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Chapter
Tool Kit Chapter 4 10/27/15
The Time Value of Money
The worksheet shown below performs most of the calculations required for Chapter 4, and it was used to create many of the chapter's tables and figures. We pasted in a few dialog boxes for specific Excel functions and features; they are shown off to the right of where they were used. We encourage students to become familiar with Excel functions. It is also useful to learn how Excel models can be used to create tables and graphs that can then be copied into Word documents, which is the way we prepared the text manuscript for submission to the publisher. That procedure is used often in business (and in business courses) to prepare reports.
Although answers to the Self-Test questions within the chapter are generally quite easy and can be worked with a calculator, we also solved them with Excel as a check and also to provide some information on the solutions for students who might have questions. The tabs at the lower part of this screen take you to the solutions for self-tests in the various sections of the chapter. Even students who are not familiar with Excel should still be able to see the solution setup and then work the problem with a calculator.
4-2 Future Values
A dollar in hand today is worth more than a dollar to be received in the future because, if you had it now, you could invest it, earn interest, and end up with more than one dollar in the future. The process of going to future values (FVs) from present values (PVs) is called compounding.
To illustrate, refer to our 3-year time line in the Figure below and assume that you plan to deposit $100 in a bank that pays a guaranteed 5% interest each year. How much would you have at the end of Year 3?
To answer this question, we show 4 methods: (1) the step-by-step using a regular calculator; (2) the formula approach using a regular calculator; (3) the financial calculator approach; and (4) the Excel approach.
Figure 4-1
Alternative Procedures for Calculating Future Values
INPUTS:
Investment = CF0 = PV = −$100.00
Interest rate = I = 5%
No. of periods = N = 3
Time Line Periods: 0 1 2 3
Cash flow: −$100.00 0 0 FV = ?
1. Step-by-Step: Multiply by (1 + I) each step $100.00 → $105.00 → $110.25 → $115.76
2. Formula: FVN = PV(1+I)N FV3 = $100(1.05)3 = $115.76
Inputs: 3 5 −100 0
3. Financial Calculator: N I/YR PV PMT FV
Output: $115.76
4. Excel Spreadsheet: FV function: FVN = =FV(I,N,0,PV)
Fixed inputs: FVN = =FV(0.05,3,0,−100) = $115.76
Cell references: FVN = =FV(C33,C34,0,C32) = $115.76
In the Excel formula, the terms are entered in this sequence: interest, periods, 0 to indicate no periodic cash flows, and then the PV. The data can be entered as fixed numbers or, better yet, as cell references.
Figure 4-2 (just below) shows how a $1 investment grows over time at different interest rates. The curves were created by solving for FV at different values for N and I. The graph shows, simultaneously, the effects of time and interest rates. The data table used to create this figure is shown to the right of the figure.
Figure 4-2
Growth of $100 at Various Interest Rates and Time Periods Future Value of $1
Periods (N) Interest Rate (I)
115.7625 I = −20% I = 0% I = 5% I = 10% I = 20%
0 $100.00 $100.00 $100.00 $100.00 $100.00
1 $80.00 $100.00 $105.00 $110.00 $120.00
2 $64.00 $100.00 $110.25 $121.00 $144.00
3 $51.20 $100.00 $115.76 $133.10 $172.80
4 $40.96 $100.00 $121.55 $146.41 $207.36
5 $32.77 $100.00 $127.63 $161.05 $248.83
6 $26.21 $100.00 $134.01 $177.16 $298.60
7 $20.97 $100.00 $140.71 $194.87 $358.32
8 $16.78 $100.00 $147.75 $214.36 $429.98
9 $13.42 $100.00 $155.13 $235.79 $515.98
10 $10.74 $100.00 $162.89 $259.37 $619.17
4-3 Present Values
Mathematically, the present value is the opposite of the future value. Instead of compounding a present value forward to find the FV, you discount the FV back to find the PV. Thus, if you know the PV, you can compound to find the FV, while if you know the FV, you can discount to find the PV.
To illustrate, refer to the time line below and assume that you will need $115.76 in 3 years. If a bank pays a guaranteed 5% interest rate each year, how much must you deposit now to have $115.76 in 3 years?
Figure 4-3
Present value of at Various Interest Rates and Time Periods
INPUTS:
Future payment = CFN = FV = $115.76
Interest rate = I = 5.00%
No. of periods = N = 3
Time Line Periods: 0 1 2 3
Cash flow: PV = ? 0 0 $115.76
1. Step-by-Step: $100.00 ← $105.00 ← $110.25 ← $115.76
2. Formula: PV = FVN/(1+I)N PV = $115.76/(1.05)3 = $100.00
Inputs: 3 5 0 115.76
3. Financial Calculator: N I/YR PV PMT FV
Output: −$100.00
4. Excel Spreadsheet: PV function: PV = =PV(I,N,0,FV)
Fixed inputs: PV = =PV(0.05,3,0,115.76) = −$100.00
Cell references: PV = =PV(C99,C100,0,C98) = −$100.00
In the Excel formula, the terms are entered in this sequence: interest, periods, 0 to indicate no periodic cash flows, and then the FV. The data can be entered as fixed numbers or, better yet, as cell references.
Figure 4-4 shows how the present value of $1 due in the future declines as either the interest rate or the time until receipt increases. The Data Table to the right provides the data used to draw the figure. At 0%, the PV of $1 always remains at $1, but at higher rates the value at the end of N years is lower the higher the rate, and at a given rate, the value declines the larger the value of N.
Figure 4-4
Present Value of $100 at Various Interest Rates and Time Periods
Present Value of $1
Periods (N) Interest Rate (I)
86.3838 I = 0% I = 5% I = 10% I = 20%
0 $100.0000 $100.0000 $100.0000 $100.0000
4 $100.0000 $82.2702 $68.3013 $48.2253
8 $100.0000 $67.6839 $46.6507 $23.2568
12 $100.0000 $55.6837 $31.8631 $11.2157
16 $100.0000 $45.8112 $21.7629 $5.4088
20 $100.0000 $37.6889 $14.8644 $2.6084
24 $100.0000 $31.0068 $10.1526 $1.2579
28 $100.0000 $25.5094 $6.9343 $0.6066
32 $100.0000 $20.9866 $4.7362 $0.2926
36 $100.0000 $17.2657 $3.2349 $0.1411
40 $100.0000 $14.2046 $2.2095 $0.0680
4-4 Finding the Interest Rate, I
Previously, we solved the basic equation to find FV and PV. However, we could just as easily solve for I or N. For example, suppose we know that a given bond has a cost of $100 and that it will return $150 after 10 years. Thus, we know PV, FV, and N, and we want to find the rate of return we would earn if we bought the bond.
INPUTS:
Present value (PV) -$100.00
Future value (FV) $150.00
No. of years (N) 10
OUTPUT:
Interest rate (I) = RATE(N,0,PV,FV)
Interest rate (I) 4.14%
4-5 Finding the Number of Years, N
Sometimes we need to know how long it will take to accumulate a given sum of money, given our beginning funds and the rate we will earn on those funds. For example, suppose we believe that we could retire comfortably if we had $1 million, and we want to find how long it will take us to reach that goal, assuming that we now have $500,000 invested at 4.5%.
INPUTS:
Present value (PV) -$500,000
Future value (FV) $1,000,000
Interest rate (I) 4.50%
OUTPUT:
No. of years (N) =NPER(I,0,PV,FV)
No. of years (N) 15.7473
4-6 Perpetuities
Perpetuities are securities that promise to make payments forever. The tale below shows how the present value of an ordinary annuity changes as the number of payments increases. Note that we cannot calculate the future value of a perpetuity because, since payments go on forever, this value would be infinitely large and thus meaningless.
Payment (PMT) $25
Interest rate (I) 5.2%
Number of Periods PV of Ordinary Annuity
1 $23.76
2 $46.35
3 $67.83
4 $88.24
5 $107.64
10 $191.18
15 $256.02
20 $306.34
25 $345.39
30 $375.70
40 $417.48
50 $442.65
60 $457.81
70 $466.94
80 $472.44
90 $475.75
100 $477.75
200 $480.75
500 $480.77
Notice in the table above that the PV of an ordinary annuity increases as the number of payments increases, as you would expect. However, the PV appears to begin leveling off. This is because the present value of a cash flow far in the future is very small and approaches zero as the time of the cash flow goes to infinity. In fact, the present value of a perpetuity can be found with a simple formula: Value = PMT / I .
Consider a British consol that pays a $25 annual payment. If interest rates are currently 5.2%, what is the value of the consol?
Payment (PMT) $25
Interest rate (I) 5.2%
Value (PV): $25 / 0.052 = $480.77
4-7 Annuities
An annuity is a series of equal cash flows. The cash flows can be at the end of the period or the beginning, but they must not change.
4-8 Future Value of an Ordinary Annuity
An ordinary annuity has regular, periodic payments that occur at the end of each period. Methods for solving the future value of an ordinary annuity are shown below.
Figure 4-5
Summary: Future Value of an Ordinary Annuity
INPUTS:
Payment amount = PMT = −$100
Interest rate = I = 5.00%
No. of periods = N = 3
1. Step-by-Step: Periods: 0 1 2 3
Cash flow: −$100 −$100 −$100
↓ ↓ ↓
↓ ↓ $100.00
Multiply each payment by ↓ └ → → $105.00
(1+I)N-t and sum these FVs to └ → → → → → → → → → $110.25
find FVAN: $315.25
2. Formula:
FVAN $315.25
Inputs: 3 5 0 −100
3. Financial Calculator: N I/YR PV PMT FV
Output: $315.25
4. Excel Spreadsheet: FV function: FVAN = =FV(I,N,PMT,PV)
Fixed inputs: FVAN = =FV(0.05,3,-100,0) = $315.25
Cell references: FVAN = =FV(C245,C246,C244,0) = $315.25
4-9 Future Value of an Annuity Due
An annuity due also has regular, periodic payments, but unlike an ordinary annuity, the payments occur at the beginning of each period.
Figure Not In Textbook
Summary: Future Value of an Annuity Due
INPUTS:
Payment amount = PMT = −$100
Interest rate = I = 5.00%
No. of periods = N = 3
1. Step-by-Step: Periods: 0 1 2 3
Cash flow: −$100 −$100 −$100
↓ ↓ ↓
↓ ↓ └ → → $105.00
Multiply each payment by ↓ └ → → → → → → → → → $110.25
(1+I)N-t and sum these FVs to └ → → → → → → → → → → → → → → → → $115.76
find FVAN: $331.01
2. Formula:
FVAN = $331.01
Inputs: Mode = BEG 3 5 0 −100
3. Financial Calculator: N I/YR PV PMT FV
Output: $331.01
4. Excel Spreadsheet: FV function: FVAN = =FV(I,N,PMT,PV,TYPE)
Fixed inputs: FVAN = =FV(0.05,3,-100,0,1) = $331.01
Cell references: FVAN = =FV(C280,C281,C279,0,1) = $331.01
4-10 Present Value of Ordinary Annuities and Annuities Due
The present value of an ordinary annuity is the sum of the PVs of the individual cash flows. Methods for solving the present value of an ordinary annuity are shown below.
Figure 4-6
Summary: Present Value of an Ordinary Annuity
INPUTS:
Payment amount = PMT = −$100
Interest rate = I = 5.00%
No. of periods = N = 3
1. Step-by-Step: Periods: 0 1 2 3
Cash flow: −$100 −$100 −$100
↓ ↓ ↓
$95.24 ← ← ┘ ↓ ↓
Divide each payment by $90.70 ←←←←←← ← ← ┘ ↓
(1+I)t and sum these PVs to $86.38 ←←←←←← ←←←←←← ← ← ┘
find PVAN: $272.32
2. Formula:
PVAN $272.32
Inputs: 3 5 −100 0
3. Financial Calculator: N I PV PMT FV
Output: 272.32
4. Excel Spreadsheet: PV function: PVAN = =PV(I,N,PMT,FV)
Fixed inputs: PVAN = =PV(0.05,3,-100,0) = $272.32
Cell references: PVAN = =PV(C315,C316,C314,0) = $272.32
PRESENT VALUE OF AN ANNUITY DUE (this table is not in text)
The difference between the present value of an ordinary annuity and an annuity due is that payments are received earlier in an annuity due.
Figure Not In Textbook
Summary: Present Value of an Annuity Due
INPUTS:
Payment amount = PMT = −$100
Interest rate = I = 5.00%
No. of periods = N = 3
1. Step-by-Step: Periods: 0 1 2 3
Cash flow: −$100 −$100 −$100
↓ ↓ ↓
$100.00 ↓ ↓
Divide each payment by $95.24 ← ← ⤶ ↓
(1+I)t and sum these PVs to $90.70 ←←←←←← ← ← ⤶
find PVAN: $285.94
2. Formula:
PVAN = = $285.94
Inputs: Mode = BEG 3 5 −$100 0
3. Financial Calculator: N I PV PMT FV
Output: 285.94
4. Excel Spreadsheet: PV function: PVAN = =PV(I,N,PMT,FV)
Fixed inputs: PVAN = =PV(0.05,3,-100,0,1) = $285.94
Cell references: PVAN = =PV(C351,C352,C350,0,1) = $285.94
4-11 Finding Annuity Payments, Periods, and Interest Rates
Fundamentally, this section is no different than previous TVM exercises. When solving for PMT, N, or I, you must be given values for the other variables, and then you solve the problem.
FINDING PMT
Suppose we need to accumulate $10,000 and have it available 5 years from now. Suppose further that we can earn a return of 6% on our savings, which are currently zero. How much must we save in each of the 5 years, assuming (a) end-of-year payments and (b) beginning-of-year payments?
No. of years (N) 5
Interest rate (I) 6%
Present value (PV) $0
Future value (FV) $10,000
a. END MODE b. BEGIN MODE
Payment (PMT) -$1,773.96 Payment (PMT) -$1,673.55
=PMT(I,N,PV,FV) =PMT(I,N,PV,FV,Type=1)
FINDING N
Suppose you decide to make end-of-year deposits, but you can only save $1,200 per year. Again assume that you would earn 6%. How long would it take you to reach your $10,000 goal?
BEGIN MODE
Interest rate (I) 6% 6%
Present value (PV) $0 $0
Payment (PMT) -$1,200 -$1,200
Future value (FV) $10,000 $10,000
No. of years (N) 6.96 6.63
=NPER(I,PMT,PV,FV,0) =NPER(I,PMT,PV,FV,1)
FINDING I
Now suppose you can only save $1,200 annually, but you still want to have the $10,000 in 5 years. What rate of return would enable you to achieve your goal?
BEGIN MODE
No. of years (N) 5 5
Present value (PV) $0 $0
Payment (PMT) -$1,200 -$1,200
Future value (FV) $10,000 $10,000
Interest rate (I) 25.78% 17.54%
=RATE(N,PMT,PV,FV,0) =RATE(N,PMT,PV,FV,1)
4-12 Uneven, or Irregular, Cash Flows
First, consider a security that pays $100 for 5 years plus a lump sum of $1,000 at the end of the 5th year. We can find the PV in several ways: (1) With a financial calculator using the step-by-step approach, or by finding the PV of the annuity plus the PV of the final $1,000 and then summing these two values, or by using the calculator's cash flow register, or (2) with Excel, using either the PV or the NPV function. We illustrate the step-by-step and the two Excel approaches below.
Figure Not Shown in Textbook
Present Value of an Annuity Plus Additional Final Payment
INPUTS:
Interest rate = I = 12.00%
No. of periods = N = 5
Payment amount = PMT = $100
Future value = FV = $1,000
1. Step-by-Step:
Periods: 0 1 2 3 4 5
PMT CFs: $100.00 $100.00 $100.00 $100.00 $100.00
Additional CF: $1,000.00
Total CFs: $100.00 $100.00 $100.00 $100.00 $1,100.00
PVs of the CFs: $89.29 $79.72 $71.18 $63.55 $624.17
PV of the CF Stream = Sum of the Individual PVs = $927.90
2. Financial Calculator:
You could enter the cash flows into the cash flow register of a financial calculator, enter I/YR, and then press the NPV key to find the answer. $927.90
3. Excel Spreadsheet: PV Function: PVAN = =PV(I,N,PMT,FV)
PV fixed inputs: PVAN = =PV(0.12,5,100,1000) = −$927.90
PV cell references: PVAN = =PV(C432,C433,C434,C435) = −$927.90
NPV Function: NPV = =NPV(I,CFs)
NPV fixed inputs: NPV = =NPV(0.12,100,100,100,100,1100) = $927.90
NPV cell references: NPV = =NPV(C432,C441:G441) = $927.90
The Excel formula ignores the initial cash flow (in Year 0). When entering a cash flow range, Excel assumes that the first value occurs at the end of the first year. As we will see later, if there is an initial cash flow, it must be added separately to complete the NPV formula result. Notice too that you can enter cash flows one-by-one, but if the cash flows appear in consecutive cells, you can enter the cell range, as we did here.
Now consider an irregular cash flow stream, where the CFs can take on any value.
Figure 4-7
Present Value of an Irregular Cash Flow Stream
INPUTS:
Interest rate = I = 12%
1. Step-by-Step:
Periods: 0 1 2 3 4 5
Cash flow: $0.00 $100.00 $300.00 $300.00 $300.00 $500.00
PVs of the CFs: $89.29 $239.16 $213.53 $190.66 $283.71
PV of the Irregular CF Stream = Sum of the Individual PVs = $1,016.35
2. Calculator: You could enter the cash flows into the cash flow register of a financial calculator, enter I/YR, and then press the NPV key to find the answer. $1,016.35
3. Excel Spreadsheet: NPV function: NPV = = NPV(I,CFs)
Fixed inputs: NPV = =NPV(0.12,100,300,300,300,500) $1,016.35
Cell references: NPV = =NPV(C467,C471:G471) $1,016.35
The Excel formula ignores the initial cash flow (in Year 0). When entering a cash flow range, Excel assumes that the first value occurs at the end of the first year. As we will see later, if there is an initial cash flow, it must be added separately to complete the NPV formula result. Notice too that you can enter cash flows one-by-one, but if the cash flows appear in consecutive cells, you can enter the cell range, as we did here.
4-13 Future Value of an Uneven Cash Flow Stream
We find the future value of uneven cash flow streams by compounding rather than discounting. The step-by-step approach works the same, but unfortunately, Excel does not have a net future value (NFV) function, although financial calculators do have this function. One way around this is to solve for the NPV and then find the FV of this amount at the end of the cash flow stream.
Figure 4-8
Future Value of an Irregular Cash Flow Stream
INPUTS:
Interest rate = I = 12%
1. Step-by-Step:
Periods: 0 1 2 3 4 5
Cash flow: $0.00 $100.00 $300.00 $300.00 $300.00 $500.00
FVs of the CFs: $157.35 $421.48 $376.32 $336.00 $500.00
FV of the Irregular CF Stream = Sum of the Individual FVs = $1,791.15
2. Calculator: You could enter the cash flows into the cash flow register of a financial calculator, enter I/YR, and then press the NFV key to find the answer. $1,791.15
3. Excel Spreadsheet Step 1. Find NPV: =NPV(C499,C503:G503) $1,016.35
Step 2. Compound NPV to find NFV: =FV(C499,G502,0,-G511) $1,791.15
4-14 Solving for I with Irregular Cash Flows
Assume that a bond will pay $100 at the end of each of the next 5 years, plus an additional $1,000 at the end of the 5th year. The cost of the bond is $927.90. What rate of return would you earn if you bought the bond?
You could find the rate of return using Excel's IRR (for "internal rate of return") function or its RATE function, as shown below. The RATE function deals with situations where we have an annuity plus a final lump sum. The IRR function deals with any cash flow pattern, and it is easier to use. You could enter a guess as to the IRR, but this is not necessary.
Finding the Interest Rate, Annuity Plus Lump Sum
INPUTS:
Annuity pmts $100
Future lump sum $1,000
Periods: 0 1 2 3 4 5
Cash Flows: -$927.90 $100 $100 $100 $100 $1,100
Excel Function Approach: Cell references: IRR = =IRR(B532:G532) 12.00%
Cell references: RATE = =RATE(G531,B528,B532,B529) 12.00%
The IRR function is used to find the rate of return on capital budgeting projects, where the firm makes a capital expenditure and then expects to receive a series of cash inflows. Figure 4-9 illustrates this calculation. Note that the IRR function can be used even if one of the post-investment cash flows is negative. Change the 4th year CF from $300 to -$100 and see the IRR drop to 2.90%. Then change it back to $300.
Figure 4-9
IRR of an Uneven Cash Flow Stream
Periods: 0 1 2 3 4 5
Cash flows: -$1,000 $100 $300 $300 $300 $500
1. Calculator: You could enter the cash flows into the cash flow register of a financial calculator and then press the IRR key to find the answer. 12.55%
2. Excel IRR Function: Cell references: IRR = =IRR(B546:G546) 12.55%
4-15 Semiannual and Other Compounding Periods
If $100 is invested in an account at an annual nominal interest rate of 12% for 1 year, what are the effective interest rates and the future values based on annual, semiannual, quarterly, monthly and daily compounding?
When you work this problem, recognize that with more compounding periods, you receive interest sooner than with annual compounding, so you will earn more "interest on interest." Therefore, you will end up with more money, and the effective interest rate will be higher, than with annual compounding.
Nominal annual rate = 12%
Amount invested = $100
Number of years = 1
Figure 4-10
Effect on $100 of Compounding More Frequently than Once a Year
Frequency of Compounding Nominal Annual Rate Number of Periods per Year (M)a Periodic Interest Rate (IPER) Effective Annual Rate (EFF%)b Future Valuec Percentage Increase in FV
Annual 12% 1 12.0000% 12.0000% $112.00
Semiannual 12% 2 6.0000% 12.3600% $112.36 0.32%
Quarterly 12% 4 3.0000% 12.5509% $112.55 0.17%
Monthly 12% 12 1.0000% 12.6825% $112.68 0.12%
Daily 12% 365 0.0329% 12.7475% $112.75 0.06%
a We used 365 days per year in the calculations.
bThe EFF% is calculated as (1 + IPER)M.
cThe Future value is calculated as $100(1 + EFF%).
ADD-ON INTEREST (Box: Truth in Lending)
Cost of Credit based on "Add-On" Interest. This table is not in the text, but the
procedure is discussed in the "Truth in Lending Box". This procedure is commonly
used by retailers, auto dealers, and many other lenders. The calculator solution is
explained in the text and also below. The Excel solution is explained just below.
Amount borrowed = Cost of TV. Disregards the advanced payment, handled separately. $3,000.00
Nominal rate 8.00%
Amount of interest = interest rate x Amt borrowed $240.00
Stated loan size = Amt borrowed + Interest $3,240.00
Number of payments 12
Payment/month -$270.00
0 1 2 3 4 5 6 7 8 9 10 11 12
Amt borrowed $3,000.00
Monthly Pmts -$270.00 -$270.00 -$270.00 -$270.00 -$270.00 -$270.00 -$270.00 -$270.00 -$270.00 -$270.00 -$270.00 -$270.00
CF time line $2,730.00 -$270.00 -$270.00 -$270.00 -$270.00 -$270.00 -$270.00 -$270.00 -$270.00 -$270.00 -$270.00 -$270.00 $0.00
IRR = periodic rate: =IRR(B598:M598) = 1.4313%
APR rate: =E600*G592 = 17.1758%
EFF%: =(1+E600)^G592-1 = 18.5945%
Before the Truth in Lending Act, auto dealers, TV dealers, and even student loan officers would
make add-on loans and just tell customers about the 8% stated rate. After 1968, such lenders were
required to also report the much higher APR rate. But lenders are still not required to report the even
higher EFF%, which is the "true" rate that borrowers should base decisions on.
We showed the cash flows above as a"horizontal" time line, but it's easier to fit the analysis on the
screen using a vertical time line, as shown below. The calculations are identical, but the vertical setup
is better from a presentation standpoint if we have more cash flows than can be shown on the screen.
Periods Borrowed Payments Monthly CFs
0 $3,000.00 -$270.00 $2,730.00
1 -$270.00 -$270.00
2 -$270.00 -$270.00
3 -$270.00 -$270.00
4 -$270.00 -$270.00
5 -$270.00 -$270.00
6 -$270.00 -$270.00
7 -$270.00 -$270.00
8 -$270.00 -$270.00
9 -$270.00 -$270.00
10 -$270.00 -$270.00
11 -$270.00 -$270.00
12 $0.00 $0.00
IRR = periodic rate: 1.4313%
APR rate: 17.176%
EFF%: 18.595%
To solve the problem with a calculator, first set the machine to BEGIN mode, then enter N = 12, PV =
3000, and PMT = -270. When you press the I/YR key to get the periodic rate, 1.431313, which you can
use to find the APR and EFF% as we did above.
4-16 Fractional Time Periods
Suppose you deposited $100 in a bank that pays a nominal rate of 10%, compounded daily, based on a 365-day year. How much would you have after 9 months?
It depends on whether interest is compounded or is simple interest.
Inputs
PV = $100
INOM = 10%
Number of days in year = 365
Number of months interest charged = 9
Compounded Interest Results
IPER = 0.02740%
Number of days interest charged (rounded up) = 274
Ending amount = $107.79
Interest owed = $7.79
Simple Interest Results
Number of days interest charged = 274
Number of years interest charged = 0.7506849
Method 1: Interest owed= amount borrowed x annual rate x number of years = $7.51
Method 2: Interest owed= amount borrowed x daily rate x number of days = $7.51
4-17 Amortized Loans
If a loan is to be repaid in equal amounts on a monthly, quarterly, or annual basis it is called an amortized loan.
The figure below illustrates the amortization process. A company borrows $100,000, with the loan to be repaid in 5 equal payments at the end of each of the next 5 years. The lender charges 6% on the balance at the beginning of each year.
With a calculator, we solve for the required payment, then we construct an amortization table as shown in The figure below. It is far easier, and less prone to errors, to construct the amortization table with Excel, as we do here.
Figure 4-11
Loan Amortization Schedule, $100,000 at 6% for 5 Years
INPUTS:
Amount borrowed: $100,000
Years: 5
Rate: 6%
Intermediate calculation:
PMT: $23,739.64 =PMT(C678,C677,−C676)
Year Beginning Amount (1) Payment (2) Interesta (3) Repayment of Principalb (2) − (3) = (4) Ending Balance (1) − (4) = (5)
1 $100,000.00 $23,739.64 $6,000.00 $17,739.64 $82,260.36
2 $82,260.36 $23,739.64 $4,935.62 $18,804.02 $63,456.34
3 $63,456.34 $23,739.64 $3,807.38 $19,932.26 $43,524.08
4 $43,524.08 $23,739.64 $2,611.44 $21,128.20 $22,395.89
5 $22,395.89 $23,739.64 $1,343.75 $22,395.89 $0.00
a Interest in each period is calculated by multiplying the loan balance at the beginning of the year by the interest rate. Therefore, interest in Year 1 is $100,000(0.06) = $6,000; in Year 2 it is $82,260.36(0.06) = $4,935.62; and so on.
b Repayment of principal is the $23,739.64 annual payment minus the interest charge for the year, $17,739.64 for Year 1.
Consider a 30-year home mortgage of $250,000 at an annual rate of 6%. How much interest will the borrower pay over the life of the loan?

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