Assume that you have two contracts to choose from: Contract A pays $5,000 per year for 5 years starting one year from today. Contract B pays $5,000 per year for 5 years starting today. The discount rate for each is 6%. Which annuity contract would you choose for your retirement? Why?
Would it be contract A?
To determine which annuity contract would be more beneficial for your retirement, we need to calculate the present value of each contract. The present value is the current value of future cash flows taking into account the time value of money.
Let's begin with Contract A, which pays $5,000 per year for 5 years starting one year from today. Since we are dealing with future cash flows, we need to discount these payments to their present value.
Step 1: Calculate the present value of each annual payment in Contract A.
Using the formula for the present value of an annuity:
PV = C * (1 - (1 + r)^(-n)) / r
where:
PV = Present Value
C = Cash flow per period ($5,000)
r = Discount rate (6% or 0.06)
n = Number of periods (5)
PV_A = $5,000 * (1 - (1 + 0.06)^(-5)) / 0.06
PV_A = $19,877.27
Step 2: Calculate the present value of the additional year's delay in payments for Contract A.
Since Contract B starts today, while Contract A starts one year later, we need to calculate the present value of having $5,000 a year today versus receiving it one year later. We can do this using the formula for the present value of a single payment:
PV = FV / (1 + r)^n
where:
FV = Future value ($5,000)
PV = Present value
r = Discount rate (6% or 0.06)
n = Number of periods (1)
PV_delay = $5,000 / (1 + 0.06)^1
PV_delay = $4,716.98
Step 3: Add the present value of the annual payments and the present value of the delay to get the total present value of Contract A.
PV_total_A = PV_A + PV_delay
PV_total_A = $19,877.27 + $4,716.98
PV_total_A = $24,594.25
Now, let's calculate the present value of Contract B, which pays $5,000 per year for 5 years starting today.
Step 1: Calculate the present value of each annual payment in Contract B using the same formula as in Step 1 of Contract A.
PV_B = $5,000 * (1 - (1 + 0.06)^(-5)) / 0.06
PV_B = $23,577.10
Comparing the two present values:
PV_total_A = $24,594.25 (Contract A)
PV_B = $23,577.10 (Contract B)
Since the present value of Contract A is higher than that of Contract B, choosing Contract A would be the more beneficial option for your retirement.