An investment will provide Nicholas with $100 at the end of each year for the next 10 years. What is the present value of that annuity if the discount rate is 8% annually? What is the present value of the above annuity if the payments are received at the beginning of each year? If you deposit those payments into an account earning 8%, what will the future value be in 10 years? What will the future value be if he opening the account with $1,000 today, and then make the $100 deposits at the end of each year?
To calculate the present value of an annuity, you can use the formula:
PV = p * (1 - (1 + r)^(-n)) / r
Where:
PV = Present value
p = Payment amount
r = Discount rate
n = Number of periods
1. Present Value with End-of-Year Payments:
In this case, Nicholas will receive $100 at the end of each year for the next 10 years, with a discount rate of 8% annually. You can plug these values into the formula:
p = $100
r = 8% = 0.08 (as a decimal)
n = 10
PV = 100 * (1 - (1 + 0.08)^(-10)) / 0.08
PV = $676.08
Therefore, the present value of the annuity, with end-of-year payments, is $676.08.
2. Present Value with Beginning-of-Year Payments:
If the payments are received at the beginning of each year, the calculation is a bit different. You need to adjust the formula to account for the different timing of the payments. The formula becomes:
PV = p * (1 - (1 + r)^(-n)) / r * (1 + r)
Using the same values as before:
p = $100
r = 8% = 0.08 (as a decimal)
n = 10
PV = 100 * (1 - (1 + 0.08)^(-10)) / 0.08 * (1 + 0.08)
PV = $729.53
Therefore, the present value of the annuity, with beginning-of-year payments, is $729.53.
3. Future Value of the Payments at 8% Interest:
If you deposit the $100 payments into an account earning 8% interest annually for 10 years, you can use the future value of an annuity formula:
FV = p * ((1 + r)^n - 1) / r
Using the same values as before:
p = $100
r = 8% = 0.08 (as a decimal)
n = 10
FV = 100 * ((1 + 0.08)^10 - 1) / 0.08
FV = $1,452.65
Therefore, the future value of the payments in 10 years will be $1,452.65.
4. Future Value with Initial Deposit:
If Nicholas opens an account with $1,000 today and makes $100 deposits at the end of each year, you need to incorporate the initial deposit into the calculation:
FV = PV * (1 + r)^n
Where:
PV = Present value
r = Interest rate
n = Number of periods
Using the same values as before:
PV = $1,000
r = 8% = 0.08 (as a decimal)
n = 10
FV = 1,000 * (1 + 0.08)^10
FV = $2,158.92
Therefore, the future value with the initial deposit will be $2,158.92.
To find the present value of the annuity, we can use the formula:
PV = PMT * [(1 - (1 + r)^(-n))/r]
where:
PV = Present Value
PMT = Payment per period
r = Discount rate per period
n = Number of periods
1. Present Value of the annuity received at the end of each year:
PMT = $100 (annual payment)
r = 8% = 0.08 (annual discount rate)
n = 10 (number of years)
Using the formula, we can calculate the present value:
PV = $100 * [(1 - (1 + 0.08)^(-10))/0.08]
PV = $100 * [(1 - 1.08^(-10))/0.08]
PV ≈ $671.75
Therefore, the present value of the annuity received at the end of each year is approximately $671.75.
2. Present Value of the annuity received at the beginning of each year:
To calculate the present value of an annuity received at the beginning of each year, we can use the same formula as above but with a slight modification. We need to adjust the payment per period (PMT) by multiplying it by (1+r) to account for the time value of money:
Adjusted PMT = PMT * (1+r)
Adjusted PMT = $100 * (1+0.08)
Adjusted PMT = $108
Now, we can calculate the present value using the formula:
PV = Adjusted PMT * [(1 - (1 + r)^(-n))/r]
PV = $108 * [(1 - (1 + 0.08)^(-10))/0.08]
PV ≈ $727.73
Therefore, the present value of the annuity received at the beginning of each year is approximately $727.73.
3. Future value of the annuity if the payments are deposited into an account earning 8% annually:
Since the payments are deposited into an account earning 8% annually, we can use the future value (FV) formula:
FV = PMT * [(1 + r)^n - 1)/r]
Using the formula with the given values:
FV = $100 * [(1 + 0.08)^10 - 1]/0.08
FV ≈ $1,469.33
Therefore, the future value of the annuity, after 10 years of depositing $100 per year into an account earning 8% annually, is approximately $1,469.33.
4. Future value of the annuity if opening the account with $1,000 today and then making $100 deposits at the end of each year:
In this case, we need to first find the future value of the initial $1,000 deposit over 10 years and then add the future value of the annuity received at the end of each year.
Future value of the initial deposit:
FV1 = PV * (1 + r)^n
FV1 = $1,000 * (1 + 0.08)^10
FV1 ≈ $2,159.27
Future value of the annuity received at the end of each year:
FV2 = PMT * [(1 + r)^n - 1)/r]
FV2 = $100 * [(1 + 0.08)^10 - 1]/0.08
FV2 ≈ $1,469.33
Total future value:
Total FV = FV1 + FV2
Total FV ≈ $2,159.27 + $1,469.33
Total FV ≈ $3,628.60
Therefore, if Nicholas deposits $1,000 today and then makes $100 deposits at the end of each year for 10 years, the future value of the account would be approximately $3,628.60.