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A B C D E F G H I J K
N3
I0.1
PMT 100
Time period 0 1 2 3
CFt100 100 100 0 Annuity PV
PV3100.00 90.91 82.64 0.00 = $273.55
I = 10%
Time period
0 1 2 3 4
0100 300 300 -50
Using the function wizard, we follow the same procedure as above, except remember to enter a "1" to
tell Excel that in this problem the payments occur at the beginning of the periods.
To find the present value of the annuity due, this problem is solved just like the previous problem,
except that the payments occur in periods 0 through 2.
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A B C D E F G H I J K
2300 247.93
3300 225.39
4-50 -34.15
$530.09
Or
Inputs
m=periods/yr 2This is the number of periods per year, m.
IPER = inom/m
IPER = 10% / 2
IPER = 5%
EFF% = (1+ INOM/M)M
EFF% = (1 + (10%/2))^2 - 1
h. (1.) Identify (a) the stated, or quoted, or nominal rate (iNom) and (b) the periodic rate (iPER).
With, the financial calculator, we could enter each of these cash flows and the discount rate, and
simply press NPV for the present value of the cash flow stream. In Excel, we can perform a similar
time period zero. However, the "NPV" function interprets the first data entry as being the cash flow in
time period one. Therefore, the initial cash flow must be added seperately. In this particular example,
the initial cash flow is zero.
Larger, because interest is earned on interest.
PV of CF stream =
compounding periods.
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A B C D E F G H I J K
EFF% = 10.25%
SEMIANNUAL AND OTHER COMPOUNDING PERIODS
h. (3.) What is the future value of $100 after 5 years under 12% annual compounding?
N 3
I0.12 FV = $140.49
PV 100
N (years x 2) 6
I (I per year/2) 0.06 FV = $141.85
PV 100
What is the FV with quarterly compounding?
N (years x 4) 12
I (I per year/4) 0.03 FV = $142.58
PV 100
N (years x 12) 36
I (I per year/12) 0.01 FV = $143.08
PV 100
N (years x 365) 1095
I (I per year/12) 0.00032877 FV = $143.32
PV 100
NBeg. Amt. Payment Interest Principal End. Amt.
1 $1,000.00 $402.11 $100.00 $302.11 $697.89
2 $697.89 $402.11 $69.79 $332.33 $365.56
3 $365.56 $402.11 $36.56 $365.56 $0.00
Note: See Columns M
Now, construct an amortization table for the loan described above.
j. (1.) What would the required payment be on a $1,000 loan that is to be repaid in three equal installments at the
end of each of the next three years if the interest rate is 10%?
I. Will the effective annual rate ever be equal to the nominal (quoted) rate? Only if the compounding period is equal
to 1 year.
j. (2.) What is the annual interest expense for the borrower, and the annual interest income for the lender, during
Year 2?
$450.00
Payment
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A B C D E F G H I J K
through R for a 30 year
mortgage example.
I0.00031054
N273
FV $108.85
Years 0 0.5 1 1.5 2 2.5 3
Periods 0 1.0 2 3.0 4 5.0 6
Cash Flow 0 100 0100 0100
Total FV = Σ = $331.80
k. On January 1, you deposit $100 in an account that pays a nominal (or quoted) interest rate of 11.33463%, with
interest added (compounded) daily. How much will you have in your account on October 1, or 9 months later? (273
days)
l. (1.) What is the value at the end of Year 3 of the following cash flow stream if the quoted interest rate is 10%,
Alternatively, you could calculate the annual effective rate and use this to find the future value of a 3-year annuity.
$0.00
$50.00
$100.00
$150.00
$200.00
$250.00
$300.00
$350.00
$400.00
$450.00
1 2 3
Year
Principal
Interest
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A B C D E F G H I J K
Periods 0 1 2 3.0 4 5.0 6
PV of CF $90.70 $82.27 $74.62
In the second approach, we use the annual effective rate to find the present value of a 3-year annuity.
PV = $247.59
0 1 2 3 4 5456
850
I0.00018538
0 1 2 3 4 5456
1000
0 1 2 3 4 5456
850 1000
what condition holds? (Hint: Think of annual compounding, when iNOM = EAR = iPER.) What would be wrong with
your answer to questions l(1) and l(2) if you used the nominal rate (10%) rather than the periodic rate (iNOM/2 = 10%/2
l. (2.) What is the PV of the same stream?
See which has the greater present value
m. Suppose someone offered to sell you a note calling for the payment of $1,000 in 15 months (or 456 days). They
offer to sell it to you for $850. You have $850 in a bank time deposit that pays a 6.76649% nominal rate with daily
compounding, which is a 7% effective annual interest rate, and you plan to leave the money in the bank unless you
buy the note. The note is not risky--you are sure it will be paid on schedule. Should you buy the note? Check the
l. (3.) Is the stream an annuity? No, because we don't have a payment for each compounding period.
Using the first approach, we find the present value of each individual cash flow using the periodic rate
and the number of periods.
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729
A B C D E F G H I J K
I 0.035646% per day
EAR 13.89% > 7% so buy the note.
16 of 30
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