Barbara wants to save money to meet 2 objectives:

i. She would like to be able to retire 25 years from now and have a pension for 15 years after that.She would like to get the first annual payment on the pension 26 years from now, she would like it to be $30,000 and she would like it to grow at the rate of 3% per year from then onwards (for15 years).

ii. She would also like to purchase a $25,000 car, 10 year from now.
The interest rate (cost of capital) Barbara can lend and borrow at is 6% per year. Assume she deposits the same amount S at the end of each year for 25 years, with the first deposit of annual savings being 1 year from now.

What is the amount S such that Barbara will exactly be able to meet her objectives (described in (i)
and (ii): pension plan and car)?

To calculate the amount S that Barbara needs to save annually for 25 years to meet her objectives, we need to break down each objective and calculate the present value of the future cash flows.

Objective i: Retirement and Pension
- Barbara wants to retire 25 years from now and have a pension for 15 years after retirement.
- She wants the first annual pension payment to be $30,000, and it should grow at a rate of 3% per year for 15 years.
- The interest rate (cost of capital) is 6% per year.

To calculate the present value of the pension payments, we can use the formula for the present value of an annuity:

PV = P * (1 - (1 + r)^(-n)) / r

Where:
PV = Present Value
P = Annual payment
r = Interest rate per period
n = Number of periods

PV(pension) = 30,000 * (1 - (1 + 0.03)^(-15)) / 0.03
PV(pension) ≈ 30,000 * (1 - 0.478) / 0.03
PV(pension) ≈ 30,000 * (0.522) / 0.03
PV(pension) ≈ $522,000

Objective ii: Car Purchase
- Barbara wants to purchase a $25,000 car 10 years from now.
- The interest rate (cost of capital) is 6% per year.

To calculate the present value of the car purchase, we can use the formula for the present value of a single cash flow:

PV = FV / (1 + r)^n

Where:
PV = Present Value
FV = Future Value
r = Interest rate per period
n = Number of periods

PV(car) = 25,000 / (1 + 0.06)^10
PV(car) ≈ 25,000 / (1.06)^10
PV(car) ≈ 25,000 / 1.790847
PV(car) ≈ $13,967.29

Now, we can calculate the amount S that Barbara needs to save annually for 25 years to meet both objectives. We need to consider the present value of the pension payments and the car purchase:

PV(total) = PV(pension) + PV(car)
PV(total) ≈ 522,000 + 13,967.29
PV(total) ≈ $535,967.29

Finally, we can calculate the annual savings required (amount S) using the formula for the present value of an ordinary annuity:

PV(total) = S * (1 - (1 + r)^(-n)) / r

Rearranging the formula, we get:

S = PV(total) * r / (1 - (1 + r)^(-n))

Substituting the values:

S ≈ 535,967.29 * 0.06 / (1 - (1 + 0.06)^(-25))
S ≈ 32,158.04 / (1 - 0.2625878)
S ≈ 32,158.04 / 0.7374122
S ≈ $43,637.61

Therefore, Barbara needs to save approximately $43,637.61 annually for 25 years to meet her retirement and pension objective as well as her car purchase objective.