Juan has an annuity that pays him $9400 at the beginning of each year. Assume the economy will grow at a rate of 3.4% annually. What is the value of the annuity if he received it now instead of over a period of 10 years?
Method1
Since the basic formulas for annuity assume payments at the end of a period, a slight adjustment is needed.
PV = 9400 +9400(1 - 1.034^-9)/.034
= 81243.05
method 2.
look at the geometric series
9400 + 9400(1.034^-1) + 9400(1.034^-2) + ... + 9400(1.034^-9)
here a = 9400, r = 1.034^-1
sum = a(1 - r^10)/(1-r)
= 9400(1 - 1/1.034)^10/(1-1/1.034)
= 9400(.28419519)/.032882011)
= 81243.05, same as before
Thank you very much
To find the present value of the annuity, we can use the present value formula for an ordinary annuity:
PV = P * (1 - (1 + r)^(-n)) / r
Where:
PV = Present Value
P = Payment per period
r = Interest rate per period
n = Total number of periods
Given:
P = $9400 (Payment per year)
r = 3.4% (Interest rate per year)
n = 10 (Total number of years)
Converting the interest rate to decimal form:
r = 3.4 / 100 = 0.034
Plugging the values into the formula:
PV = 9400 * (1 - (1 + 0.034)^(-10)) / 0.034
Now let's calculate this:
PV = 9400 * (1 - (1.034)^(-10)) / 0.034
PV = 9400 * (1 - 0.694646652) / 0.034
PV = 9400 * 0.305353348 / 0.034
PV = 8525.35294117647
Therefore, the value of the annuity if Juan received it now instead of over a period of 10 years is approximately $8525.35.
To calculate the value of the annuity if Juan received it now instead of over a period of 10 years, we can use the present value formula for annuities. The formula is:
PV = PMT × [1 - (1 + r)^(-n)] / r
where:
PV = Present Value (Value of the annuity)
PMT = Payment per period ($9400)
r = Interest rate per period (3.4% or 0.034)
n = Number of periods (10 years)
Let's calculate the value of the annuity using the formula:
PV = $9400 × [1 - (1 + 0.034)^(-10)] / 0.034
First, let's simplify the expression in the brackets:
[1 - (1 + 0.034)^(-10)] = [1 - (1.034)^(-10)]
Calculating (1.034)^(-10):
(1.034)^(-10) ≈ 0.696580573
Substituting this value back into the original equation:
PV = $9400 × [1 - 0.696580573] / 0.034
Simplifying further:
PV ≈ $9400 × 0.303419427 / 0.034
PV ≈ $83,289.35
Therefore, the value of the annuity if Juan received it now instead of over a period of 10 years would be approximately $83,289.35.