Time Value of Money

Review of Basic Concepts

Types of problems

• Single Sum. One sum ($1) will be received or paid either in the– Present (Present Value of a Single

Sum or PV)– Future (Future Value of a Single Sum

or FV)

PV FV

Types of Annuity Problems

Ordinary annuity (OAOA)A series of equal payments (or rents) received or paid at the

end of a period, assuming a constant rate of interest.

PV-OA (Present value of an ordinary annuity)

PV-OA

PMT PMTPMTPMT PMT

Types of Annuity Problems

Ordinary annuity (OAOA)A series of equal payments (or rents) received or paid at the

end of a period, assuming a constant rate of interest.

FV-OA (Future value of an ordinary annuity)

FV-OA

PMT PMTPMTPMT PMT

Types of Problems

Annuity Due (ADAD)A series of equal payments (or rents) received or

paid at the beginning of a period, assuming a constant rate of interest.

FV-ADFV-AD (future value of an annuity due) (future value of an annuity due)

FV-AD

PMTPMT PMTPMT PMT 0

Note: Each rent or payment is discounted (interest removed) one less period under a FV-AD.

Types of Problems

Annuity Due (ADAD)A series of equal payments (or rents) received or paid at the

beginning of a period, assuming a constant rate of interest.

PV-AD PV-AD (present value of an annuity due)(present value of an annuity due)

PV-AD

PMTPMT PMTPMT PMT 0

Note: Each rent or payment compounds (interest added) one more period in a PV-AD

Deferred Annuities

PMTPMT PMT

0 54321

d = 2n = 3

This is an ordinary annuity of 3 periods deferred for 2 periods.

We could find either the PV or the FV of the annuity.

1515

Calculation Variables

• There will always be at least four variables in any present or future value problem. Three of the four will be known and you will solve for the fourth.– Single sum problems:

• n = number of compounding periods• i = interest rate

• PV = Value today of a single sum ($1)

• FV = Value in the future of a single sum ($1)• PMT = 0 (important it using PV calculator!)

Annuity Problems• n = number of payments or rents• i = interest rate• PMT = Periodic payment (rent) received

or paid– And either:

• FV of an annuity (OA or AD) = Value in the future of a series of future payments

– OR• PV of an annuity (OA or AD) = Value today

of a series of payments in the futureWhen we know any three of the four amounts, we can solve for the fourth!

i and n must match!• The “n” refers to periods not necessarily

defined as years! The period may be annual, semi-annual, quarterly or another time frame.

• The “n” and the “i” must match. That is, if the time period is semi-annual then so must the interest rate.

• Interest rates are assumed to be annual unless otherwise stated so you may have to adjust the rate to match the time period.

Single Sum Formulas

•FV = (1+i)n

•PV = FV (1+i)n

Single Sum Formulas

•FV = (1+i)n

•PV = FV (1+i)n

• Present value calculators are generally no more expensive than those that do nth powers and nth roots!

Annuity Formulas

•FV-OA =

•

PV-OA = PMT

i

1

(1 + i)n1 -

(1 + i) - 1

iPMT

Formulas vs. Tables

• Before fancy calculators, people had no easy way to compute nth roots and raise numbers to the nth power.

• So they created tables for of sums of $1 or annuities of $1.

• The values on the table, I call the “interest factor” or IF.

• So we have PVIF (for n and i)and the PVIF-OA (for n and i) and so forth

Using the tables

horizontally for the

““ii””

The tables are the result of the required multiplications and division at various “n” and “i” and are to be read

vertically for the ““nn””

and

24

Study the tables . . • They are very logical.

– All sums in the future are worth LESS in the present.

• All factors on the present value of a single sum table are less than one.

All present sums are worth more than themselves in the future.

• All factors on the future value of a single sum table are greater than one.

– Notice how the factors change dramatically as the “i” increases and the “n” lengthens!

“Formulas” for the IF Tables• PV = FV * IF {IF from PV of $ table}• FV = PV * IF {IF from FV of $ table}• PV-OA = PMT * IF {IF from PV-OA table}• FV-OA = PMT * IF {IF from FV-OA table}• PMT = (PV-OA) / IF {IF from PV-OA table}• or • PMT = (FV-OA) / IF {from FV-OA table}

“IF” stands for “interest factor” from the appropriate n row and i column of the table

Adjustments to ordinary annuity tables

Rules for annuity dues and deferred annuities

Conversion to Annuity Due• To find IF for FV-AD: Add one to the

number of periods and look up IF on table. Then subtract one from the interest factor listed.

• To find IF for PV-AD: Subtract one from the number of periods and look up IF on table. Add one to the interest factor.

• Or look up the IF on the appropriate table and multiply by (1 + i).

Ordinary Annuity Example

• Suppose I must make three payments of $500, each at the end of each of the next three years. The interest rate is 8%. How much should I set aside today to have the required payments?

• This is an ordinary annuity:

PV-OA = PMT * (PVIF-OA n,i) where n = 3 payments and i = 8% PV-OA = $500 * 2.5771 = $1,289

Page 480

Stop and think . . .

• If the first payment comes immediately instead of at the end of the first year,

• Will the present value be

• MORE or • LESS?

36

Annuity Due Example

• If the first payment comes immediately, this would be an annuity due problem.

• We can use one of the formulas to adjust the IF – the easiest to memorize is the “multiply by (1+i)” rule:

PV-AD = PMT (PVIF-OA n,i)(1+i) where n = 3 payments and i = 8% PV-AD = $500 (2.5771)(1 + .08) = $1,391

36

Annuity Due Example

• Alternative adjustment to the IF table is even easier – at least if you write the method at the top of your table!

• Look up IF for (n-1) and add 1:

PVIF-OA (n=2, i=8%) = 1.7833 + 1 = 2.7833

PV-AD = $500 (2.7833) = $1,392

Annuity Due Example

• This second method is also the “logical” decision you would make from looking at the time-line.

FV-AD

PMTPMT PMTPMT PMT 0

This is an ordinary annuity of 4 periods (n-1)

The first payment comes immediately and therefore is NOT discounted!

Deferred Annuities

PMTPMT PMT

0 54321

d = 2n = 3

When using the tables, there are some short-cuts for doing deferred annuities

Using PV Tables – Deferred Annuity• Let d = number of periods deferred and n =

number of periodic payments• Look up (d+n) on the appropriate table. Look

up d on the same table. Subtract the smaller interest factor from the larger to get the deferred annuity IF.

• Or look up interest factor for n periods on appropriate annuity table. Then look up interest factor from the corresponding “lump sum” table for d. Multiply the two interest factors together to get the deferred annuity IF.

Deferred Annuity Example

• Let i = 12%• Find the present value of the

annuity due:

$100$100 $100

0 54321

d = 2n = 3

$100$100 $100

0 54321

d = 2n = 3

Alternative 1 – Work as two part problem

Find present value of ordinary annuity at end of year 2. Then discount it back to beginning of year 1

PV-OA IF(n=3, i=12%) = 2.4018

2.4018 * $100 = $240.18

PVIF (n=2, i=12%) = .7972

$240.18 * .7972 = $191.47

$100$100 $100

0 54321

d = 2n = 3

Alternative 2 – Adjust the ordinary annuity table IF:Look up PV-OA IF for (d+n) and then subtract the PV-OA IF for d

PV-OA IF(n=5,i=12%) = 3.6048PV-OA IF(n=2,i=12%) = -1.6901Adjusted IF = 1.9147

$100 * 1.9147 = $191.47

$100$100 $100

0 54321

d = 2n = 3

Alternative 3 – Adjust the ordinary annuity table IF:Look up PV-OA IF for n and then multiply by the PV IF for d

PV-OA IF(n=3,i=12%) = 2.4018PV IF (n=2,i=12%) = 0.7972Adjusted IF = 1.9147

$100 * 1.9147 = $191.47

Deferred Annuities

PMTPMT PMT

0 54321

d = 2n = 3

If it is a FV problem, this is pretty much the only way to analyze the facts.

However . . . .

FV

Deferred Annuities

PMTPMT PMT

0 54321

d = 2n = 3

We could do it the same way if we wanted to compute the present value, but we could also analyze it as an annuity due problem.

PV

Deferred Annuities

Then it would be 3 annuity due payments and the period before the annuity starts would be 3 periods instead of 2.

PMTPMT PMT

0 54321

d = 3n = 3

PV6

Note that in the last period we have zero left which earns no interest

at any interest rate

Deferred Annuities

Now see if you can work the problem (with the tables) if you analyzed the annuity as having the first payment happen immediately.

PMTPMT PMT

0 54321

d = 3n = 3

PV6

Spreadsheet demo• We’ll look at using

Excel functions to solve lease problems

• =NPV• =IRR• =PMT• =PV• =FV