Present Value of an Annuity with Annual Payments

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Dr. Craig RuffDepartment of FinanceJ. Mack Robinson College of BusinessGeorgia State University

© 2014 Craig Ruff

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The present value of an annuity is asking what is the value of this annuity at the start of the first period. As the future value of an annuity is the sum of a set of future values, the present value of an annuity is simply the sum of a set of present values.

Example: Using the same set of cash flows as the example of a future value of an annuity with annual payments, suppose now youwant to determine the present value of a $100 annuity/year that lasts for 3 years. The rate is again 10%, compounded annually. Again. on a time-line, the cash flows would look something like this, with the first payment being made at the end of the first year:

Present Value of an Annuity with Annual Payments

0 1

100

2

100

3

100

1st payment made at end of 1st year

2nd payment made at end of 2nd year

3rd payment made at end of 3rd year

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First calculate the present value (at t=0) of the payment made at the end of the first period (t=1)…$90.90

Next, find the present value (at t=0) of the payment made at the end of the second period (t=2)…$82.64.

Next, find the present value (at t=0) of the payment made at the end of the third period (t=3)…$75.13.

Finally, sum the three pieces to arrive at the PV of the annuity of $248.68 (90.90+82.64+75.13).

0 1

100

2

100

3

100

100

(1.1)190.90

100

(1.1)282.64

100

(1.1)375.13

© 2014 Craig Ruff

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Like the FV of an annuities, there are comparable summation and computational formulas:

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Buttons Numbers to Enter  PV ???? 248.68

FV 0

I 10  N 3  PMT -100  

The only thing tricky here is to remember that the FV=0. Given that you have told your calculator that there are three payments of $100 (N=3 and PMT=-100), then to put a value in for the FV would tell your calculator that there is some cash flow that is not really there.

Working with the same example: What is the present value of a $100 annuity/year that lasts for 3 years. The rate is again 10%,compounded annually.

On the calculator, you would solve for this as

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Buttons Numbers to Enter  PV ???? 256.198

FV -100

I 10  N 2  PMT -100  

This would lead you to the wrong answer.

The N=2 and PMT=-100 tells your calculator that there are two payments of$100, one at t=1 and another at t=2.

What about FV=-100? This tells your calculator that there is a third payment.Note, though, that the entry for N is also telling the calculator where in timethe FV amount (if there is one) is showing up. So, with N=2,we are telling thecalculator that the FV amount of $100 is showing up at t=2, which is one yeartoo early for the third payment. Thus, we end up with the incorrect answer of256.198, when the correct answer is 248.68.

Using the same example, what if you tell the calculator that there are two cash flows of $100 and the FV=$100?

© 2014 Craig Ruff

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Quick Comment:

Often we refer to the rate in future value examples as the ‘compounding rate’ and the rate in present value examples as the‘discounting rate.’ Please don’t get caught up on this terminology; there is typically only one rate in a problem and you may use thatrate to move money forward (future value) or move money backward (present value).

For instance, in the example of the 3 year, $100/year annuity. The rate was 10% in both the PV of annuity and FV of an annuityexamples. Recall that the FV of the annuity was $331 and the present value of the annuity is $248.68. Since we are using the samecash flows and the same rate, then these numbers must be linked.

If you were to take the $248.68 and move it forward three years as a single sum at 10%, you would get the $331. That is:

Or, if you were to take the $331 and move it backwards three years as a single sum at 10%, you would get the $248.68. That is:

248.68 (1.1)3 331

331

(1.1)3 248.68

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Examples

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Buttons Numbers to Enter  PV -100000

FV 0

I 10  N 3  PMT ????   40,211.48

So, the annual loan payment on this oddmortgage is $40,211.48

As an example, suppose you plan on taking out an odd mortgage. You plan to borrow $100,000 today. In exchange, you will pay the bankthree equal end-of-year payments. Each payment will include principal and interest. When the final payment is made at t=3, the loan will becompletely paid off (principal and interest). Assume the rate is 10%, compounded annually.

We refer to this type of loan as a self-amortizing loan. Car loans and home mortgages are typically self-amortizing loans.

The payment is found by calculating the PV of an annuity.

While we will go into more detail on this later in the course, for now, you can think of this as finding the annuity payment such that thepresent value of what the bank is giving up today ($100,000) is equal to the PV of what it is getting (the three-year annuity). That is, we wantto find the annuity payments that make the PV of the annuity equal to the loan amount…. $100,000 = PV of Annuity Worth $100,000

Example: Present Value of an Annuity

Bank gives uptoday…

Value today of payments the bankwill hopefully get back in the future…

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Using this same example, let’s look at what we refer to as an amortization schedule.

Example: Present Value of an Annuity

At t=1, the interest amountequals the starting loanbalance times the rate.

At t=2, the interest amountequals the loan balance at theend of t=1 of $69,788.52times the rate. Etc.

The principal reduction equalsthe payment of 40,211.48minus the interest amount foreach year.

The remaining balance is the balance from the year before less that year’s principal reduction.

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Using this same example, let’s look at what we refer to as an amortization schedule.

Example: Present Value of an Annuity

Notice that the principal reduction increases each year and theinterest expense decreases each year.

This is logical. With the first payment, you are decreasing theamount of the loan to $69,788.52. Thus, in the second year,because you have less of a loan, you owe less interest. Etc.

However, since your payment is fixed through time, as theportion of the payment going toward covering interest expensedecreases, the proportion going toward principal reductionmust be increasing.

Notice that when the finalpayment is made, then theloan is completely paid off(plus interest).

© 2014 Craig Ruff

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Buttons Numbers to Enter  PV -1000000

FV 0

I 6  N 30  PMT ????   72,648.911

So, the annual withdrawal amount is $72,648.911

As another example, suppose you are about to retire and have $1,000,000 saved up. You estimate that you will live anadditional 30 years. You plan to make equal, annual withdraws from this wealth at the end of each of the next 30 years,with the first withdrawal made in one year. Assuming you can earn 6% on your wealth, how much can you withdraw eachyear so that you completely exhaust your wealth with the last withdrawal at t=30?

The withdrawal amount is found by calculating the PV of an annuity. Here, you want to find the annuity payment so thatthe present value of the annuity equals the $1,000,000.

Example: Present Value of an Annuity

© 2014 Craig Ruff

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As we didbefore,modelingthese cashflows out onaspreadsheetcan beIntuitivelyuseful….

Example: Present Value of an Annuity

© 2014 Craig Ruff