Q#1) Seekie is 50 years old and his salary next year will be $80,000. He forecasts that his salary will increase at a steady rate of 4% per year until her retirement at age 60. (a) If the interest rate is 8% per year, what is the PV of these future salary payments? (b) If Seekie saves 4% of his salary each year and invests these savings at an interest rate of 8%, how much will he have saved by age 60?
To find the present value (PV) of Seekie's future salary payments, we need to discount each payment by the interest rate. Let's break down the calculation step by step:
(a) To calculate the present value of the future salary payments, we will use the formula for the present value of an annuity:
PV = C * ((1 - (1 + r)^-n) / r)
Where:
PV = Present value of the future salary payments
C = Cash flow (salary payment)
r = Interest rate
n = Number of periods
In this case, the cash flow (salary payment) for each year is $80,000. The interest rate is 8%, and the number of periods is the difference between Seekie's current age (50) and his retirement age (60), which is 10 years.
Using the formula:
PV = $80,000 * ((1 - (1 + 0.08)^-10) / 0.08)
Calculating this, we get:
PV = $80,000 * ((1 - 1.08^-10) / 0.08)
= $80,000 * ((1 - 0.4632) / 0.08)
= $80,000 * (0.5368 / 0.08)
= $80,000 * 6.71
= $537,680
Therefore, the present value of the future salary payments is $537,680.
(b) To calculate how much Seekie will have saved by age 60, we need to calculate the future value (FV) of his annual savings. Again, we can use the formula for the future value of an annuity:
FV = P * ((1 + r)^n - 1) / r
Where:
FV = Future value of Seekie's savings
P = Annual savings amount
r = Interest rate
n = Number of periods
In this case, Seekie saves 4% of his salary each year and invests it at an interest rate of 8%. We can calculate his annual savings as 0.04 * $80,000 = $3,200.
The number of periods is the same as in part (a), which is 10 years.
Using the formula:
FV = $3,200 * ((1 + 0.08)^10 - 1) / 0.08
Calculating this, we get:
FV = $3,200 * ((1.08^10 - 1) / 0.08)
= $3,200 * (2.1589 - 1) / 0.08)
= $3,200 * (1.1589 / 0.08)
= $3,200 * 14.4863
= $46,355.36
Therefore, Seekie will have saved approximately $46,355.36 by age 60.