Part I

 

Answer the following questions

 

 

1.     What is the difference between an annuity and a perpetuity? (10 points)

2.     What is the difference between an ordinary annuity and an annuity due? (10 points)

 

Part II

Calculate the value that is requested for each problem. The calculations and formulas of each exercise must be presented with the answer.

1.     José Luis has the opportunity to make an investment that requires a payment of $750 per year for the next twelve years. If the investment is made at an interest rate of 8%, what is the value of that investment today?

2.     is the present value of an investment that guarantees a payment of $22,500 per year for the next five years if the annual compound interest rate is 15%?

3.     Roberto Homar is planning to invest $25,000 annually for the next seven years, an investment that will pay you an annual compound interest of 11.4%. How much money will Roberto have at the end of seven years?

4.     Thomas is 25 years old and planning to invest $3,000 annually in an IRA that pays 9.75% annual compound interest until she retires at age 65. How much money will Cecilia have for retirement?

5.     How much would you pay today for a perpetuity of $1,000 a year if the prevailing interest rate is 5.25%?

6.     How much would you buy a preferred stock that pays $5 of annual dividend if you require a yield of 12% annual interest?

7.     Determine how much you would pay in total, interest, and principal, at the end of your 30-year mortgage at 3.99% monthly compound interest if the monthly payment is $675.00.

8.     How much would you have to pay monthly for the car you want to buy if the cost of the car is $30,000 and they fund it at 5.75% of the compound interest for six years without having to pay soon?

9.     How much money will you have paid at the end of ten years if you pay $10,000 per equipment lease at the beginning of each year and the annual compound interest rate is 4.75%?

10. From the information of the previous exercise, calculate how much you'll pay at the end of the ten years, if you pay $10,000 annually at the end of each year instead of the beginning of the year. Which option suits you best? 

Part I
(1) Perpetuity is a constant payment received forever, while an annuity is a constant payment received for a given period of time

(2) Ordinary annuity is a constant payment made at the end of the period, while annuity due is a constant payment made at the beginning of the period

Part II
(1) Value of investment today                = $5,652.06

(2) Present value of investment              = $ 75,423.497

(3) Amount accumulated in seven years        = $247,609.95

 (4) The amount at Cecilias retirement       = $1,240,676.41

(5) The amount to pay for perpetuity         = $19,047.62

(6) The value of the preferred stock         = $41.67

(7) The total payment on the mortgage        = $243,000
The total payment principal on the mortgage  = $141,557.22
The total interest on the mortgage           = $101,442.78

(8) The monthly payment for the car          = $493.65

(9) The total amount paid on annuity due     = $130,226.15

(10) The total amount paid on ordinary annuity = $124,320.91
The best option is to make the lease payments at the end of each year

Step-by-step explanation

Part I
(1) The difference between an annuity and a perpetuity
Perpetuity is a constant payment received forever, while an annuity is a constant payment received for a given period of time (Cheng, 2020). 

(2) The difference between an ordinary annuity and an annuity due
Ordinary annuity is a constant payment made at the end of the period, while annuity due is a constant payment made at the beginning of the period (Heyford, 2010)

Part II
(1) Calculate the value of investment today
Value investment today = Payment per year * (1 -(1 + r)^-n)/r
                       = $750 * (1 -(1 + 8%)^-12)/8%
                       = $5,652.06

Value of investment today is $5,652.06

(2) Calculate the present value of investment
PV  of investment = Payment per year * (1 -(1 + r)^-n)/r
                  = $22,500 * (1 -(1 + 15%)^-5)/15%
                  = $ 75,423.49

Present value of investment is $ 75,423.497

(3) Calculate the amount accumulated in seven years

Amount accumulated = Annual investment * ((1 + r)^n - 1)/r
                   = $25,000 * ((1 + 11.4%)^7 - 1)/11.4%
                   = $247,609.95

Amount accumulated in seven years is $247,609.95

 (4) Calculate the amount at Cecilias retirement

Amount at retirement = Annual investment * ((1 + r)^n - 1)/r
                     = $3,000 * ((1 + 9.75%)^40 - 1)/9.75%
                     = $1,240,676.41

The amount at Cecilias retirement is $1,240,676.41

(5) Calculate the amount to pay for perpetuity
Amount to pay = Annual income / Interest rate
              = $1,000 / 5.25%
              = $19,047.62

The amount to pay for perpetuity is $19,047.62

(6) Calculate the value of the preferred stock

Preferred stock value = Annual dividend / Interest rate
                      = $5 / 12%
                      = $41.67

The value of the preferred stock is $41.67

(7) Calculate the total payment on the mortgage

Monthly rate     r = 0.33% » 3.99% / 12
No of payments   n = 360 » 30 * 12

Total payment = Monthly payment * No of payments
              = $675.00 * 360
              = $243,000

The total payment on the mortgage is $243,000

Calculate the principal on the mortgage

Principal = Monthly payment * (1 -(1 + r)^-n)/r
          = $675.00 * (1 -(1 + 0.33%)^-360)/0.33%
          = $141,557.22

The total payment principal on the mortgage is $141,557.22

Calculate the total interest on the mortgage

Total interest = Total payment - Principal
               = $243,000 - $141,557.22
               = $101,442.78

The total interest on the mortgage is $101,442.78

(8) Calculate the monthly payment for the car

Monthly rate r = 0.48% » 5.75% / 12
Period n = 72 » 6 * 12

Monthly payment = Purchase cost / ((1 - (1 + r)^-n)/r)
                = $30,000 / ((1 - (1 + 0.48%)^-72)/0.48%)
                = $493.65

The monthly payment for the car is $493.65

(9) Calculate the total amount paid on annuity due 

Total amount paid = Annual payment * ((1 + r)^n - 1)/ r * (1 + r)
                  = $10,000 * ((1 + 4.75%)^10 - 1)/4.75% * (1 + 4.75%)
                  = $130,226.15

The total amount paid on annuity due is $130,226.15

(10) Calculate the total amount paid on ordinary annuity

Total amount paid = Annual payment * ((1 + r)^n - 1)/ r
                  = $10,000 * ((1 + 4.75%)^10 - 1)/4.75%
                  = $124,320.91

The total amount paid on ordinary annuity is $124,320.91

The best option is to make the lease payments at the end of each year

Referrences 
Cheng, J. (2020). Annuity Derivation Vs. Perpetuity Derivation: What is the Difference? Ivestopedia.

Heyford, S. C. (2010). Calculating Present and Future Value of Annuities. Investopedia.