Notesale: Turn your study into money

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1
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The account credits interest at an annual effective interest rate of i
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Calculate X
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I is equal to the interest (discount) rate and N number of
duration
For 40 years,
X = 100[(1 + i)^40 + (1 + i)^36 + · · ·+ (1 + i)^4]
=[100 × (1+i)^4 × (1 - (1 + i)^40]/1 − (1 + i)^4
For 20 years,
Y = A(20) = 100[(1+i)^20+(1+i)^16+· · ·+(1+i)^4]
Using X = 5Y (5 times the accumulated amount in the account at the ned of 20 years) and using
a difference of squares on the left side gives
1 + (1 + i)^20 = 5
so (1 + i)^20 = 4
so (1 + i)^4 = 4^0
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319508
Hence X = [100 × (1 + i)^4 × (1 − (1 + i)^40)] / 1 − (1 + i)^4
= [100×1
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3195
X = 6194
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A perpetuity costs 77
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The
perpetuity pays 1 at the end of year 2, 2 at the end of year 3,
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After year (n+1), the payments remain constant at n
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5%
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1000 is deposited into Fund X, which earns an annual effective rate of 6%
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At the
end of the tenth year, the fund is depleted
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Determine
the accumulated value of Fund Y at the end of year 10
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A perpetuity-immediate pays 100 per year
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Each subsequent annual payment will be 8% greater than the preceding
payment
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Calculate X
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up to infinity=100/r
In this case interest rate of perpetuity is =8%=0
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08=$1250
This amount is exchanged for another perpetuity paying X at the end of 6th year
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08*X
Payment at end of 8th year =8% more than the payment in 7th year
=1
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08*X=(1
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08^3)*X
The 25 year annuity will have last payment =(1
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08
N=Year of Cash Flow
We calculate the Present Value at end of year5 of the new annuity (with payment of X in the
first year)and lasting for another 25 years
Since this is exchanged with the original constant payment perpetuity of $100 per year
(Having a Present Value of $1250 at end of year5),
The Present Value of this new annuity should be =$1250
Present Value of new annuity at end of year
5=(X/1
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08)/(1
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08^2))/(1
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+(((X*(1
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08^25)=$1250
(X/1
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08)+(X/1
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+(X/1
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(25 times)=$1250
25X/1
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08)/25=$54
5
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During the first 5
years, the payment is constant and equal to 10
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For year 6 and all future years, the current year's payment is K%
larger than the previous year's payment
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2%,
the perpetuity has a present value of 167
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Calculate K, given K < 9
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According to the given data we have:
i = 0
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5 = 10a5] at
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092]
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092 component gone from the problem, we have:

128
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for infinity}

You can turn this into a geometric progression by pulling out
10 * [(1+k)/1
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then you're left with 1 + (1+k)/1
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092^2
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Since the problem says k <
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you know that (1+k)/1
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Therefore, you'll have 1/(1-(1+k)/1
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That geometric sum * (10*v^6*(1+K)) then has to equal your constant, 128
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After dividing 128
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84
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84 = (1+k)*[1/(1-(1+k)/1
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04 so k =
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To accumulate 8000 at the end of 3n years, deposits of 98 are made at the end of each
of the first n years and 196 at the end of each of the next 2n years
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You are given (1 + ir = 2 Determine i
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Olga buys a 5-year increasing annuity for X
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The nominal interest rate is 9% convertible quarterly
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8
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Interest is credited at a force of interest
20,000
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After 10 +t years, the account is worth

9
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Brian receives the first n payments, Colleen
receives the next n payments, and Jeff receives the remaining payments
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Calculate K

10
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At the same annual effective rate of i, the present value of a perpetuity paying 1 at the
end of each 4-month period, with first payment at the end of 4 months, is X
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11An insurance company has an obligation to pay the medical costs for a claimant
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The claimant is expected to live an additional 20 years
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Find the present value of the obligation if the annual interest rate is 5%
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A man turns 40 today and wishes to provide supplemental retirement C income of
3000 at the beginning of each month starting on his 65th birthday
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The fund earns a nominal rate
of 8% compounded monthly
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65
of income at the C beginning of each month starting immediately and continuing as long
as he "' survives
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65 = is the number of thousands required to provide the desired monthly retirement
benefit because each 1000 provides 9
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Thus, 310,881 is the capital required at age 65 to provide the desired monthly
retirement benefit
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73
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They plan to contribute X at each of their daughter's 1st
through 17th birthdays to fund the four 50,000 withdrawals
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Which of the following expressions does NOT represent a definition for

?

15
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Seth,
Susan, and Lori share the perpetuity such that Seth receives the payments of X for the
first n years and Susan receives the payments of X for the next m years, after which Lori
receives all the remaining payments of X
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The Present Value of the perpetuity for Susan is denoted by;
=

because it is the value after n periods multiplied by the payments received for m

periods
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The present value of a series of 50 payments starting at 100 at the end of the first year
and increasing by 1 each year thereafter is equal to X
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Calculate X
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The present value is
equal to the first payment divided by the annual effective interest rate
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Payments
continue forever
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Each payment is equal to the interest earned on the principal
I is only true for a level of perpetuity-immediate, and false in general
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III may be true for some level
perpetuities, but it is clearly not true for an increasing perpetuity – for example, an increasing
perpetuity due has a present value of 1 d2 and its first payment is 1, clearly not the interest on
its present value, even if we calculate the interest as the discount rate applied to the present
value
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18
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At an annual effective interest rate of 6%, the present value is equal to X
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19
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Calculate i
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An investor wishes to accumulate 10,000 at the end of 10 years by making
level deposits at the beginning of each year
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The interest is
immediately reinvested at an annual effective interest rate of 8%
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21A discount electronics store advertises the following financing arrangement: "We don't
offer you confusing interest rates
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" The first payment is due on the date of sale and
the remaining eleven payments at monthly intervals thereafter
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Solution
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The customer effectively borrows 100 and pays off that loan in twelve
monthly payments of 10 at the beginning of each monthly period
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Then
Using a financial calculator, we determine that j ≈
0
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03503153)^ 12 − 1 ≈ 51
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An annuity pays 1 at the end of each year for n years
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776
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476
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23
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Her monthly payment is 50 for the first 2 years, 100 for the next 2 years, and
150 for the last 2 years
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The annual
effective interest rate is i, and the monthly effective interest rate is j
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Matthew makes a series of payments at the beginning of each year for 20 years
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Each subsequent payment through the tenth year increases by 5%
from the previous payment
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Calculate the present value of these payments at the time the
first payment is made using an annual effective rate of 7%
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A company deposits 1000 at the beginning of the first year and 150 at the beginning
of each subsequent year into perpetuity
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The first payment is 100
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Calculate the company's yield rate for this transaction
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Megan purchases a perpetuity-immediate for 3250 with annual payments of 130
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Calculate P
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For 10,000, Kelly purchases an annuity-immediate that pays 400 quarterly for the next
10 years
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28
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These payments
earn interest at the end of each year at an annual effective rate of 8%
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At the end of 20 years, the
accumulated value of the 20 payments and the reinvested interest is 5600
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29
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Assuming an annual effective interest rate of
10%, calculate X
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John borrows 10,000 for 10 years at an annual effective interest rate of 10%
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45 at the end of
each year
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The deposits to the sinking fund are equal to 1,627
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Determine the balance in the sinking fund immediately after repayment of the loan
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Each of the first
ten payments equals 150% of the amount of interest due
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The lender charges interest at an annual effective rate of 10%
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3
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It can be
repaid under the following two options: (i) Equal annual payments at an annual effective
rate of 8
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(ii) Installments of 200 each year plus interest on the unpaid balance at an
annual effective rate of i
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Determine i
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A loan is amortized over five years with monthly payments at a nominal interest rate of
9% compounded monthly
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Each succeeding monthly payment will be 2% lower than the prior
payment
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5
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5% ii)
sinking fund method in which the lender receives an annual effective rate of 8% and the
sinking fund earns an annual effective rate of j Both methods require a payment of X to
be made at the end of each year for 20 years
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6
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• Seth has interest accumulated over the five years and pays
all the interest and principal in a lump sum at the end of five years
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• Lori repays her loan with 10 level payments at the end of every six month period
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7
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The
amount of interest paid in period t plus the amount of principal repaid in period t + 1
equals X
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Seth borrows X for four years at an annual effective interest rate of 8%, to be repaid
with equal payments at the end of each year
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12
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9
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With the lOth payment, the
borrower pays an extra 1000, and then repays the balance over 10 years with a revised
annual payment
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Calculate the amount of the revised
annual payment
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Lori borrows 10,000 for 10 years at an annual effective interest rate of 9%
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The total payments made by Lori over the 10-year period is X
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A bank customer takes out a loan of 500 with a 16% nominal interest rate convertible
quarterly
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Calculate the
amount of principal in the fourth payment
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That 4% applied to the principal of 500 gives 20 as
the interest due every quarter
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Answer A
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A loan is repaid with level annual payments based on an annual effective interest rate
of 7%
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Calculate the
amount of interest paid in the 18th payment



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