annuities and sinking funds

Annuities and Sinking Funds

Sinking Funds

A sinking fund is an account into which periodic deposits are made.

Sinking Funds


Usually, the deposits are made either monthly or quarterly, although the formula allows for any number of deposits, so long as they are regular.

Sinking Funds


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.

Sinking Funds

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 second deposit, made February 1st, will only earn interest for 11 months.

The third deposit will only earn interest for 10 months, etc…

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?


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.

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

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).

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?


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.


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:

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).


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.

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:


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=mt and i=r/m ).

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?

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

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.

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.

A word about calculators:

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)


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.

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.

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.

Annuities

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.

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.

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.

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.

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.


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.

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

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?

Fortunately there is a formula that we can use to answer questions like this, it is the payment formula for an annuity:

Fortunately there is a formula that we can use to answer questions like this, it is the payment formula for an annuity:

In this formula, i and n are defined as they were in a sinking fund.

For example, suppose you have $10,000 in an annuity that pays 12% interest, compounded monthly. You would like to make equal monthly withdrawals (PMT) for 5 years. How much will each withdrawal be?


Here, i=0.12/12=0.01 and n=12*5=60.


Here, i=0.12/12=0.01 and n=12*5=60.

Thus, each monthly withdrawal will be $222.44.

Notice that 222.44x60=13,346.40, which is more than the $10,000 you initially had in the annuity. The interest allows you to get an extra $3,346.40 over the course of 5 years.

Notice that 222.44x60=13,346.40, which is more than the $10,000 you initially had in the annuity. The interest allows you to get an extra $3,346.40 over the course of 5 years.

Or, looking at it another way, if you just put $10,000 in your drawer and took 60 equal amounts out of it over the course of 5 years (12 months x 5 years = 60 equal withdrawals), you would only be able to take out $166.67 each month. The earned interest makes the monthly payments larger.

In some cases, you may be interested in finding the present value of annuity, based on the monthly payments received from the annuity.

In some cases, you may be interested in finding the present value of annuity, based on the monthly payments received from the annuity.

For example, suppose you would like to make equal monthly withdrawals of $500 over the course of 10 years from an annuity that pays 6% interest, compounded monthly. What is the present value of the annuity (that is, how much money do you need to start the annuity)?

Here, we use the formula for the present value of an annuity.

Here, we use the formula for the present value of an annuity.

Plugging in our numbers, we get

Thus, we will need a present value of $45,036.73 in our annuity in order to be able to withdraw $500 per month for 10 years.

Thus, we will need a present value of $45,036.73 in our annuity in order to be able to withdraw $500 per month for 10 years.

Again, notice that if the account didn’t earn any interest, you would need a present value of 500x12x10=60000 dollars to be able to take out $500 per month for 10 years. The fact that we only need $45,036.73 is due to the fact that we earn compound interest.

A mortgage is a type of annuity. It is an annuity that a bank gets from you.


When you want to buy a house, chances are you don’t have all the money that you need.



A bank is (hopefully) willing to loan you the money that you need. The loan that they give you is an annuity.



A bank is (hopefully) willing to loan you the money that you need. The loan that they give you is an annuity.

If you look at it from the bank’s point of view, they are putting a large sum of money into an account that pays periodic interest.

The bank takes monthly withdrawals from the annuity. These monthly withdrawals are the payments that the bank receives from you, the borrower.


We can use the payment formula for an annuity to compute how much your monthly payment will be on a mortgage.


We can use the payment formula for an annuity to compute how much your monthly payment will be on a mortgage.

For example, suppose you want to borrow $200,000 with a 30 year mortgage to buy a house. If the interest rate is 5%, what will your monthly mortgage payments be?

We use the payment formula for an annuity to calculate our monthly payment: The present value is 200,000.

We use the payment formula for an annuity to calculate our monthly payment. The present value is 200,000.


Thus, the monthly mortgage payments will be $1,073.64.


Thus, the monthly mortgage payments will be $1,073.64.

The surprising thing about mortgages is how much you actually end up paying for a house that supposedly costs $200,000.

It is fairly simple to figure out how much you have actually paid the bank – you just multiply your monthly payments by the number of months that you pay.


In this case, that amounts to 1,073.64x12x30=386,510.40. So for a house that costs $200,000, you end up paying $386,510.40. And the interest rate here is only 5%!


In this case, that amounts to 1,073.64x12x30=386,510.40. So for a house that costs $200,000, you end up paying $386,510.40. And the interest rate here is only 5%!

Really there is no avoiding this, unless you can afford to buy a house without borrowing.

Another interesting annuity is a lottery annuity. When you see the jackpot billboard for a lottery, the payout that they are advertising is for a 20 year annuity.

Another interesting annuity is a lottery annuity. When you see the jackpot billboard for a lottery, the payout that they are advertising is for a 20 year annuity.

For example, if you see that the billboard says the jackpot is currently $20 million, that is the total amount that you would receive, including interest, over the course of 20 years.

A lottery winner who chooses the lump-sum option would not receive $20 million dollars, since they would be taking all of the money right now, and thus it would not include the 20 years of interest that the lottery would pay to a person who chooses the annuity option.

A lottery winner who chooses the lump-sum option would not receive $20 million dollars, since they would be taking all of the money right now, and thus it would not include the 20 years of interest that the lottery would pay to a person who chooses the annuity option.

So, if you are lucky enough to win the lottery which would you choose, the annuity option or the lump-sum option?

Before you decide, consider that if you take the annuity option, the lottery is giving you their own interest rate, which is usually fairly low (around 4% or less per year). If you think you could earn more interest than that with $20 million, then you should choose the lump-sum option.

Before you decide, consider that if you take the annuity option, the lottery is giving you their own interest rate, which is usually fairly low (around 4% or less per year). If you think you could earn more interest than that with $20 million, then you should choose the lump-sum option.

Most lottery jackpot winners choose the lump-sum, but they are probably not thinking too much about interest rates when they do so!

To summarize, we had two kinds of accounts: sinking funds and annuities.


To put these into perspective, consider what a typical person does in their life. They get a job and save a certain amount of money each month for their working careers.


To put these into perspective, consider what a typical person does in their life. They get a job and save a certain amount of money each month for their working careers.

The money they save each month is put into a sinking fund, and it earns interest as described earlier. Typically a person will do this for 25-35 years.

When a person decides to retire, ideally they have saved a large amount of money in their sinking fund.

When a person decides to retire, ideally they have saved a large amount of money in their sinking fund.

When they retire, the sinking fund changes into an annuity. The person is no longer putting money into the account every month – they have now switched to taking money out of the account each month during retirement.

Of course, this scenario is a greatly simplified version of a person’s life. Most people don’t keep the same job for life, and even if they do they generally get raises and are able to save more money as time goes by. Also, interest rates for their sinking funds and annuities will probably change over time.

Of course, this scenario is a greatly simplified version of a person’s life. Most people don’t keep the same job for life, and even if they do they generally get raises and are able to save more money as time goes by. Also, interest rates for their sinking funds and annuities will probably change over time.

However, this is a typical cycle – put money into a sinking fund during your working life, and then take it out of an annuity during retirement.

For more information on sinking funds and annuities, see the worksheet for section 2.3 at my website, http://www.math.utep.edu/faculty/cmmundy

http://www.math.utep.edu/faculty/cmmundy

For more information on sinking funds and annuities, see the worksheet for section 5.3 at my website, http://www.math.utep.edu/faculty/cmmundy

Also, read through the examples worksheet on Blackboard.

http://www.math.utep.edu/faculty/cmmundy

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