Question

Perpetuity X has annual payments of 1,2,3,... at the end of each year. Perpetuity Y has annual payments of q, q, 2q, 2q, 3q, 3q, ... at the end of each year. The present value of X is equal to the present value of Y at an annual effective interest rate of 10%. Calculate q.

I'm new to perpetuities but basically understand how perpetuities work. I also have a formula for perpetuities that increase every year. I just can't figure out how to break down the "q, q, 2q, 2q, 3q, 3q, ..." portion that increases every other year so please be sure to elaborate how that portion of the problem is broken down.

Answer #1

You can consider this perpetuity as two perpetuities:

First perpetuity with cash flow as q,2q,3q,4q....in year 1,3,5,7
respectively

1,3,5,7 can be considered as 1,2,3,4 with interest rate compounded
biennially instead of annually=1.1^2-1=21%

Present value at t=1:=(q/21%+q/(21%*21%))*1.21

Present value at t=0:=(q/21%+q/(21%*21%))*1.1

Second perpetuity with cash flow as q,2q,3q,4q....in year
2,4,6,8 respectively

2,4,6,8 can be considered as 1,2,3,4 with interest rate compounded
biennially instead of annually=1.1^2-1=21%

Present value at t=2:=(q/21%+q/(21%*21%))*1.21

Present value at t=0:=(q/21%+q/(21%*21%))

Sum=57.61904762*q

Calculus is needed.
Perpetuity X has level payments of $220 at the end of
each year. Perpetuity Y also has end-of-year payments but
they begin at $11 and increase by $11 each year. Find the rate of
interest which will make the difference in present values between
these two perpetuities a maximum. (Round your answer to two decimal
places.)

1. A perpetuity-due has monthly payments in this pattern: Q, 2Q,
3Q, Q, 2Q, 3Q, Q, 2Q, 3Q, . . . The present value of the perpetuity
is $700,000 and the effective annual discount rate is 6%. Find
Q.
2. A 30 year annuity-immediate has first payment $1200 and each
subsequent payment increases by 0.5%. The payments are monthly and
the annual effective rate is 8%. Find the accumulated value of the
annuity at the end of 30 years.
3....

A perpetuity with payments of 1 at the end of each year has a
present value of 40. A 10-year annuity pays X at the beginning of
each year. Assuming the same effective interest rate, the present
values of the perpetuity and the 10-year annuity are equal. Find
X.

1. Perpetuities in arithmetic progression. If a perpetuity has
first payment P and each payment increases by Q, then its present
value, one period before the first payment, is P/i + Q/i^2 Using
this formula, find the present value of a perpetuity-immediate
which has annual payments with first payment $360 and each
subsequent payment increasing by $40, at annual interest rate
1.3%.
The answer should be ($264,378.70).
2. Filip buys a perpetuity-immediate with varying annual
payments. During the first 5...

A perpetuity will make $1000 payments at the end of each year
starting from year 5 (i.e., first payment will occur at the end of
year 5). If the discount rate is 7%, what is the present value of
this perpetuity? Round your answer to the nearest dollar; do not
include the $ sign (i.e., if the answer is $8,300.3562, enter it as
8,300).

A perpetuity pays $1000 at the end of every month for 11 months
of each year. At the end of the 12th month of each year, it pays
double that amount. If the effective ANNUAL rate is 10.4%, what is
the present value of this perpetual annuity?

A perpetuity will make payments of $100,000 every third year,
with the first payment occurring three years from now. The annual
nominal interest rate convertible quarterly is 8%. Find the present
value of this perpetuity.
(I did this problem, just want to check if I did it correctly
because the answer doesn't look right to me, not sure what I did
incorrectly, I got PV = 372,800.47)

Charities A, B and C split level payments of X at the end of
each year forever. For the first n years, two-thirds of the payment
will be given to Charity A and the other one-third of the payment
to Charity B. Thereafter, all payments will be directed to Charity
C. If the annual effective interest rate for the perpetuity is 10%,
then the proportion of the present value going to Charity C is
0.4240976184 If the present value of...

John Smith will receive annual payments of $800 at the end of
each year for 12 years. The first payment will be received in year
4. What is the present value of these payments if the discount rate
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You will receive annual payments of $5,000 at the end of each
year for 10 years, but the first payment will be received in year
3. What is the present value of these payments if the discount rate
is 8 percent? $30,260.49 $26,633.40 $28,251.12 $24,387.13

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