8. Consumer Mathematics. The Mathematics of Everyday Life. Annuities. 8.4. Calculate the future value of an ordinary annuity. Perform calculations regarding sinking funds. Annuities. An annuity is an interest-bearing account into which we make a series of payments of the same size.
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8 Consumer Mathematics The Mathematics of Everyday Life
Annuities 8.4 • Calculate the future value of an ordinary annuity. • Perform calculations regarding sinking funds.
Annuities An annuityis an interest-bearing account into which we make a series of payments of the same size. If one payment is made at the end of every compounding period, the annuity is called an ordinary annuity. The future value of an annuityis the amount in the account, including interest, after making all payments.
Annuities Suppose that in January you begin making payments of $100 at the end of each month into an account paying 12% yearly interest compounded monthly. (continued on next slide)
Annuities If we compute how much each deposit contributes to the account and sum these amounts, we will have the value of the annuity on July 1. We may express the value of this annuity as Factoring out 100, we can write this in the form
Annuities • Example: • Show that • Solution:Multiply by . This shows that
Annuities From the previous example, we have Returning to our earlier example, we have and multiplying by $100, we obtain
Annuities • Example: A payment of $50 is made at the end of each month into an account paying a 6% annual interest rate, compounded monthly. How much will be in that account after 3 years? • Solution: • We see that R= 50, = and n= (continued on next slide)
Annuities Using the formula for finding the future value of an ordinary annuity, we get
Sinking Funds You may want to save regularly to have a fixed amount available in the future. The account that you establish for your deposits is called a sinking fund. Because a sinking fund is a special type of annuity, it is not necessary to find a new formula. We can use the formula for calculating the future value of an ordinary annuity that we have stated earlier. In this case, we will know the value of Aand we will want to find R.
Sinking Funds • Example: Assume that you wish to save $1,800 in a sinking fund in 2 years. The account pays 6% compounded quarterly and you will also make payments quarterly. What should be your • monthly payment? • Solution:Recall the formula for finding the future value of an ordinary annuity: (continued on next slide)
Sinking Funds We see that A= 1,800, = and n=
Sinking Funds • Example: Suppose you have decided to retire as soon as you have saved $1,000,000. Your plan is to put $200 each month into an ordinary annuity that pays an annual interest rate of 8%. In how many years will you be able to retire? • Solution: • We see that A= 1,000,000, = and R= 200. (continued on next slide)
Sinking Funds We solve this equation for n.