Asked by Jazmin Chacon on Jun 30, 2024
Verified
You are comparing two annuities with equal present values. The applicable discount rate is 7.5%. One annuity pays $5,000 on the first day of each year for twenty years. How much does the second annuity pay each year for twenty years if it pays at the end of each year?
A) $4,651
B) $5,075
C) $5,000
D) $5,375
E) $5,405
Annuities
Financial products that pay out a fixed stream of payments to an individual, primarily used as an income stream for retirees.
Discount Rate
The interest rate charged by central banks on loans they give to commercial banks.
Present Values
The present equivalent of a future money sum or cash flow series, calculated using a particular rate of return.
- Absorb the basic tenets of the time value of money and the compounding of interest.
- Utilize the theories of expanding annuities and perpetuities for financial decision-making.
Verified Answer
PV = PMT * ((1 - (1 / (1 + r)^n)) / r)
Where PV is the present value, PMT is the payment amount, r is the discount rate, and n is the number of periods.
For the first annuity, we know that the present value is equal to the payments multiplied by the present value factor:
PV = $5,000 * ((1 - (1 / (1 + 0.075)^20)) / 0.075)
PV = $52,846.20
Since we know the present value of both annuities is the same, we can use this value to solve for the payment amount of the second annuity:
PV = PMT * ((1 - (1 / (1 + r)^n)) / r)
$52,846.20 = PMT * ((1 - (1 / (1 + 0.075)^20)) / 0.075)
PMT = $5,375.20
Therefore, the second annuity pays $5,375 at the end of each year for twenty years. The best choice is D.
Learning Objectives
- Absorb the basic tenets of the time value of money and the compounding of interest.
- Utilize the theories of expanding annuities and perpetuities for financial decision-making.
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