase a new car and you are willing to pay $20,000. If you can invest at 10% per year and you currently have $15,000, how long will it be before you have enough money to pay cash for the car?
t = ln(20,000 / 15,000) / ln(1.1) = 3.02 years
35
Number of Periods – Example 1Slide37
Suppose you want to buy a new house. You currently have $15,000 and you figure you need to have a 10% down payment plus an additional 5% of the loan amount for closing costs. Assume the type of house you want will cost about $150,000 and you can earn 7.5% per year, how long will it be before you have enough money for the down payment and closing costs?
36
Number of Periods – Example 2Slide38
When might you want to compute the number of periods?
Suppose you want to buy some new furniture for your family room. You currently have $500 and the furniture you want costs $600. If you can earn 6%, how long will you have to wait if you don’t add any additional money?
37
Example 3Slide39
38
Suppose you need to come up with $40,000 for a world cruise. You have only $20,000. You know you can earn 15% interest per year. How many years do you have to wait?
Example 4Slide40
Many financial calculators are available online
Click on the web surfer to go to Investopedia’s web site and work the following example:You need $50,000 in 10 years. If you can earn 6% interest, how much do you need to invest today?You should get $27,919.74
39
Work the Web ExampleSlide41
Use the following formulas for TVM calculations
FV(rate,nper,pmt,pv)PV(rate,nper,pmt,fv)
RATE(nper,pmt,pv,fv)
NPER(rate,pmt,pv,fv)
The formula icon is very useful when you can’t remember the exact formulaClick on the Excel icon to open a spreadsheet containing four different examples.40Spreadsheet ExampleSlide42
41
Table 5.4Slide43
You have $10,000 to invest for five years.
How much additional interest will you earn if the investment provides a 5% annual return, when compared to a 4.5% annual return?How long will it take your $10,000 to double in value if it earns 5% annually?
What annual rate has been earned if $1,000 grows into $4,000 in 20 years?
42
Comprehensive ProblemSlide44
Discounted Cash Flow Valuation
43
6Slide45
Be able to compute the future value of multiple cash flows
Be able to compute the present value of multiple cash flowsBe able to compute loan payments
Be able to find the interest rate on a loan
Understand how interest rates are quoted
Understand how loans are amortized or paid off44Key Concepts and SkillsSlide46
45
Multiple Cash Flows
Future Value Calculations
So far, we have looked at problems involving only a single cash flow.
This is unrealistic – most business investments will involve multiple cash flows over time.
We need a method for coping with such streams of cash flows!Slide47
46
Suppose we receive $2,000 each year starting next year for 5 years – what is FV?
Future value calculated by compounding forward one period at a time or each cash flow (to time 5) separately.Slide48
Assume interest rates are 8%.
You make 3 deposits to your bank account:$1,200 today
$1,400 one year later.
$1,000 two years later.
How much will you have in three years time?47Example 1Slide49
48
Doing Future Value Calculations
Calculate what each cash flow will be worth at the specified future date and add up these future values.
0
1
2
3
$1,200
$1,400
$1,000
$1,080.00 = $1,000 x 1.08
$1,632.96 = $1,400 x (1.08)
2
$4,224.61
FV in Year 3:
$1,511.65 = $1,200 x (1.08)
3
Slide50
Find the value at year 3 of
the following cash flows: Year 0 cash flow is 7000, year 1, 2, and 3 cash flows are 4000 each. Today (year 0): FV = 7000(1.08)
3
= 8,817.98
Year 1: FV = 4,000(1.08)2 = 4,665.60Year 2: FV = 4,000(1.08) = 4,320Year 3: value = 4,000Total value in 3 years = 8,817.98 + 4,665.60 + 4,320 + 4,000 = 21,803.58Value at year 4 = 21,803.58(1.08) = 23,547.87
49Example 2Slide51
Suppose you invest $500 in a mutual fund today and $600 in one year. If the fund pays 9% annually, how much will you have in two years?
FV = 500(1.09)2 + 600(1.09) = 1,248.05
50
Example 3Slide52
How much will you have in 5 years if you make no further deposits?
First way:FV = 500(1.09)5 + 600(1.09)
4
= 1,616.26
Second way – use value at year 2:FV = 1,248.05(1.09)3 = 1,616.2651Example 3 ContinuedSlide53
Suppose you plan to deposit $100 into an account in one year and $300 into the account in three years. How much will be in the account in five years if the interest rate is 8
%?
52
Example 4Slide54
53
PV calculation - what would we do to calculate PV of a $1,000 payment in the next 5 years?Slide55
You wish to buy a car making three installments:
$8,000 today$4,000 one year later.
$4,000 two years later.
How much money would you have to place in an account today to generate this stream of cash flows?
54PV- Example 1Slide56
55
Present Value Calculations
You would need to place $15,133.06 in an account today to generate the desired cash flows:
0
1
2
-$8,000
-$4,000
-$4,000
$8,000.00
$4,000 / (1.08) = $3,703.30
$4,000 / (1.08)
2
= $3,429.36
$15,133.06
PV today:
Slide57
56
Example 2: Find the PV of each cash flows and add them
0
1
2
3
4
200
400
600
800
178.57
318.88
427.07
508.41
1,432.93Slide58
You can use the PV or FV functions in Excel to find the present value or future value of a set of cash flows
Click on the Excel icon for an example
57
Multiple Cash Flows Using a SpreadsheetSlide59
You are considering an investment that will pay you $1,000 in one year, $2,000 in two years and $3000 in three years. If you want to earn 10% on your money, how much would you be willing to pay
?
58
Example 3Slide60
Another way to use the financial calculator for uneven cash flows is to use the cash flow keys
Usually you press CF and enter the cash flows beginning with year 0.
In Excel you can use the NPV function.
59
Multiple Uneven Cash Flows – Using the CalculatorSlide61
Suppose you are looking at the following possible cash flows: Year 1 CF = $100; Years 2 and 3 CFs = $200; Years 4 and 5 CFs = $300. The required discount rate is 7%
What is the value of the cash flows at year 5?What is the value of the cash flows today?What is the value of the cash flows at year 3?
60
Question