Annuities

Let's get the definitions out of the way first.

Annuity: A form of investment that involves regular contributions.

Future Value: The future value of an annuity is equal to the sum of the regular contributions to the annuity plus the compound interest earned on these contributions.
Note that the money deposited per period is deposited at the END of the period and consequently does not earn interest for that period.

Present Value: The present value of an annuity is the sum of money that could be deposited NOW to be equal to the same amount as the future value at the end of the investment period.

We will be using the compound interest formula for our calculations in this section:
A = P(1 + r)n where
A = amount i.e. principal + interest at the end of the investment period.
P = principal i.e. amount invested.
r = rate of interest as a decimal i.e. percent rate divided by 100.
n = number of interest periods.

Example 1:
Calculate the future value of an annuity whereby $1000 is contributed annually for 5 years at an interest rate of 6% per annum.

Answer
Year 1: $1000 is deposited at the end of the year and so earns 6% compound interest for 4 (not 5) years.
A = P(1 + r)n
A = 1000(1.06)4 = $1262.48
Year 2: $1000 is deposited at the end of the year and so earns 6% compound interest for 3 years.
A = P(1 + r)n
A = 1000(1.06)3 = $1191.02
Year 3: $1000 is deposited at the end of the year and so earns 6% compound interest for 2 years.
A = P(1 + r)n
A = 1000(1.06)2 = $1123.60
Year 4: $1000 is deposited at the end of the year and so earns 6% compound interest for 1 year.
A = P(1 + r)1
A = 1000(1.06)4 = $1060.00
Year 5: $1000 is deposited at the end of the year just before the annuity matures and so earns no interest.
A = $1000.00

Future Value = $1262.48 + $1191.02 + $1123.60 + $1060.00 + $1000 = $5637.10

Example 2:
The future value of an annuity is $5637.10. If the interest rate is 6% per annum, and the term of the annuity is 5 years, what is the present value of the annuity?

Answer
The present value of an annuity is the sum of money that could be deposited NOW to be equal to the same amount as the future value at the end of the investment period.
A = P(1 + r)n
5637.10 = P(1.06)5
P = 5637.10/(1.06)5
P = 4212.37
The present value is $4212.37

Example 3:
Izabella borrowed $4212.37 to buy a computer. The loan was to be repaid with annual repayments of $1000.00 at a reducible interest rate of 6% per annum. Complete the following table to show that it takes 5 years to pay off the loan.

YearAmount owing at
beginning of year.($)
   Interest($)   Amount owing + Interest($)Amount owing + Interest - Repayment
i.e. Amount owing at end of year($)
Year 1    
Year 2    
Year 3    
Year 4    
Year 5    

Answer.
YearAmount owing at
beginning of year.($)
   Interest($)   Amount owing + Interest($)Amount owing + Interest - Repayment
i.e. Amount owing at end of year($)
Year 14212.37252.744465.113465.11
Year 23465.11207.913673.022673.02
Year 32673.02160.382833.401833.40
Year 41833.40110.001943.40943.40
Year 5943.4056.601000.000.00

By dividing all quantities by 4212.37 the table above can be adjusted to show values per dollar borrowed at 6% p.a. over 5 years.
YearAmount owing at
beginning of year.($)
   Interest($)   Amount owing + Interest($)Amount owing + Interest - Repayment
i.e. Amount owing at end of year($)
Year 11.000000.060001.060000.82260
Year 20.822600.0493570.871960.063456
Year 30.634560.0380740.672640.43524
Year 40.435240.0261140.461360.22182
Year 50.221820.0134370.237400.00000

Exercise:
David borrowed $20000 to buy a car. The loan was to be repaid in 5 annual instalments at 6%p.a. reducible interest.
(i)Use the table above to calculate the amount of David's instalments.
(ii) What is the total amount that David paid for the car?
(iii) How much interest did David pay?

Answer:
(i)Repayment per dollar borrowed = 1.06000 - 0.82260 = $0.2374
For $20000 the annual repayments are 20000 x 0.2374 = $4748.00

(ii) Total amount = $4748 x 5 = $ 23740

(iii) Interest = $23740 - $20000 = $3740

Exercises

(a)Calculate the future value of an annuities whereby $1 is contributed annually for 5 years at an interest rate of (i) 10% per annum.
(ii) 5% per annum
(iii) 3% per annum

(b) Calculate the present value of the future values found in part (a).

Now to make up some tables.

Use the compound interest formula to calculate the future value of $1.00 invested annually in an annuity paying 3% per annum compound interest for
(i) 1year
(ii) 2 years
(iii) 3 years
(iv) 4 years
(v) 5 years
Answers
Note that the money deposited per period is deposited at the END of the period and consequently does not earn interest for that period.
1 year: A = P(1 + r)n = 1(1.03)0 = 1.00
2 years: 1.00 + 1(1.03)1 1 + 1.03 = 2.03
3 years: 2.03 + 1(1.03)2 = 2.03 + 1.0609 = 3.0909
4 years: 3.0909 + 1(1.03)3 = 3.0909 + 1.09273 = 4.18363
5 years: 4.18363 + 1(1.03)4 = 4.18363 + 1.12551 = 5.30914

Future Value Table

Future Value of $1
Period3%5%10%
11.00000                                             
22.03000              
33.09090              
44.18363              
55.30914              
Exercise
Complete the above table for interest rates of 5% and 10%.

Completed Table

Future Value of $1
Period3%5%10%
11.000001.000001.00000
22.030002.050002.10000
33.090903.152503.31000
44.183634.310134.64100
55.309145.525636.10510
Now to calculate present values:
The present value of an annuity is the sum of money that could be deposited NOW to be equal to the same amount as the future value at the end of the investment period.

Use the formula: PV = FV/(1+r)n to calculate the present value of the future values shown in the 3% column of the future values table above.

Period 1 = 1.00000/1.031 = 0.97087
Period 2 = 2.03000/1.032 = 1.91347
Period 3 = 3.09090/1.033 = 2.82861
Period 4 = 4.18363/1.034 = 3.71710
Period 5 = 5.30914/1.035 = 4.57971

Putting these values into a table gives the pollowing:

Present Value Table

Present Value of $1
Period3%5%10%
10.97087                                             
21.91347              
32.82861              
43.71710              
54.57971              
Exercise
Complete the above table for interest rates of 5% and 10%.

Completed Table

Present Value of $1
Period3%5%10%
10.970870.952380.90909
21.913471.859411.73554
32.828612.723252.48685
43.717103.545953.16987
54.579714.329483.79079
There are other formulae that can be used to calculate future and present values but they are beyond the scope of this course.

Present value tables can be used to calculate loan repayments. All you have to do is to divide the amount of the loan by the present value for the corresponding period and interest rate.
Example:
David borrowed $5000 to pay for a holiday. The loan was to be repaid in 4 annual instalments at 10% reducible interest. What was the amount of David's annual repayments?

Complete the table below to show that it takes 4 years to pay off the loan.
YearAmount owing at
beginning of year.($)
   Interest($)   Amount owing + Interest($)Amount owing + Interest - Repayment
i.e. Amount owing at end of year($)
Year 15000   
Year 2    
Year 3    
Year 4    

Answer:
From the present value table above the intersection of the period 4 with the interest rate of 10% is 3.16987.
$5000/3.16987 = $1577.35
YearAmount owing at
beginning of year.($)
   Interest($)   Amount owing + Interest($)Amount owing + Interest - Repayment
i.e. Amount owing at end of year($)
Year 1500050055003922.65
Year 23922.65392.274314.922737.57
Year 32737.57273.763011.321433.97
Year 41433.97143.401577.370.02
There is a slight difference due to rounding off.

Q.1.
Chris borrowed $4000 to pay for a TV. The loan was to be repaid in 2 annual instalments at 10% reducible interest. What was the amount of Chris's annual repayments?

Complete the table below to show that it takes 2 years to pay off the loan.
YearAmount owing at
beginning of year.($)
   Interest($)   Amount owing + Interest($)Amount owing + Interest - Repayment
i.e. Amount owing at end of year($)
Year 14000   
Year 2    

Q.2. Jordan borrowed $15000 to pay for a caravan. The loan was to be repaid in 5 annual instalments at 3% reducible interest. What was the amount of Jordan's annual repayments?

Complete the table below to show that it takes 5 years to pay off the loan.
YearAmount owing at
beginning of year.($)
   Interest($)   Amount owing + Interest($)Amount owing + Interest - Repayment
i.e. Amount owing at end of year($)
Year 115000   
Year 2    
Year 3    
Year 4    
Year 5    

Q.3. Elizabeth borrowed $8000 to pay for a cruise. The loan was to be repaid in 3 annual instalments at 5% reducible interest. What was the amount of Elizabeth's annual repayments?

Complete the table below to show that it takes 3 years to pay off the loan.
YearAmount owing at
beginning of year.($)
   Interest($)   Amount owing + Interest($)Amount owing + Interest - Repayment
i.e. Amount owing at end of year($)
Year 18000   
Year 2    
Year 3    

Q.4.
Izzy borrowed $10000 to buy a car.. The loan was to be repaid in 5 annual instalments at 5% reducible interest. What was the amount of Izzy's annual repayments?

Complete the table below to show that it takes 5 years to pay off the loan.
YearAmount owing at
beginning of year.($)
   Interest($)   Amount owing + Interest($)Amount owing + Interest - Repayment
i.e. Amount owing at end of year($)
Year 110000   
Year 2    
Year 3    
Year 4    
Year 5    

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