Question

1. You have decided to retire. You would like to receive equal retirement payments each year for the next 10 years. You want to receive your first retirement payment in one year. You currently have $1,000,000 of savings. You expect to receive another $1,000,000 in 5 years. You can earn 10 percent interest compounded annually on your savings. How large will your annual retirement payment be?

2. What is the Annual Percentage Rate (APR) if the Effective Annual Rate (EAR) is 10% and your money is compounded quarterly?

3. You take out a $100,000 five-year loan with annual payments. You will pay 10% interest compounded annually. How much will your annual payments be? How much total interest will you pay?

4. You want to buy a car in five years and a boat in ten years. Today, the car and boat cost $5,000 and $10,000, respectively. The price of the boat and car will grow by 3% each year. How much money do you need to put away each year (equal payments at the end of the year) for the next three years, to have enough money saved to buy the boat and car? You will receive 10% interest compounded annually.

Please help me with the setup of these problems to solve correctly.

Thanks!

Answer 1

1. Let the retirement payments received be P each year

Number of years = n = 10

Interest Rate = r = 10%

Present Value of Future Payments = PV = P(1+r)^n-1 +....+ P(1+r)² + P(1+r) + P = P[(1+r)ⁿ -1]/r = P[(1+0.10)¹⁰ -1]/0.10 = 15.937P

Present Value of amount to be received after 5 years = 1000000/(1+0.10)⁵ = $620921.323

Hence, Present Value of total Savings = 1000000 + 620921.323 = $1620921.323

HEnce, 15.937P = 1620921.323

=> P = $101708.06

2. EAR = 10%

Compounding Frequency = m = 4

EAR = (1 + APR/m)^m - 1 = (1 + 0.10/4)⁴ - 1 = 0.1038 or 10.38%

3. Let the quarterly payments be X

Loan Amount = P = $100000

Quarterly Interest Rate = r = 10/4 = 2.5%

Number of periods = N = 5*4 = 20 quarters

he sum of present value of quarterly payments must be equal to the value of the loan amount

=> X/(1+r) + X/(1+r)² +....+ X/(1+r)^N = P

=> X[1- (1+r)^-N]/r = P

=> X = rP(1+r)^N/[(1+r)^N-1]

Hence, Quarterly Payments =  rP(1+r)^N/[(1+r)^N-1]

= 0.025*100000(1+0.025)²⁰/[(1+0.025)²⁰-1]

= $6414.71

4. Let amount saved for the next 3 years be P

Interest Rate = r = 3%

Number of years = n = 3

Present Value of amount saved = P/(1+r) + P/(1+r)² + P/(1+r)³ = P[1- (1+r)^-3]/r = P[1- (1+0.03)^-3]/0.03 = 2.829P

Sum of Present Value of car and boat = 5000/(1+0.03)⁵ + 10000/(1+0.03)¹⁰ = $11753.983

Hence, present value of amount invested = Sum of Present Value of car and boat

=> 2.829P = 11753.983

=> P = $4154.82