This document provides an overview of time value of money concepts including simple and compound interest, present and future value, and annuities. Some key points include:
- Compound interest earns interest on interest, resulting in higher total interest compared to simple interest over time.
- The future value of a single deposit or investment can be calculated using a formula that compounds the principal by the interest rate over multiple periods.
- The present value of a future amount can be calculated by discounting the future value back using the interest rate.
- Annuities represent a series of equal periodic payments or receipts, and their future or present value can be calculated using annuity formulas that take into account all
What is the 'Time Value of Money - TVM'
The time value of money (TVM) is the idea that money available at the present time is worth more than the same amount in the future due to its potential earning capacity. This core principle of finance holds that, provided money can earn interest, any amount of money is worth more the sooner it is received. TVM is also referred to as present discounted value.
BREAKING DOWN 'Time Value of Money - TVM'
Money deposited in a savings account earns a certain interest rate. Rational investors prefer to receive money today rather than the same amount of money in the future because of money's potential to grow in value over a given period of time. Money earning an interest rate is said to be compounding in value.
BREAKING DOWN 'Compound Interest'
Compound Interest Formula
Compound interest is calculated by multiplying the principal amount by one plus the annual interest rate raised to the number of compound periods minus one.The total initial amount of the loan is then subtracted from the resulting value.
this is a lecture on time value of money which explains the topic time value of money in a very easy and simple way... it also explains some examples on the topic... plus definition of rate of return, real rate of return, inflation premium, nominal interest rate,market risk, maturity risk,liquidity risk,and default risk,
What is the 'Time Value of Money - TVM'
The time value of money (TVM) is the idea that money available at the present time is worth more than the same amount in the future due to its potential earning capacity. This core principle of finance holds that, provided money can earn interest, any amount of money is worth more the sooner it is received. TVM is also referred to as present discounted value.
BREAKING DOWN 'Time Value of Money - TVM'
Money deposited in a savings account earns a certain interest rate. Rational investors prefer to receive money today rather than the same amount of money in the future because of money's potential to grow in value over a given period of time. Money earning an interest rate is said to be compounding in value.
BREAKING DOWN 'Compound Interest'
Compound Interest Formula
Compound interest is calculated by multiplying the principal amount by one plus the annual interest rate raised to the number of compound periods minus one.The total initial amount of the loan is then subtracted from the resulting value.
this is a lecture on time value of money which explains the topic time value of money in a very easy and simple way... it also explains some examples on the topic... plus definition of rate of return, real rate of return, inflation premium, nominal interest rate,market risk, maturity risk,liquidity risk,and default risk,
CFA LEVEL 1- Time Value of Money_compressed (1).pdfAlison Tutors
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This document focuses on End of Chapter questions and commonly asked questions under Quantitative methods (Time Value of Money ) . The mainly asked questions include :
-calculation and interpretation of Future Value and Present Value of a single sum of money , an ordinary annuity, annuity due, a perpetuity (PV only) and a series of unequal cash flows.
-demonstration of the use of timelines in modeling and solving time value of money
This is a very interesting topic!
CHAPTER 9Time Value of MoneyFuture valuePresent valueAnn.docxtiffanyd4
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CHAPTER 9
Time Value of Money
Future value
Present value
Annuities
Rates of return
Amortization
9-โน#โบ
1
Time lines
Show the timing of cash flows.
Tick marks occur at the end of periods, so Time 0 is today; Time 1 is the end of the first period (year, month, etc.) or the beginning of the second period.
CF0
CF1
CF3
CF2
0
1
2
3
I%
9-โน#โบ
2
Drawing time lines
100
100
100
0
1
2
3
I%
3 year $100 ordinary annuity
100
0
1
2
I%
$100 lump sum due in 2 years
9-โน#โบ
3
Future Value of Money
If you deposit $1,000 today at 10%, how much will you have after 15 years?
Interest($) = Principal โ Interest Rate(%)
Simple Interest
The original principal stays the same.
There is no interest on interest. The interest is only on the original principal.
Compound Interest
The principal changes through time.
There is โinterest on interestโ. The interest is on the new principal.
9-โน#โบ
9-โน#โบ
9-โน#โบ
Simple Interest
Interest($) = Principal($) โ Interest Rate(%) = V0 โ I
V1 = V0 + Interest = V0 + V0 โ I = V0(1 + I)
V2 = V1 + Interest = V1 + V0 โ I = V0(1 + I) + V0 โ I
= V0(1 + I + I) = V0(1 + 2I)
V3 = V2 + Interest = V2 + V0 โ I = V0(1 + 2I) + V0 โ I
= V0(1 + 2I + I) = V0(1 + 3I)
.
.
Vn = V0(1 + nI)
FVn = PV(1 + nI)
9-โน#โบ
Compound Interest
Interest ($) = Principal ($) โ Interest Rate (%) = V โ I
V1 = V0 + Interest = V0 + V0 โ I = V0(1 + I)
V2 = V1 + Interest = V1 + V1 โ I = V1(1 + I)
V3 = V2 + Interest = V2 + V2 โ I = V2(1 + I)
V2 = V1(1 + I) = V0(1 + I)(1 + I) = V0(1 + I)2
V3 = V2(1 + I) = V0(1 + I)2(1 + I) = V0(1 + I)3
Vn = V0 (1 + I)n
FVn = PV(1 + I)n = PVโFVIF
V2 = V1 + Interest = V1 + (V0 + Interest) โ I
9-โน#โบ
Example
What is the future value of $20 invested for 2 years at 10%?
Simple: FV = PV(1+nI)
= 20(1+2I) = 20(1+0.2) = $24
Compound: FV = PV(1+I)n
= 20(1+I)2 = 20(1+0.1)2 = $24.2
What is the future value of $20 invested for 100 years at 10%?
Simple: FV = 20(1+ ) =
Compound : FV = 20(1.1)100 = 275,612.25
9-โน#โบ
The Power of Compounding
The Value of Manhattan
In 1626, the land was bought from American Indians at $24.
In 2018, value = $24(1+I)392
9-โน#โบ
Solving for FV:
The formula method
Solve the general FV equation:
FVN = PVโ(1 + I)N = PV โ FVIF
FV15 = PVโ(1 + I)15 = $1,000โ(1.10)15 = $4,177.25
= $1,000โ4.177 = $4,177
(Table A)
9-โน#โบ
Present Value of Money
If you want to have $4,177.25 after 15 years, how much do you have to deposit today at 10%?
9-โน#โบ
PV = ?
4,177.25
Present Value of Money
Finding the PV of a cash flow or series of cash flows is called discounting (the reverse of compounding).
0
1
2 โฆ
15
10%
9-โน#โบ
13
Solving for PV:
The formula method
Solve the general FV equat.
Time lines
Future value / Present value of lump sum
FV / PV of annuity
Perpetuities
Uneven CF stream
Compounding periods
Nominal / Effective / Periodic rates
Amortization
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2. 2
The Time Value of MoneyThe Time Value of Money
The Interest Rate
Simple Interest
Compound Interest
Amortizing a Loan
3. 3
Obviously, $10,000 today$10,000 today.
You already recognize that there is
TIME VALUE TO MONEYTIME VALUE TO MONEY!!
The Interest RateThe Interest Rate
Which would you prefer -- $10,000$10,000
todaytoday or $10,000 in 5 years$10,000 in 5 years?
4. 4
TIMETIME allows you the opportunity to
postpone consumption and earn
INTERESTINTEREST.
Why TIME?Why TIME?
Why is TIMETIME such an important
element in your decision?
5. 5
Types of InterestTypes of Interest
Compound InterestCompound Interest
Interest paid (earned) on any previous
interest earned, as well as on the
principal borrowed (lent).
Simple InterestSimple Interest
Interest paid (earned) on only the original
amount, or principal borrowed (lent).
6. 6
Simple Interest FormulaSimple Interest Formula
FormulaFormula SI = P0(i)(n)
SI: Simple Interest
P0: Deposit today (t=0)
i: Interest Rate per Period
n: Number of Time Periods
7. 7
SI = P0(i)(n)
= $1,000(.07)(2)
= $140$140
Simple Interest ExampleSimple Interest Example
Assume that you deposit $1,000 in an
account earning 7% simple interest for
2 years. What is the accumulated
interest at the end of the 2nd year?
8. 8
FVFV = P0 + SI
= $1,000 + $140
= $1,140$1,140
Future ValueFuture Value is the value at some future
time of a present amount of money, or a
series of payments, evaluated at a given
interest rate.
Simple Interest (FV)Simple Interest (FV)
What is the Future ValueFuture Value (FVFV) of the
deposit?
9. 9
The Present Value is simply the
$1,000 you originally deposited.
That is the value today!
Present ValuePresent Value is the current value of a
future amount of money, or a series of
payments, evaluated at a given interest
rate.
Simple Interest (PV)Simple Interest (PV)
What is the Present ValuePresent Value (PVPV) of the
previous problem?
11. 11
Assume that you deposit $1,000$1,000 at
a compound interest rate of 7% for
2 years2 years.
Future ValueFuture Value
Single Deposit (Graphic)Single Deposit (Graphic)
0 1 22
$1,000$1,000
FVFV22
7%
12. 12
FVFV11 = PP00 (1+i)1
= $1,000$1,000 (1.07)
= $1,070$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.
Future ValueFuture Value
Single Deposit (Formula)Single Deposit (Formula)
13. 13
FVFV11 = PP00 (1+i)1
= $1,000$1,000 (1.07)
= $1,070$1,070
FVFV22 = FV1 (1+i)1
= PP00 (1+i)(1+i) = $1,000$1,000(1.07)(1.07)
= PP00 (1+i)2
= $1,000$1,000(1.07)2
= $1,144.90$1,144.90
You earned an EXTRA $4.90$4.90 in Year 2 with
compound over simple interest.
Future ValueFuture Value
Single Deposit (Formula)Single Deposit (Formula)
14. 14
FVFV11 = P0(1+i)1
FVFV22 = P0(1+i)2
General Future ValueFuture Value Formula:
FVFVnn = P0 (1+i)n
or FVFVnn = P0 (FVIFFVIFi,n) -- See Table ISee Table I
General FutureGeneral Future
Value FormulaValue Formula
etc.
15. 15
FVIFFVIFi,n is found on Table I at the end
of the book or on the card insert.
Valuation Using Table IValuation Using Table I
Period 6% 7% 8%
1 1.060 1.070 1.080
2 1.124 1.145 1.166
3 1.191 1.225 1.260
4 1.262 1.311 1.360
5 1.338 1.403 1.469
16. 16
FVFV22 = $1,000 (FVIFFVIF7%,2)
= $1,000 (1.145)
= $1,145$1,145 [Due to Rounding]
Using Future Value TablesUsing Future Value Tables
Period 6% 7% 8%
1 1.060 1.070 1.080
2 1.124 1.145 1.166
3 1.191 1.225 1.260
4 1.262 1.311 1.360
5 1.338 1.403 1.469
17. 17
TVM on the CalculatorTVM on the Calculator
Use the highlighted row
of keys for solving any
of the FV, PV, FVA,
PVA, FVAD, and PVAD
problems
N: Number of periods
I/Y: Interest rate per period
PV: Present value
PMT: Payment per period
FV: Future value
CLR TVM: Clears all of the inputs
into the above TVM keys
18. 18
Using The TI BAII+ CalculatorUsing The TI BAII+ Calculator
N I/Y PV PMT FV
Inputs
Compute
๏Focus on 3Focus on 3rdrd
row of keys (will berow of keys (will be
displayed in slides as shown above)displayed in slides as shown above)
19. 19
Entering the FV ProblemEntering the FV Problem
Press:
2nd
CLR TVM
2 N
7 I/Y
-1000 PV
0 PMT
CPT FV
20. 20
N: 2 periods (enter as 2)
I/Y: 7% interest rate per period (enter as 7 NOT .07)
PV: $1,000 (enter as negative as you have โlessโ)
PMT: Not relevant in this situation (enter as 0)
FV: Compute (Resulting answer is positive)
Solving the FV ProblemSolving the FV Problem
N I/Y PV PMT FV
Inputs
Compute
2 7 -1,000 0
1,144.90
21. 21
Julie Miller wants to know how large her deposit
of $10,000$10,000 today will become at a compound
annual interest rate of 10% for 5 years5 years.
Story Problem ExampleStory Problem Example
0 1 2 3 4 55
$10,000$10,000
FVFV55
10%
22. 22
Calculation based on Table I:
FVFV55 = $10,000 (FVIFFVIF10%, 5)
= $10,000 (1.611)
= $16,110$16,110 [Due to Rounding]
Story Problem SolutionStory Problem Solution
Calculation based on general formula:
FVFVnn = P0 (1+i)n
FVFV55 = $10,000 (1+ 0.10)5
= $16,105.10$16,105.10
23. 23
Entering the FV ProblemEntering the FV Problem
Press:
2nd
CLR TVM
5 N
10 I/Y
-10000 PV
0 PMT
CPT FV
24. 24
The result indicates that a $10,000
investment that earns 10% annually
for 5 years will result in a future value
of $16,105.10.
Solving the FV ProblemSolving the FV Problem
N I/Y PV PMT FV
Inputs
Compute
5 10 -10,000 0
16,105.10
25. 25
We will use the โโRule-of-72Rule-of-72โโ..
Double Your Money!!!Double Your Money!!!
Quick! How long does it take to
double $5,000 at a compound rate
of 12% per year (approx.)?
26. 26
Approx. Years to Double = 7272 / i%
7272 / 12% = 6 Years6 Years
[Actual Time is 6.12 Years]
The โRule-of-72โThe โRule-of-72โ
Quick! How long does it take to
double $5,000 at a compound rate
of 12% per year (approx.)?
27. 27
The result indicates that a $1,000
investment that earns 12% annually
will double to $2,000 in 6.12 years.
Note: 72/12% = approx. 6 years
Solving the Period ProblemSolving the Period Problem
N I/Y PV PMT FV
Inputs
Compute
12 -1,000 0 +2,000
6.12 years
28. 28
Assume that you need $1,000$1,000 in 2 years.2 years.
Letโs examine the process to determine
how much you need to deposit today at a
discount rate of 7% compounded
annually.
0 1 22
$1,000$1,000
7%
PV1PVPV00
Present ValuePresent Value
Single Deposit (Graphic)Single Deposit (Graphic)
30. 30
PVPV00 = FVFV11 / (1+i)1
PVPV00 = FVFV22 / (1+i)2
General Present ValuePresent Value Formula:
PVPV00 = FVFVnn / (1+i)n
or PVPV00 = FVFVnn (PVIFPVIFi,n) -- See Table IISee Table II
General PresentGeneral Present
Value FormulaValue Formula
etc.
31. 31
PVIFPVIFi,n is found on Table II at the end
of the book or on the card insert.
Valuation Using Table IIValuation Using Table 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
32. 32
PVPV22 = $1,000$1,000 (PVIF7%,2)
= $1,000$1,000 (.873)
= $873$873 [Due to Rounding]
Using Present Value TablesUsing Present Value Tables
Period 6% 7% 8%
1 .943 .935 .926
2 .890 .873 .857
3 .840 .816 .794
4 .792 .763 .735
5 .747 .713 .681
33. 33
N: 2 periods (enter as 2)
I/Y: 7% interest rate per period (enter as 7 NOT .07)
PV: Compute (Resulting answer is negative โdepositโ)
PMT: Not relevant in this situation (enter as 0)
FV: $1,000 (enter as positive as you โreceive $โ)
Solving the PV ProblemSolving the PV Problem
N I/Y PV PMT FV
Inputs
Compute
2 7 0 +1,000
-873.44
34. 34
Julie Miller wants to know how large of a
deposit to make so that the money will
grow to $10,000$10,000 in 5 years5 years at a discount
rate of 10%.
Story Problem ExampleStory Problem Example
0 1 2 3 4 55
$10,000$10,000
PVPV00
10%
35. 35
Calculation based on general formula:
PVPV00 = FVFVnn / (1+i)n
PVPV00 = $10,000$10,000 / (1+ 0.10)5
= $6,209.21$6,209.21
Calculation based on Table I:
PVPV00 = $10,000$10,000 (PVIFPVIF10%, 5)
= $10,000$10,000 (.621)
= $6,210.00$6,210.00 [Due to Rounding]
Story Problem SolutionStory Problem Solution
36. 36
Solving the PV ProblemSolving the PV Problem
N I/Y PV PMT FV
Inputs
Compute
5 10 0 +10,000
-6,209.21
The result indicates that a $10,000
future value that will earn 10%
annually for 5 years requires a
$6,209.21 deposit today (present
value).
37. 37
Types of AnnuitiesTypes of Annuities
Ordinary AnnuityOrdinary Annuity: Payments or receipts
occur at the end of each period.
Annuity DueAnnuity Due: Payments or receipts
occur at the beginning of each period.
An AnnuityAn Annuity represents a series of equal
payments (or receipts) occurring over a
specified number of equidistant periods.
38. 38
Examples of AnnuitiesExamples of Annuities
Student Loan Payments
Car Loan Payments
Insurance Premiums
Mortgage Payments
Retirement Savings
39. 39
Parts of an AnnuityParts of an Annuity
0 1 2 3
$100 $100 $100
(Ordinary Annuity)
EndEnd of
Period 1
EndEnd of
Period 2
Today EqualEqual Cash Flows
Each 1 Period Apart
EndEnd of
Period 3
40. 40
Parts of an AnnuityParts of an Annuity
0 1 2 3
$100 $100 $100
(Annuity Due)
BeginningBeginning of
Period 1
BeginningBeginning of
Period 2
Today EqualEqual Cash Flows
Each 1 Period Apart
BeginningBeginning of
Period 3
41. 41
FVAFVAnn = R(1+i)n-1
+ R(1+i)n-2
+
... + R(1+i)1
+ R(1+i)0
Overview of anOverview of an
Ordinary Annuity -- FVAOrdinary Annuity -- FVA
R R R
0 1 2 nn n+1
FVAFVAnn
R = Periodic
Cash Flow
Cash flows occur at the end of the period
i% . . .
42. 42
FVAFVA33 = $1,000(1.07)2
+
$1,000(1.07)1
+ $1,000(1.07)0
= $1,145 + $1,070 + $1,000
= $3,215$3,215
Example of anExample of an
Ordinary Annuity -- FVAOrdinary Annuity -- FVA
$1,000 $1,000 $1,000
0 1 2 33 4
$3,215 = FVA$3,215 = FVA33
7%
$1,070
$1,145
Cash flows occur at the end of the period
43. 43
Hint on Annuity ValuationHint on Annuity Valuation
The future value of an ordinary
annuity can be viewed as
occurring at the endend of the last
cash flow period, whereas the
future value of an annuity due
can be viewed as occurring at
the beginningbeginning of the last cash
flow period.
44. 44
FVAFVAnn = R (FVIFAi%,n)
FVAFVA33 = $1,000 (FVIFA7%,3)
= $1,000 (3.215) = $3,215$3,215
Valuation Using Table IIIValuation Using Table 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
45. 45
N: 3 periods (enter as 3 year-end deposits)
I/Y: 7% interest rate per period (enter as 7 NOT .07)
PV: Not relevant in this situation (no beg value)
PMT: $1,000 (negative as you deposit annually)
FV: Compute (Resulting answer is positive)
Solving the FVA ProblemSolving the FVA Problem
N I/Y PV PMT FV
Inputs
Compute
3 7 0 -1,000
3,214.90
46. 46
FVADFVADnn = R(1+i)n
+ R(1+i)n-1
+
... + R(1+i)2
+ R(1+i)1
= FVAFVAnn (1+i)
Overview View of anOverview View of an
Annuity Due -- FVADAnnuity Due -- FVAD
R R R R R
0 1 2 3 n-1n-1 n
FVADFVADnn
i% . . .
Cash flows occur at the beginning of the period
47. 47
FVADFVAD33 = $1,000(1.07)3
+
$1,000(1.07)2
+ $1,000(1.07)1
= $1,225 + $1,145 + $1,070
= $3,440$3,440
Example of anExample of an
Annuity Due -- FVADAnnuity Due -- FVAD
$1,000 $1,000 $1,000 $1,070
0 1 2 33 4
$3,440 = FVAD$3,440 = FVAD33
7%
$1,225
$1,145
Cash flows occur at the beginning of the period
48. 48
FVADFVADnn = R (FVIFAi%,n)(1+i)
FVADFVAD33 = $1,000 (FVIFA7%,3)(1.07)
= $1,000 (3.215)(1.07) = $3,440$3,440
Valuation Using Table IIIValuation Using Table 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
49. 49
Solving the FVAD ProblemSolving the FVAD Problem
N I/Y PV PMT FV
Inputs
Compute
3 7 0 -1,000
3,439.94
Complete the problem the same as an โordinary annuityโ
problem, except you must change the calculator setting
to โBGNโ first. Donโt forget to change back!
Step 1: Press 2nd
BGN keys
Step 2: Press 2nd
SET keys
Step 3: Press 2nd
QUIT keys
50. 50
PVAPVAnn = R/(1+i)1
+ R/(1+i)2
+ ... + R/(1+i)n
Overview of anOverview of an
Ordinary Annuity -- PVAOrdinary Annuity -- PVA
R R R
0 1 2 nn n+1
PVAPVAnn
R = Periodic
Cash Flow
i% . . .
Cash flows occur at the end of the period
51. 51
PVAPVA33 = $1,000/(1.07)1
+
$1,000/(1.07)2
+
$1,000/(1.07)3
= $934.58 + $873.44 + $816.30
= $2,624.32$2,624.32
Example of anExample of an
Ordinary Annuity -- PVAOrdinary Annuity -- PVA
$1,000 $1,000 $1,000
0 1 2 33 4
$2,624.32 = PVA$2,624.32 = PVA33
7%
$ 934.58
$ 873.44
$ 816.30
Cash flows occur at the end of the period
52. 52
Hint on Annuity ValuationHint on Annuity Valuation
The present value of an ordinary
annuity can be viewed as
occurring at the beginningbeginning of the
first cash flow period, whereas
the present value of an annuity
due can be viewed as occurring
at the endend of the first cash flow
period.
53. 53
PVAPVAnn = R (PVIFAi%,n)
PVAPVA33 = $1,000 (PVIFA7%,3)
= $1,000 (2.624) = $2,624$2,624
Valuation Using Table IVValuation Using Table 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
54. 54
N: 3 periods (enter as 3 year-end deposits)
I/Y: 7% interest rate per period (enter as 7 NOT .07)
PV: Compute (Resulting answer is positive)
PMT: $1,000 (negative as you deposit annually)
FV: Not relevant in this situation (no ending value)
Solving the PVA ProblemSolving the PVA Problem
N I/Y PV PMT FV
Inputs
Compute
3 7 -1,000 0
2,624.32
55. 55
PVADPVADnn = R/(1+i)0
+ R/(1+i)1
+ ... + R/(1+i)n-1
= PVAPVAnn (1+i)
Overview of anOverview of an
Annuity Due -- PVADAnnuity Due -- PVAD
R R R R
0 1 2 n-1n-1 n
PVADPVADnn
R: Periodic
Cash Flow
i% . . .
Cash flows occur at the beginning of the period
56. 56
PVADPVADnn = $1,000/(1.07)0
+ $1,000/(1.07)1
+
$1,000/(1.07)2
= $2,808.02$2,808.02
Example of anExample of an
Annuity Due -- PVADAnnuity Due -- PVAD
$1,000.00 $1,000 $1,000
0 1 2 33 4
$2,808.02$2,808.02 = PVADPVADnn
7%
$ 934.58
$ 873.44
Cash flows occur at the beginning of the period
57. 57
PVADPVADnn = R (PVIFAi%,n)(1+i)
PVADPVAD33 = $1,000 (PVIFA7%,3)(1.07)
= $1,000 (2.624)(1.07) = $2,808$2,808
Valuation Using Table IVValuation Using Table 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
58. 58
Solving the PVAD ProblemSolving the PVAD Problem
N I/Y PV PMT FV
Inputs
Compute
3 7 -1,000 0
2,808.02
Complete the problem the same as an โordinary annuityโ
problem, except you must change the calculator setting
to โBGNโ first. Donโt forget to change back!
Step 1: Press 2nd
BGN keys
Step 2: Press 2nd
SET keys
Step 3: Press 2nd
QUIT keys
59. 59
1. Read problem thoroughly
2. Determine if it is a PV or FV problem
3. Create a time line
4. Put cash flows and arrows on time line
5. Determine if solution involves a single
CF, annuity stream(s), or mixed flow
6. Solve the problem
7. Check with financial calculator (optional)
Steps to Solve Time ValueSteps to Solve Time Value
of Money Problemsof Money Problems
60. 60
Julie Miller will receive the set of cash
flows below. What is the Present ValuePresent Value
at a discount rate of 10%10%?
Mixed Flows ExampleMixed Flows Example
0 1 2 3 4 55
$600 $600 $400 $400 $100$600 $600 $400 $400 $100
PVPV00
10%10%
61. 61
1. Solve a โpiece-at-a-timepiece-at-a-timeโ by
discounting each piecepiece back to t=0.
2. Solve a โgroup-at-a-timegroup-at-a-timeโ by first
breaking problem into groups
of annuity streams and any single
cash flow group. Then discount
each groupgroup back to t=0.
How to Solve?How to Solve?
65. 65
Use the highlighted
key for starting the
process of solving a
mixed cash flow
problem
Press the CF key
and down arrow key
through a few of the
keys as you look at
the definitions on
the next slide
Solving the Mixed FlowsSolving the Mixed Flows
Problem using CF RegistryProblem using CF Registry
66. 66
Defining the calculator variables:
For CF0: This is ALWAYS the cash flow occurring
at time t=0 (usually 0 for these problems)
For Cnn:* This is the cash flow SIZE of the nth
group of cash flows. Note that a โgroupโ may only
contain a single cash flow (e.g., $351.76).
For Fnn:* This is the cash flow FREQUENCY of the
nth group of cash flows. Note that this is always a
positive whole number (e.g., 1, 2, 20, etc.).
Solving the Mixed FlowsSolving the Mixed Flows
Problem using CF RegistryProblem using CF Registry
* nn represents the nth cash flow or frequency. Thus, the
first cash flow is C01, while the tenth cash flow is C10.
67. 67
Solving the Mixed FlowsSolving the Mixed Flows
Problem using CF RegistryProblem using CF Registry
Steps in the Process
Step 1: Press CF key
Step 2: Press 2nd
CLR Work keys
Step 3: For CF0 Press 0 Enter โ keys
Step 4: For C01 Press 600 Enter โ keys
Step 5: For F01 Press 2 Enter โ keys
Step 6: For C02 Press 400 Enter โ keys
Step 7: For F02 Press 2 Enter โ keys
68. 68
Solving the Mixed FlowsSolving the Mixed Flows
Problem using CF RegistryProblem using CF Registry
Steps in the Process
Step 8: For C03 Press 100 Enter โ keys
Step 9: For F03 Press 1 Enter โ keys
Step 10: Press โ โ keys
Step 11: Press NPV key
Step 12: For I=, Enter 10 Enter โ keys
Step 13: Press CPT key
Result: Present Value = $1,677.15
69. 69
General Formula:
FVn = PVPV00(1 + [i/m])mn
n: Number of Years
m: Compounding Periods per Year
i: Annual Interest Rate
FVn,m: FV at the end of Year n
PVPV00: PV of the Cash Flow today
Frequency ofFrequency of
CompoundingCompounding
70. 70
Julie Miller has $1,000$1,000 to invest for 2
years at an annual interest rate of
12%.
Annual FV2 = 1,0001,000(1+ [.12/1])(1)(2)
= 1,254.401,254.40
Semi FV2 = 1,0001,000(1+ [.12/2])(2)(2)
= 1,262.481,262.48
Impact of FrequencyImpact of Frequency
71. 71
Qrtly FV2 = 1,0001,000(1+ [.12/4])(4)(2)
= 1,266.771,266.77
Monthly FV2 = 1,0001,000(1+ [.12/12])(12)(2)
= 1,269.731,269.73
Daily FV2 = 1,0001,000(1+[.12/365])(365)(2)
= 1,271.201,271.20
Impact of FrequencyImpact of Frequency
72. 72
The result indicates that a $1,000
investment that earns a 12% annual
rate compounded quarterly for 2
years will earn a future value of
$1,266.77.
Solving the FrequencySolving the Frequency
Problem (Quarterly)Problem (Quarterly)
N I/Y PV PMT FV
Inputs
Compute
2(4) 12/4 -1,000 0
1266.77
73. 73
Solving the FrequencySolving the Frequency
Problem (Quarterly Altern.)Problem (Quarterly Altern.)
Press:
2nd
P/Y 4 ENTER
2nd
QUIT
12 I/Y
-1000 PV
0 PMT
2 2nd
xP/Y N
CPT FV
74. 74
The result indicates that a $1,000
investment that earns a 12% annual
rate compounded daily for 2 years will
earn a future value of $1,271.20.
Solving the FrequencySolving the Frequency
Problem (Daily)Problem (Daily)
N I/Y PV PMT FV
Inputs
Compute
2(365) 12/365 -1,000 0
1271.20
75. 75
Solving the FrequencySolving the Frequency
Problem (Daily Alternative)Problem (Daily Alternative)
Press:
2nd
P/Y 365 ENTER
2nd
QUIT
12 I/Y
-1000 PV
0 PMT
2 2nd
xP/Y N
CPT FV
76. 76
Effective Annual Interest Rate
The actual rate of interest earned
(paid) after adjusting the nominal
rate for factors such as the number
of compounding periods per year.
(1 + [ i / m ] )m
- 1
Effective AnnualEffective Annual
Interest RateInterest Rate
77. 77
Basket Wonders (BW) has a $1,000
CD at the bank. The interest rate
is 6% compounded quarterly for 1
year. What is the Effective Annual
Interest Rate (EAREAR)?
EAREAR = ( 1 + 6% / 4 )4
- 1
= 1.0614 - 1 = .0614 or 6.14%!6.14%!
BWโs EffectiveBWโs Effective
Annual Interest RateAnnual Interest Rate
78. 78
Converting to an EARConverting to an EAR
Press:
2nd
I Conv
6 ENTER
โ โ
4 ENTER
โ CPT
2nd
QUIT
79. 79
1. Calculate the payment per period.
2. Determine the interest in Period t.
(Loan balance at t-1) x (i% / m)
3. Compute principal paymentprincipal payment in Period t.
(Payment - interest from Step 2)
4. Determine ending balance in Period t.
(Balance - principal paymentprincipal payment from Step 3)
5. Start again at Step 2 and repeat.
Steps to Amortizing a LoanSteps to Amortizing a Loan
80. 80
Julie Miller is borrowing $10,000$10,000 at a
compound annual interest rate of 12%.
Amortize the loan if annual payments are
made for 5 years.
Step 1: Payment
PVPV00 = R (PVIFA i%,n)
$10,000$10,000 = R (PVIFA 12%,5)
$10,000$10,000 = R (3.605)
RR = $10,000$10,000 / 3.605 = $2,774$2,774
Amortizing a Loan ExampleAmortizing a Loan Example
81. 81
Amortizing a Loan ExampleAmortizing a Loan Example
End of
Year
Payment Interest Principal Ending
Balance
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]
82. 82
The result indicates that a $10,000 loan
that costs 12% annually for 5 years and
will be completely paid off at that time will
require $2,774.10 in annual payments.
Solving for the PaymentSolving for the Payment
N I/Y PV PMT FV
Inputs
Compute
5 12 10,000 0
-2774.10
83. 83
Using the AmortizationUsing the Amortization
Functions of the CalculatorFunctions of the Calculator
Press:
2nd
Amort
1 ENTER
1 ENTER
Results:
BAL = 8,425.90 โ
PRN = -1,574.10 โ
INT = -1,200.00 โ
Year 1 information only
84. 84
Using the AmortizationUsing the Amortization
Functions of the CalculatorFunctions of the Calculator
Press:
2nd
Amort
2 ENTER
2 ENTER
Results:
BAL = 6,662.91 โ
PRN = -1,763.99 โ
INT = -1,011.11 โ
Year 2 information only
85. 85
Using the AmortizationUsing the Amortization
Functions of the CalculatorFunctions of the Calculator
Press:
2nd
Amort
1 ENTER
5 ENTER
Results:
BAL = 0.00 โ
PRN =-10,000.00 โ
INT = -3,870.49 โ
Entire 5 Years of loan information
86. 86
Usefulness of AmortizationUsefulness of Amortization
2.2. Calculate Debt OutstandingCalculate Debt Outstanding -- The
quantity of outstanding debt
may be used in financing the
day-to-day activities of the firm.
1.1. Determine Interest ExpenseDetermine Interest Expense --
Interest expenses may reduce
taxable income of the firm.