The Time Value of Money Annuities

Revised 4/7/17

Savings & Funding Plans

We often want to accumulate a future balance of money or meet a future obligation. We call this a Sinking Fund. Or, we want to pay-off an obligation like a Loan, doing so in a regular, systematic way.

Annuities • An annuity is a “fixed periodic” payment or

deposit: … for example, $ 500 per month for 36 months or $2,000 per year for 20 years.

These payments can be made at the beginning, or at the end, of the financing period:

a) Annuities “Due” are payments made at the beginning of the period;

b) “Ordinary” Annuities are payments made at the end of the period.

Ordinary Annuities vs. Annuities Due

If you win the Lottery, you receive an Annuity Due because you get the first payment now.

If you borrow (to buy an automobile), your payments are an Ordinary Annuity because your 1st payment is due next month, not the day you borrow.

Annuities An annuity is defined according to its

parameters. We speak of annuities by:

1. Term of the annuity – in years or months;

2. Amount of the annuity, and;

3. The expected annual rate of return on the money saved/invested or paid.

For example: A 5-year, $1,000, 5% annuity means that we save $1,000/year for 5 years, and the funds earn 5% per year.

What is the Future Value of an Annuity?

If you Saved $10,000 per year for 5 years how much would you have in 5 years?

It depends. •At the very least, you should have

$10,000 x 5 = $50,000. • However, if the money ears >0% interest, you would have > $50,000. •And, the sooner you started saving, the more you would have.

The future value of $10,000 per year (ordinary annuity style) for 5 years, earning

5% per year:

Year Begin Earn 5% Total ANNUITY Year End

$ 0 $ 0 $ 10,000

$ 10,000

$ 10,000 (1.05) $ 10,500 $ 10,000 $ 20,500

$ 20,500 (1.05)

$ 21,525 $ 10,000

$ 31,525

$ 31,525 (1.05)

$ 33,101 $ 10,000

$ 43,101

$ 43,101 (1.05)

$ 45,256 $ 10,000

$ 55,256

A Shorter Way?

The first $ 10,000 would earn 4 years @ 5% This is because it is deposited at the end of year 1.

The second $ 10,000 would earn 3 years @ 5% This is because it is deposited at the end of year 2.

The third $ 10,000 would earn 2 years @ 5% The fourth $ 10,000 would earn 1 year @ 5% The fifth $ 10,000 would be your last deposit • This looks like the sum of four calculations

using FV Factors x $10,000 each plus the last payment.

The FV Annuity Table is just a sum of FVF’s

• Below are the FVF’s from Table 1 where r = 5% and t= 5 years.

• If we start w/ 1.0 then add the first four from Table 1, we have a new Factor, the FVF of an annuity.

Time 5.0%

0 to 1 1.0000

1 to 2 1.0500

2 to 3 1.1025

3 to 4 1.1576

4 to 5 1.2155

5.5256

FVF’s

The $10,000/year for 5 years @ 5 percent

• We know that a 5 year $10,000 annuity at 5% has a FV of $ 55,256 because

Time 5.0%

0 to 1 1.0000

1 to 2 1.0500

2 to 3 1.1025

3 to 4 1.1576

4 to 5 1.2155

5.5256 x $10,000 = $55,256

$10,000 x FVFA(5%, 5) = $55,256

The FVFA(r, t)

Table 3 is a Table of Future Value Factors for Annuities written FVFA (r, t)

Each Factor, is defined by its rate “r” and its time in years “t”:

FVFA(r, t) = [ FVF(r, t) -1] /r FVFA(r, t) = [ [(1+r) t -1] /r ]

These are called Future Value Factors of Annuities

FVFA(r, t) = [ FVF(r, t) -1] /r

FVFA(5%, 5) = [ [(1+5%) 5 -1] / 5% ]

FVFA(5%, 5) = [ [(1.05) 5 -1] / 0.05 ]

FVFA(5%, 5) = [ [1.2763 -1] / 0.05 ]

FVFA(5%, 5) = [ [0.2763] / 0.05 ]

FVFA(5%, 5) = 5.5256

The 5 year @ 5 percent FVFA

Formulation: the Future Value of an annuity is

Annuity x FVFA (r, t) = Future Value [Dollars/Year] x [ [(1+r) t -1] /r ] = Dollars

We can rearrange this:

Annuity = Future Value / [ [(1+r) t -1] /r ]

Or, not as useful

FVFA = [ [(1+r) t -1] /r ] = Future Value / Annuity

Reverse the problem. Say that you want $55,256 in 5 years. Interest rates are 5%.

How much would you need to save each year?

Use this formulation: Annuity = Future Value / [ [(1+r) t -1] /r ] And insert the data Annuity = $55,256 / FVFA(5%, 5) Annuity = $55,256 / 5.5256 $10,000 = $55,256 / 5.5256

The FVFA • The FVFA’s will always be a

number greater than the number of years, since FVF’s are each > 1.

Lending

When we make a loan, we trade a single sum of

money today, a (PV) in return for a series of

periodic future payments (FV’s).

This is making a Loan …

• Lend today (a large single amount) and receive

payments in the future that repay the Loan, it’s PV.

Borrowing

When we take a loan, we trade a future series of

periodic payments (FV’s) for a the amount lent to

us today (PV).

This is taking a Loan …

• Borrow today (a large single amount) and pay

payments (FV’s) through future months that repay

the Loan, it’s PV.

The PV of an Annuity In the prior Slides, we calculated the FV of $10,000/year for 5 years at 5 percent and found that that future value was $55,256, i.e. we calculated:

$A x FVFA(r, t) = $FV $10,000 x [[1-(1+r) t -1] /r ] = $FV

$10,000 x 5.5256 = $ 55,256 Let’s reverse this process by asking - what is the PV of $10,000/year for 5 years at 5 percent?

It will be < $50,000 because each of the five $10,000 installments are paid in the future, so they need to be discounted.

Calculate the PV of each $10,000 payment by

using individual PVF’s 1-5 years at 5%.

Interest = r 5.00%

Periods = t PVFs Installments Present Value

1 0.9524 $ 10,000 $ 9,524

2 0.9070 $ 10,000 $ 9,070

3 0.8638 $ 10,000 $ 8,638

4 0.8227 $ 10,000 $ 8,227

5 0.7835 $ 10,000 $ 7,835

4.329 $ 43,295

The PV of an Annuity Rather than run the sum of five products, we can factor-out the $10,000 and sum the PVFs, then find the product.

= $10,000 x ( Σ (1.05)-t ) for t=1 to 5

= $10,000 x PFVA (r=5%, t=5)

= $10,000 x 4.329

= $43,295

Present Value Annuity Factors

Table 4is constructed using this formula Each Factor, called a PVFA, is defined by

its rate “r” and its time in years “t”: PVFA(r, t) = [ 1- PVF(r, t)] /r PVFA(r, t) = [[1-(1+r) -t] /r ]

These are called Present Value Factors of Annuities

Calculating the Present Value Annuity Factor

Table 4 is constructed using this formula PVFA(r, t) = [ 1- PVF(r, t)] /r

Each Factor, called a PVFA, is defined by its rate “r” and its time in years “t”:

For the previous example: PVFA(5%, 5) = [ 1- PVF(r, t)] /r

PVFA(5%, 5) = [[1-(1.05) –5 /0.05 ] PVFA(5%, 5) = [[1-0.7835] /0.05 ] PVFA(5%, 5) = [[0.2165] /0.05 ]

PVFA(5%, 5) = 4.3295

The PV of an Annuity The present value of an annuity is the Annuity x the PVFA(r, t). In our first example it is:

= $10,000 x PVFA(5%,5)

= $10,000 x 4.329 where we find 4.329 by

calculating it.

= $43,295

The PV of an Annuity in Reverse In the prior example, we found that the PV of a 5 year $10,000 ordinary annuity at 5% is $43,295. Reversing that, let’s ask: What is the FV of a 5 year $10,000 ordinary

annuity at 5%? Formulate it: FV(Annuity) = Annuity x FVFA(5%, 5) FV(Annuity) = $ 10,000 x 5.5256 FV (A =$10,000) = $ 55,526

The PV of an Annuity in Reverse What do we have now? • We have a Present Value of $ 43,295. • We have a Future Value of $ 55,256. • These are linked by 5 years? • What is the CAGR that bring the two values

and the term of 5 years in line? Formulate this:

The CAGR = [(FV/PV)^(1/5)] -1 CAGR = [(55,256/43,295)^0.20] -1

CAGR = [(1.276)^0.20]-1 CAGR = 1.05 -1 = 5 percent

A Car Loan You want to purchase a new car. You find the car and negotiate a price, $35,000

all-in. You make a $5,000 down payment and borrow

the remaining $30,000 from your credit union at 6% over 4 years.

What is a good approximation for your monthly payment?

Car Loan Let’s find your annual payment – just as we

did earlier - and divide by twelve (months). • You are borrowing a PV = $30,000. • You will pay-back this $30,000 PV with a 4

year 6% ordinary annuity. • The PV of your four annual payments must be

equal to $30,000. That is …. And this is IMPORTANT.

• You must return the same PV that you borrowed.

Car Loan • You must return the same PV that you

borrowed. • This means that the annual payments will be

somewhat greater than $ 7,500, which is $30,000 / 4.

• In other words, because you will be paying-off the loan with future dollars, the lender will need more than $30,000 of them.

• If you understand this, corporate finance is in your hands.

Formulation

PVFA’s transform a series of future payments or deposits into a Present Value.

Annuity x PVFA (r, t) = Present Value Annuity = PV / PVFA(r, t) Annuity = PV / [[1-(1+r) -t] /r ] Annual Payment = $30,000 / PVFA(6%, 4)

Car Loan Example, con’t Here is the formula we need to solve:

Payment = $30,000 / PVFA(6%, 4) Present Value of $1 Annuity Table of Factors

1.00% 2.00% 3.00% 4.00% 5.00% 6.00%

Periods

1 0.9901 0.9804 0.9709 0.9615 0.9524 0.9434

2 1.9704 1.9416 1.9135 1.8861 1.8594 1.8334

3 2.9410 2.8839 2.8286 2.7751 2.7232 2.6730

4 3.9020 3.8077 3.7171 3.6299 3.5460 3.4651

Car Loan Example, con’t Insert the PVFA(6%, 4) = 3.4651 into the formula:

Annual Payment x 3.4652 = $30,000 Annual Payment = $30,000 / 3.4652

$ 8,657.74 = $30,000 / 3.4652

This means that an approximation of your monthly payment is:

$ 8,657.74 / 12 = $ 721.48

Car Loan Example proves that 1) The Loan Payments on the 4 year 6% $30,000 loan

are $ 8,657.74 per year, and 2) $ 8,657.74 per year are the annual future payments

that return a present value of $30,000 @ 6%. 3) Thus, the loan payments are the present value of a

$30,000 loan meaning the Lender gets $30,000 of present value from four future payments of $ 8,658 each.

4) Therefore, if our calculations are correct, the Lender should be indifferent between:

a) lending to you at 6% for 4 years; b) putting the $30,000 in the bank at 6%. Let’s check this ….

We will compare: 1) the Lender’s Future Value of a single $30,000 deposited for 4 years @6% and 2) the Future Value of $ 8,658 per year for 4 years @6%. Bear in mind that someone with $30,000, and a 4 year investment horizon, has two choices: (a) put it all in the bank earning 6% per year, or (b) or lend it to you for 4 years expecting 4 annual payments.

Comparing the Lender’s two choices are no different in Future Value:

FV Factors

$30,000 in the Bank

Compound-ing

Pay-ments

Payments Compounded

0 1.0000 30,000 0 0

1 1.0600 31,800 8,658 8,658

2 1.1238 33,708 8,658 9,177

3 1.1910 35,730 8,658 9,727

4 1.2625 37,874 8,658 10,311

Total

37,874

4th payment

3rd payment

2nd payment

1st payment

Future Value of all four Payments

Choice (a) Choice (b)

Analysis of Car Loan Example

Your approximate monthly payment on the 48 month, 6%, $ 30,000 car loan is

$ 721.48.

Annuities w/ Monthly Compounding

To get a more precise calculation for the car loan payment, we need to make some adjustments to the basic formula.

= $ 30,000 / PVFA (r /12, t x12) = $ 30,000 / PVFA (0.005, 48) = $ 30,000 / [1- (1.005) -48] / 0.005 = $ 30,000 / [1- 0.7870] / 0.005 = $ 30,000 / 0.2130 / 0.005 = $ 30,000 / 42.58 = $ 704.55 per month

Some Observations • The monthly payment calculated w/ adjusted, i.e.

more precise, parameters: 4 years x 12 months = 48 months, and

6%/12 = 0.5% interest is smaller than the approximate payment. • How many total dollars will be paid in consideration

of this loan? You will pay 48 x $ 704.55 = $ 33,818

• What are the financing costs, i.e. the interest your will pay on the loan?

Interest = Total paid less Principal $ 3,818 = $ 33,818 - $ 30,000

Formulation Review

PVFA’s transform a series of future payments or deposits into a Present Value.

Annuity x PVFA (r, t) = Present Value Deposit x [ 1- PVF(r, t)] /r] = Present Value Payment x [[1-(1+r) -t] /r ] = Present Value

Formulation & Transformation • We know how much we want to borrow – the $PV and • We know how many years or months “t” we’d like to have to pay it back

and • The market gives us the “r” So • What are our loan payments? Use this: Payment x PVFA(r, t) = Loan Thus:

Payment = Loan / PVFA(r, t)