Review of Basic Concepts 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 9 9 10 9 11 11 11 11 10

Types of Annuity Problems Ordinary annuity (OA) 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 12 12 12 12 12

Types of Annuity Problems Ordinary annuity (OA) 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 12 12 12 12 12

FV-AD (future value of an annuity due) Types of Problems Annuity Due (AD) A series of equal payments (or rents) received or paid at the beginning of a period, assuming a constant rate of interest. FV-AD (future value of an annuity due) Note: Each rent or payment is discounted (interest removed) one less period under a FV-AD. FV-AD PMT 11 11 11 13 13 13 13 13 13

PV-AD (present value of an annuity due) Types of Problems Annuity Due (AD) A series of equal payments (or rents) received or paid at the beginning of a period, assuming a constant rate of interest. PV-AD (present value of an annuity due) Note: Each rent or payment compounds (interest added) one more period in a PV-AD PV-AD PMT 11 11 11 13 13 13 13 13 13

Annuity Due vs. Ordinary Annuity The difference between an ordinary annuity and an annuity due is that: Given the same i, n and periodic payment, the annuity due will always yield a greater present value (less interest removed) and a greater future value (more interest added). 14 12 14 12 14 14 14 14

Deferred Annuities PMT 5 4 3 2 1 d = 2 n = 3 5 4 3 2 1 d = 2 n = 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.

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!) 15 15 15 13 13 15 15 15 15 15

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 future When we know any three of the four amounts, we can solve for the fourth! 16 14 16 14 16 16 16 16

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. 20 18 20 18 20 20 20 20

Single Sum Formulas FV = (1+i)n PV = FV (1+i)n

FV = (1+i)n PV = FV (1+i)n Single Sum Formulas Present value calculators are generally no more expensive than those that do nth powers and nth roots!

Annuity Formulas FV-OA = PV-OA = (1 + i) - 1 PMT i 1 1 - PMT (1 + i)n

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 22 20 22 20 22 22 22 22

Using the tables vertically for the “n” The tables are the result of the required multiplications and division at various “n” and “i” and are to be read vertically for the “n” and horizontally for the “i” 22 20 22 20 22 22 22 22

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! 24 24 24 24

“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 Page 480 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 33 31 33 36 36 36 36

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?

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 33 31 33 36 36 36 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 36 33 31 33 36 36 36 36

Annuity Due Example This second method is also the “logical” decision you would make from looking at the time-line. FV-AD PMT This is an ordinary annuity of 4 periods (n-1) The first payment comes immediately and therefore is NOT discounted!

Deferred Annuities PMT 5 4 3 2 1 d = 2 n = 3 5 4 3 2 1 d = 2 n = 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 $100 5 4 3 2 1 d = 2 n = 3 Let i = 12% Find the present value of the annuity due:

Alternative 1 – Work as two part problem $100 5 4 3 2 1 d = 2 n = 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 5 4 3 2 1 d = 2 n = 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%) = 3.6048 PV-OA IF(n=2,i=12%) = -1.6901 Adjusted IF = 1.9147 $100 * 1.9147 = $191.47

$100 5 4 3 2 1 d = 2 n = 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%) = 2.4018 PV IF (n=2,i=12%) = 0.7972 Adjusted IF = 1.9147 $100 * 1.9147 = $191.47

Deferred Annuities FV PMT 5 4 3 2 1 d = 2 n = 3 5 4 3 2 1 d = 2 n = 3 If it is a FV problem, this is pretty much the only way to analyze the facts. However . . . .

Deferred Annuities PV PMT 5 4 3 2 1 d = 2 n = 3 5 4 3 2 1 d = 2 n = 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.

Note that in the last period we have zero left which earns no interest at any interest rate Deferred Annuities PMT 5 4 3 2 1 d = 3 n = 3 PV 6 Then it would be 3 annuity due payments and the period before the annuity starts would be 3 periods instead of 2.

Deferred Annuities PV PMT 5 4 3 2 1 d = 3 n = 3 6 5 4 3 2 1 d = 3 n = 3 PV 6 Now see if you can work the problem (with the tables) if you analyzed the annuity as having the first payment happen immediately.

Spreadsheet demo We’ll look at using Excel functions to solve lease problems =NPV =IRR =PMT =PV =FV