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Annuities and Sinking Funds

Annuities and Sinking Funds. Sinking Funds. Sinking Funds. A sinking fund is an account into which periodic deposits are made. Sinking Funds. A sinking fund is an account into which periodic deposits are made.

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Annuities and Sinking Funds

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  1. Annuities and Sinking Funds

  2. Sinking Funds

  3. Sinking Funds • A sinking fund is an account into which periodic deposits are made.

  4. Sinking Funds • A sinking fund is an account into which periodic deposits are made. • Usually, the deposits are made either monthly or quarterly, although the formula allows for any number of deposits, so long as they are regular.

  5. Sinking Funds • A sinking fund is an account into which periodic deposits are made. • Usually, the deposits are made either monthly or quarterly, although the formula allows for any number of deposits, so long as they are regular. • Deposits made into sinking funds earn compound interest, and for this course we assume the interest is compounded at the same frequency that the deposits are made.

  6. With a sinking fund, each deposit will earn interest for a different length of time.

  7. With a sinking fund, each deposit will earn interest for a different length of time. • For example, suppose you deposit $100 per month for one year into an account that earns 10% interest, compounded monthly.

  8. With a sinking fund, each deposit will earn interest for a different length of time. • For example, suppose you deposit $100 per month for one year into an account that earns 10% interest, compounded monthly. • The first deposit (suppose it’s made January 1st) will earn interest for the full year.

  9. With a sinking fund, each deposit will earn interest for a different length of time. • For example, suppose you deposit $100 per month for one year into an account that earns 10% interest, compounded monthly. • The first deposit (suppose it’s made January 1st) will earn interest for the full year. • The second deposit, made February 1st, will only earn interest for 11 months.

  10. With a sinking fund, each deposit will earn interest for a different length of time. • For example, suppose you deposit $100 per month for one year into an account that earns 10% interest, compounded monthly. • The first deposit (suppose it’s made January 1st) will earn interest for the full year. • The second deposit, made February 1st, will only earn interest for 11 months. • The third deposit will only earn interest for 10 months, etc…

  11. The last deposit you make, on December first, will only earn interest for one month.

  12. The last deposit you make, on December first, will only earn interest for one month. • The question is, how much money will be in the account at the end of the year?

  13. The last deposit you make, on December first, will only earn interest for one month. • The question is, how much money will be in the account at the end of the year? • Fortunately, there is a formula that will answer that question for us.

  14. Suppose a sinking fund account has an annual interest rate of r compounded m times per year, so that i=r/m is the interest rate per compounding period. If you make a payment of PMT at the end of each period, then the future value after t years, or n=mt periods, will be

  15. Suppose a sinking fund account has an annual interest rate of r compounded m times per year, so that i=r/m is the interest rate per compounding period. If you make a payment of PMT at the end of each period, then the future value after t years, or n=mt periods, will be • Note that the formula we use in this class is for end of period deposits (so for example, monthly deposits would be made on the last day of the month).

  16. As an example, suppose we deposit $100 per month (at the end of each month) into an account paying 10% interest, compounded monthly, for 2 years. How much would we have at the end of the 2 years?

  17. As an example, suppose we deposit $100 per month (at the end of each month) into an account paying 12% interest, compounded monthly, for 2 years. How much would we have at the end of the 2 years? • In this case, r=0.12, m =12, and t =2. This means that i=r/m=0.12/12=0.01, and n=12*2=24.

  18. As an example, suppose we deposit $100 per month (at the end of each month) into an account paying 12% interest, compounded monthly, for 2 years. How much would we have at the end of the 2 years? • In this case, r=0.12, m =12, and t =2. This means that i=r/m=0.12/12=0.01, and n=12*2=24. • Plugging these values into the formula gives:

  19. As an example, suppose we deposit $100 per month (at the end of each month) into an account paying 12% interest, compounded monthly, for 2 years. How much would we have at the end of the 2 years? • In this case, r=0.12, m =12, and t =2. This means that i=r/m=0.12/12=0.01, and n=12*2=24. • Plugging these values into the formula gives:

  20. As an example, suppose we deposit $100 per month (at the end of each month) into an account paying 12% interest, compounded monthly, for 2 years. How much would we have at the end of the 2 years? • In this case, r=0.12, m =12, and t =2. This means that i=r/m=0.12/12=0.01, and n=12*2=24. • Plugging these values into the formula gives:

  21. Thus, at the end of two years, the account will have $2,697.35.

  22. Thus, at the end of two years, the account will have $2,697.35. • Compare this to how much you would have if you just put $100 per month into a drawer every month for two years ($100*24=$2,400).

  23. Thus, at the end of two years, the account will have $2,697.35. • Compare this to how much you would have if you just put $100 per month into a drawer every month for two years ($100*24=$2,400). • The difference, $297.35, is the earned interest. Remember that each deposit earns interest for a different amount of time, but the formula takes this into account and gives the cumulative amount.

  24. Sometimes, we may want to know how much we need to deposit periodically in order to have a certain amount of money in the future.

  25. Sometimes, we may want to know how much we need to deposit periodically in order to have a certain amount of money in the future. • For this, we can rearrange the sinking fund formula slightly to get:

  26. Sometimes, we may want to know how much we need to deposit periodically in order to have a certain amount of money in the future. • For this, we can rearrange the sinking fund formula slightly to get: • This formula will tell us how much a periodic payment needs to be to have a future value of FV in t years (where n=mtand i=r/m ).

  27. For example, suppose we want to have $2,000 at the end of two years. We find an account (a sinking fund) that will pay 6% interest, compounded monthly. How much do we need to deposit into the account each month in order to have our $2,000 in two years?

  28. We can use the payment formula for a sinking fund to answer this question:

  29. We can use the payment formula for a sinking fund to answer this question:

  30. We can use the payment formula for a sinking fund to answer this question:

  31. We can use the payment formula for a sinking fund to answer this question:

  32. We can use the payment formula for a sinking fund to answer this question: • Thus, we must deposit $78.64 every month in order to have $2000 at the end of 2 years.

  33. Again, notice what happens without compounding. If we were simply to put $78.64 into our drawer every month for two years, we would have only 78.64x24=1887.36 dollars. The extra $112.64 comes from the total accumulated interest of all of the monthly deposits, taking into account that each deposit will earn less interest than earlier deposits since they don’t earn interest for as much time.

  34. A word about calculators:

  35. A word about calculators: Obviously you have to use a calculator to use these formulas. When you enter the previous example into a TI-83 (or any TI calculator), it must be entered as follows, if you want to enter the whole thing at once:

  36. A word about calculators: Obviously you have to use a calculator to use these formulas. When you enter the previous example into a TI-83 (or any TI calculator), it must be entered as follows, if you want to enter the whole thing at once: • 2000*(0.06/12)/((1+0.06/12)^(12*2)-1)

  37. A word about calculators: Obviously you have to use a calculator to use these formulas. When you enter the previous example into a TI-83 (or any TI calculator), it must be entered as follows, if you want to enter the whole thing at once: • 2000*(0.06/12)/((1+0.06/12)^(12*2)-1) • The parentheses must be exactly where you see them here. This is to make sure that the order of operations is satisfied.

  38. Annuities

  39. Annuities • An annuity is an account which pays compound interest, from which periodic withdrawals are made. In this course, we only deal with annuities in which the withdrawals are made with the same frequency as the compounding period.

  40. Annuities • An annuity is an account which pays compound interest, from which periodic withdrawals are made. In this course, we only deal with annuities in which the withdrawals are made with the same frequency as the compounding period. • Some common annuities are mortgages, retirement funds, and lottery winnings.

  41. Consider the difference between a sinking fund and an annuity. A sinking fund is an account which you put money into, and an annuity is an account which you take money out of.

  42. Consider the difference between a sinking fund and an annuity. A sinking fund is an account which you put money into, and an annuity is an account which you take money out of. • For an annuity, you must have a relatively large sum of money if you want to be able to take monthly withdrawals of any worthwhile amount.

  43. Consider the difference between a sinking fund and an annuity. A sinking fund is an account which you put money into, and an annuity is an account which you take money out of. • For an annuity, you must have a relatively large sum of money if you want to be able to take monthly withdrawals of any worthwhile amount.

  44. The usual way people will accumulate enough money to have an annuity is by saving for retirement. Ideally, you save money for several years and then when you retire, you would like the money that you’ve saved to earn interest, even as you take out monthly (or other periodic) withdrawals.

  45. The usual way people will accumulate enough money to have an annuity is by saving for retirement. Ideally, you save money for several years and then when you retire, you would like the money that you’ve saved to earn interest, even as you take out monthly (or other periodic) withdrawals. • For a few very lucky people, the money for an annuity can come from winning a lottery or other large gambling win.

  46. Regardless of how you get the money to start an annuity, they all work in essentially the same way.

  47. Regardless of how you get the money to start an annuity, they all work in essentially the same way. • You start with some amount of money, and you make periodic withdrawals of equal amounts of money.

  48. Regardless of how you get the money to start an annuity, they all work in essentially the same way. • You start with some amount of money, and you make periodic withdrawals of equal amounts of money. • When you start the annuity, the entire initial amount of money is earning interest. When you take your first withdrawal, the money that you withdraw is no longer earning interest.

  49. Each withdrawal that you take no longer earns interest, but the money that remains in the account continues to earn interest.

  50. Each withdrawal that you take no longer earns interest, but the money that remains in the account continues to earn interest. • A question you will need to answer is: Given a starting amount (a present value) of an annuity, at a given interest rate, how much can you withdraw each month from the annuity, so that there is nothing left after some amount of time has passed?

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