You have the opportunity to invest in several annuities. Which of the following
ID: 2818354 • Letter: Y
Question
You have the opportunity to invest in several annuities. Which of the following 10-year annuities has the greatest present value (PV)? Assume that all annuities earn the same positive interest rate. An annuity that pays $1,000 at the beginning of each year An annuity that pays $500 at the beginning of every six months An annuity that pays $500 at the end of every six months An annuity that pays $1,000 at the end of each year An ordinary annuity selling at $16,496.00 today promises to make equal payments at the end of each year for the next six years (N). If the annuity's appropriate interest rate (1) remains at 5.00% during this time, the annual annuity payment (PMT) will be WI You just won the lottery. Congratulations! The jackpot is $35,000,000, paid in six equal annual payments. The first payment on the lottery jackpot will be made today. In present value terms, you really won assuming annual interest rate of 5.00%.Explanation / Answer
1.
For immediate annuity or annuity due type of investment, we need to deposit money at the beginning of the each periods. In order to compute PV of annuity due we need to add the immediate(one) cash flow to present value of future periodic cash flows remained as n -1 number of periods. First cash flow is not discounted and discounting starts from 2nd cash flow. So the PV of annuity due is higher than PV of ordinary annuity with same interest rate and same number of time periods. Among the annuity dues as in the option 1st and 2nd, higher the beginning payment higher is the PV.
We can compute the PV of each stream assuming 10 % interest rate as:
1st
2nd
3rd
4th
Periodic cash flow
$ 1,000
$ 500
$ 1,000
$ 500
No. of years
10
10
10
10
No. of periods
10
20
10
20
Interest rate
10 %
10 %
10 %
10 %
Present value
$6,759.02
$6,542.66
$6,144.57
$6,231.11
PV is highest for 1st cash flow streams of $ 1,000 payments at beginning of each period.
Hence option “1st: An annuity that pays $ 1,000 at the beginning of each year” is the correct answer.
2.
Annual cash flow can be computed using PV of cash flows as:
PV = C x PVIFA (i, n)
C = PV = C / PVIFA (i, n)
PV = Present value of annuity = $ 16,496
C = Periodic cash flow
i = Rate of interest = 5 % p.a.
n = No. of periods = 6
C = $ 16,496 / PVIFA (5 %, 6)
C = $ 16,496 / 5.0757
C = $ 3,250
3.
Jackpot annual payment is = $ 35,000,000/6 = $ 5,833,333.3333
PV of annuity due can be computed as:
PV = C x PVIFAD (i, n)
C = Periodic cash flow = $ 5,833,333.3333
i = Rate of interest = 5 % p.a.
n = No. of periods = 6
PV = $ 5,833,333.3333 x PVIFAD (5 %, 6)
PV = $ 5,833,333.3333 x 5.3295
PV = $ 31,088,750
4.
Rate of return can be computed using present value of annuity and periodic cash flow as:
PV = C x PVIFA (i, n)
$ 2,709.77 = $ 800 x PVIFA (i, 4)
PVIFA (i, 4) = $ 2,709.77/ $ 800
PVIFA (i, 4) = 3.3872
Using PVIFA table we can find out that 3.3872 is in the 4th periodic row of the 7 % interest rate
Hence rate of return is 7 % p.a.
5.
No of periods needs to deposit, can be computed using future value of annuity and periodic payment as:
FV = P x [(1+r) n – 1/r]
FV = Future value of annuity = $ $ 4,526,674
P = Periodic cash flow = $ 100,000
r = Rate per period = 7 % or 0.07 p.a.
n = Numbers of periods
$ 4,526,674 = $ 100,000 x [(1+0.07) n - 1 /0.07]
$ 4,526,674 = $ 100,000 x [(1.07) n - 1 /0.07]
[(1.07) n - 1/0.07] = $ 4,526,674 /$ 100,000
[(1.07) n - 1/0.07] = 45.26674
(1.07) n – 1 = 45.26674 x 0.07
(1.07) n – 1 = 3.1686718
(1.07) n = 1 + 3.1686718
(1.07) n = 4.1686718
Taking log of both sides and solving for n, we get:
n x log 1.07 = log 4.1686718
n x 0.02938377769 = 0.61999770441
n = 0.61999770441/0.02938377769 = 21.09999983
n = 21.1 years
Keanu needs to deposit 21.1 years to reach the goal.
1st
2nd
3rd
4th
Periodic cash flow
$ 1,000
$ 500
$ 1,000
$ 500
No. of years
10
10
10
10
No. of periods
10
20
10
20
Interest rate
10 %
10 %
10 %
10 %
Present value
$6,759.02
$6,542.66
$6,144.57
$6,231.11
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