A contractual arrangement requires payments of $3,000 on 1 July 2020 and $8,000 on 1 July 2023. What is an equivalent single amount payable on 1 July 2017 at an interest rate of 12% p.a. compounding half-yearly? Group of answer choices $6,456.07 $6,090.64 $6,188.39 $8,158.54
Question
A contractual arrangement requires payments of 8,000 on 1 July 2023. What is an equivalent single amount payable on 1 July 2017 at an interest rate of 12% p.a. compounding half-yearly?
Group of answer choices
$6,456.07
$6,090.64
$6,188.39
$8,158.54
Solution
To calculate the equivalent single amount payable on 1 July 2017, we need to find the present value of the two future payments.
The present value (PV) of a future payment can be calculated using the formula:
PV = FV / (1 + r)^n
where:
- FV is the future value (the amount of the future payment)
- r is the interest rate per period (12% or 0.12 divided by 2)
- n is the number of periods (the number of half-years between the payment date and 1 July 2017)
For the $3,000 payment on 1 July 2020, n is 6 half-years. Substituting the given values into the formula, we get:
PV1 = $3,000 / (1 + 0.12/2)^6
For the $8,000 payment on 1 July 2023, n is 12 half-years. Substituting the given values into the formula, we get:
PV2 = $8,000 / (1 + 0.12/2)^12
The equivalent single amount payable on 1 July 2017 is the sum of these two present values:
PV = PV1 + PV2
Calculating these expressions, we find that the equivalent single amount payable on 1 July 2017 is approximately 6,456.07.
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