Given an effective annual interest rate of 3%, compute:
a) The future value of a sequence of quarterly payments of 700€ made at the beginning of each quarter for the next 10 years.
b) The present value of a stream of payments starting at 500€ per semester (increasing by a cumulative 4% each semester), where the first the payment takes place today and the last payment will occur in 6 years' time.
c) The present value of a perpetuity of 300€ per month, with the first payment taking place in 4 years.
Guide On Rating System
Vote
a) To calculate the future value of a sequence of quarterly payments, we can use the formula for the future value of an ordinary annuity:
FV = P * [(1+r)^n - 1] / r
where:
FV is the future value
P is the payment amount per period
r is the interest rate per period
n is the number of periods
In this case, the payment amount per quarter is 700€, the interest rate per quarter can be calculated by dividing the annual interest rate by the number of quarters in a year (3% / 4 = 0.75%), and the number of quarters is 10 years * 4 quarters per year = 40 quarters.
FV = 700 * [(1 + 0.0075)^40 - 1] / 0.0075
FV ≈ 33,890.1646€
So, the future value of the sequence of quarterly payments is approximately 33,890.1646€.
b) To calculate the present value of a stream of payments with increasing amounts, we can use the formula for the present value of a growing annuity:
PV = P * (1 - (1 + r)^-n) / (r - g)
where:
PV is the present value
P is the initial payment amount
r is the interest rate per period
n is the number of periods
g is the growth rate per period
In this case, the initial payment amount is 500€, the interest rate per semester can be calculated by dividing the annual interest rate by the number of semesters in a year (3% / 2 = 1.5%), the number of semesters is 6 years * 2 semesters per year = 12 semesters, and the growth rate per semester is 4% / 2 = 2%.
PV = 500 * (1 - (1 + 0.015)^-12) / (0.015 - 0.02)
PV ≈ 4,851.7044€
So, the present value of the stream of payments is approximately 4,851.7044€.
c) To calculate the present value of a perpetuity, we can use the formula for the present value of a perpetuity:
PV = P / r
where:
PV is the present value
P is the payment amount per period
r is the interest rate per period
In this case, the payment amount per month is 300€, the interest rate per month can be calculated by dividing the annual interest rate by the number of months in a year (3% / 12 = 0.25%).
PV = 300 / 0.0025
PV = 120000€
So, the present value of the perpetuity is 120,000€.
Got a question with my answer?
Cheapmath.com is completely free!
Asking question is free.
Getting answers is free.
+99