Unit 7 Seminar: Annuities Prof. Otis D. Jackson

 Find the future value of:  an ordinary annuity using the simple interest formula  an ordinary annuity using a $1.00 ordinary annuity future value table  an annuity due using the simple interest formula  an annuity due using a $1.00 ordinary annuity future value table  an annuity using a formula 2

 Calculate the value of a growing account subject to periodic investments of payments.  Some examples include:  Retirement funds  College education  Vacation  Company’s future investment in capital expenses 3

 Annuity payment: a payment made to an investment fund each period at a fixed interest rate.  Sinking fund payment: a payment made to an investment fund each period at a fixed interest rate to yield a predetermined future value.  Annuity certain: an annuity paid over a guaranteed number of periods. 4

 Contingent annuity: an annuity paid over an uncertain number of periods.  Ordinary annuity: an annuity for which payments are made at the end of each period.  Annuity due: an annuity for which payments are made at the beginning of each period. 5

1. Find the end-of-period principal. First end-of-period principal = annuity payment 2. For each remaining period in turn: End-of-period principal = annuity payment + [previous end-of-period principal * (1 + period interest rate)]. 3. Identify the last end-of-period principal as the future value. Future value = last end-of-period principal 6

What is the FV of an annual ordinary annuity of $1,000 for 3 years at 4% annual interest? 7  End-of-year 1 = $1,000 (no interest earned Y1)  End-of-year 2 = $1,000 + [$1,000 ( )] = $2,040  End of year 3 = $1,000 + [$2,040 (1.04)] = $3,  The future value is $3,

Find the future value and total interest of an ordinary annuity with annual payments of $5,000 at 2.9% annual interest after four years. Remember our formulas: First end-of-period principal = annuity payment Next end-of-period principal = Annuity payment + [previous end-of-period principal * (1 + period interest rate)] Future value = last end-of-period principal 8

 End-of-year 1 = $5,000 (no interest earned Y1)  End-of-year 2 = $5,000 + [$5,000 (1.029)] = $10,145  End of year 3 = $5,000 + [$10,145 (1.029)] = $15,  End of year 4 = $5,000 + [$15, (1.029)] = $20, = Future Value 9

 4 years depositing $5,000 per year  4 * $5,000 = $20,000  Future Value – principal = Interest paid  $20, $20,000 = $  Interest = $

 Find the value of an annuity after two years of $1,500 invested semi-annually at 4% annual interest. Then find the interest paid. 11

 We have two periods per year (semi annual) and 2 years of interest 2 yrs * 2 periods/year = 4 periods  Interest is 4% annually. For each period then it would be 2%. 4÷2 = 2% interest per period 12

 End-of-period 1 = $1,500 (no interest earned P1)  End-of-Period 2 = $1,500 + [$1,500 (1.02)] = $3,030  End of Period 3 = $1,500 + [$3,030 (1.02)] = $4,  End of Period 4 = $1,500 + [$4, (1.02)] = $6, = Future Value 13

 Future value = $6,  Invested amount = $1,500 * 4 periods = $6,000  Interest = $6, $6,000 = $

 Using Table 1 from the lecture notes 1. Select the periods row corresponding to the number of interest periods. 2. Select the rate per period column corresponding to the period interest rate. 3. Locate the value in the cell where the periods row intersects with the rate-per-period column. 4. Multiply the annuity payment by the table from step 3. 15

Using Table-1, find the FV of a semiannual ordinary annuity of $6,000 for five years at 6% annual interest, compounded semiannually.  5 years x 2 periods per year = 10 periods  6% annual interest rate / 2 periods per year = 3% period interest rate  See Table-1 for 10 periods at 3% =  FV = $6,000 x = $68,784  The future value of this annuity is $68,

Use Table-1 to find the accumulation phase future value and total interest of an ordinary annuity of $4,000 for eight years at 2% annual interest. 17

First pull out all information needed: $4,000 for 8 2% annually Gives us 8 periods at 2% Go to Table-1: down to 8 periods and over to 2% gives us the factor: $4,000 * = $34,332 (future value) Future Value – Principal = Interest $34,332 – ($4,000 * 8) = Interest $34,332 - $32,000 = $2,332 = Interest 18

Use Table-1 to find the accumulated amount and total interest of an ordinary annuity with semi-annual payments of $6,000 for five years at 4% annual interest. 19

Pull out all important information: $6,000 paid semi-annually for 5 4% annual interest Find each period interest: 4÷2 = 2% Number of periods: 5 years * 2 periods per year = 10 periods Going to Table-1: 10 periods at 2%: gives us a factor of $6,000 x = $65,700 = future value To find the interest: 10 periods x $6,000 per = $60,000 = Principal FV – Principal = $65,700 - $60,000 = $5,700 = Interest 20

 The difference between an ordinary annuity and an annuity due is whether you made the first payment immediately (annuity due) or at the end of the first period (ordinary annuity).  With the same numbers, the FV of an annuity due will always be greater than the FV of an ordinary annuity. This is because you gain interest immediately rather than waiting until the end of the first period. 21

1. Find the first end-of-month period principal: multiply the annuity payment by the sum of 1 and the period interest rate. Because you are paying the principal at the beginning of the month, you earn interest for that month. 2.For each remaining period in turn, find the next end- of-period principal: (previous end of period principal + annuity payment) * (1 + period interest rate) 3.Identify the last end-of-period principal as the future value. 22

Find the FV of annuity due of $1,000 for three years at 4% annual interest. Find the total investment and total interest earned.  End-of-Y 1 value = $1,000 * ( ) = $1,040.  End-of-Y 2 value = ($1,000 + $1,040) * 1.04 = $2,  End-of-Y 3 value = $3, * 1.04 = $3,  The future value of this annuity is $3,  The interest earned = $3, – $3,000 = $

1. Select the periods row corresponding to the number of interest periods. 2. Select the rate-per-period column corresponding to the period interest rate. 3. Locate the value in the cell where the periods row intersects the rate-per-period column. 4. Multiply the annuity payment by the table value from step 3. This is equivalent to an ordinary annuity. 5. Multiply the amount that is equivalent to an ordinary annuity by the sum of 1 and the period interest rate to adjust for the extra interest that is earned on an annuity due. 24 Future value = annuity payment * table value * (1 + period interest rate) Note: Same table as before but a different final calculation!

Using Table-1, find the FV of a quarterly annuity due of $2,800 for four years at 8% annual interest, compounded quarterly.  4 years x 4 periods per year = 16 periods  8% annual interest rate ÷ 4 periods per year = 2%  Table-1 value for 16 periods at 2% =  FV = Annuity pmt * table value * (1 + period interest rate)  FV = $2,800 * * 1.02 = $52, The future value of this annuity is $52,

26  R is the period rate expressed as a decimal equivalent.  N is the number of periods.  PMT is the amount of the annuity payment.

27 Find the future value of an ordinary annuity with annual payments of $5,000 at 2.9% annual interest after four years. PMT = $5,000 R = 2.9% annual interest = N = 4 years FV = $5000 *[(( [( ) ^ 4] - 1) ÷ )] FV = $5000 *[(( [(1.029) ^ 4] - 1) ÷ )] FV = $5000 *[( – 1) ÷ 0.029] FV = $5000 *[ ÷ 0.029] FV = $5000 *(4.177) = $20,886.94

28 * (1 + R)  R is the period rate expressed as a decimal equivalent.  N is the number of periods.  PMT is the amount of the annuity payment.

29 Find the future value of an annuity due with annual payments of $5,000 at 2.9% annual interest after four years. PMT = $5,000 R = N = 4 years FV = $20, * ( ) FV = $20, * (1.029) FV = $21, * (1 + R)

 Find the sinking fund payment using a $1.00 sinking fund payment table.  Find the present value of an ordinary annuity using a $1.00 ordinary annuity present value table.  Find the sinking fund payment or the present value of an annuity using a formula. Basically just asking how much do you need to invest each year to have a desired amount in the future. 30

1. Select the periods row corresponding to the number of interest periods. 2. Select the rate-per-period column corresponding to the period interest rate. 3. Locate the value in the cell where the periods row intersects the rate-per-period column. 4. Multiply the table value from step 3 by the desired future value Sinking fund payment = FV * Table-2 value The table value will be less than 1 because you are wondering what portion of the value you need today. 31

Using Table-2, find the annual sinking fund payment (SFP) required to accumulate $140,000 in 12 years at 6% annual interest rate.  Table-2 indicates that a 12-period value at 6% is equal to  SFP = $140,000 x = $8,302  A sinking fund payment of $8,302 is required at the end of each year for 12 years at 6% to yield the desired $140,

Use Table-2 to find the annual sinking fund payment required to accumulate $100,000 in 10 years at 4% annual interest. 33

Use Table-2 to find the annual sinking fund payment required to accumulate $100,000 in 10 years at 4% annual interest.  Number of periods: 10  Interest rate: 4%  Find the table value where 10 periods and 4% intersect:  Multiply the desired FV by the table value  SFP = $100,000 x = $8,330  The annual sinking fund payment required to accumulate $100,000 in 10 years is $8,330 34

 Using Table-3 in your lecture notes, locate the given number of periods and the given rate per period.  Multiply the table value times the periodic annuity payment. PV of annuity = periodic annuity payment * table value Think of retirement or trusts. How much do you set aside now to receive payments each year? 35

Use Table-3 to find the present value of a semiannual ordinary annuity of $3,000 for seven years at 6% annual interest, compounded semiannually.  7 years * 2 periods per year = 14 periods  6% annual interest rate ÷ 2 periods per year = 3% period interest rate  Table-3: row 14 periods, column 3% is  PV annuity = $3,000 x (table factor)= $33,900  By investing $33,900 now at 6% interest, compounded semiannually, you can receive an annuity payment of $3,000 twice a year for seven years. 36

37  Reminder of what to complete for Unit 7 by Tuesday at Midnight:  Discussion = initial response to one question + 2 reply posts  MML assignment  Instructor graded assignment (download from doc sharing)  Seminar quiz if you did not attend, came late, or left early