Management 3Quantitative Methods

The Time Value of MoneyPart 2

Scenario #2 – the PVof a series of future deposits

We can trade single sums of money today (PV)

for multiple payments (FV’s) paid-back periodically in the future:

a) Borrow today (a single amount) and make payments (periodically in the future) to repay the Loan.

Annuities• An annuity is a “fixed” periodic payment

or deposit:1. $ 1,000 per year/month for 36 months.

• 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.

Annuities If you win the Lottery, you receive an

Annuity Due because you get the first payment now.

If you borrow (take a mortgage), you agree to pay an Ordinary Annuity because your 1st payment is not due the day you borrow, but one month later.

The Annuity Tables

• The PVFA – present value factor annuity – Table is a sum of the PVF’s up to any point in Table 3. This will always be less than the number of years, since PVF’s are each < 1.

• The FVFA – future value factor annuity – Table is a sum of the FVF’s up to any point in Table 4. This will always be greater than the number of years, since FVF’s are each > 1.

Annuity Factors

Table 3 is constructed using this formula

Each PVFA (r, t)= [ 1- PVF(r, t)] / r= [ 1- (1+r) -t] / r

These are called Present Value Factors of Annuities

and are found on the PVFA Table 3.

Annuity Factors

Table 4 is constructed using this formula

Each FVFA (r, t)= [ FVF(r, t) -1] /r= [(1+r) t -1] /r These are called Future Value Factors of

Annuitiesand are found on the FVFA Table 4

The PV of an AnnuityWe can calculate the PV of an Annuity by determining

the PV of each payment, which would be tedious – there could be dozens of calculations.

The fact that the Annuity amount is constant allows us to factor-out the payment from the series of PVFs.

• For example: the PV of $1,000 per year for 10 years

= $1,000 x ( (1.10)-t ) for t=1 to 10

= $1,000 x PFVA (r=10%, t=10)= $1,000 x 6.144 from Table 3 = $6,144

This means that if you gave someone $ 6,144 today (and rates were 10%), then they should repay you $ 1,000 per year for 10 years.

Annuities Monthly Compounding

What is the PV of $100 per month for 3 years @ 6%?

PV of $100 for 36 months ½ % per month= $ 100 x PVFA (r /12, t x12)= $ 100 x PVFA (0.005, 36)= $ 100 x [1- 1/(1.005) 36] / 0.005 There is no Table for these calculations, unless you make one yourself, so you will need to calculate it. = $ 100 x [1- 0.1227] / 0.005 = $ 100 x 32.87 = $ 3,287

Thus, if you borrowed $ 3,287 today and agreed to repay the loan over 36 months at 6% interest – you payments would be $100 each month.

Scenario #2 : the FV of a series of future deposits

We deposit multiple small sums of money regularly (FV’s) to achieve a single large accumulation (FV) in the future:

a) Save an amount each year to achieve a future goal.

The FV of an AnnuityWe can calculate the FV of an Annuity by

determining the FV of each payment, but this too would tedious.

For example: The FV of $1,000 per year (ordinary annuity) for 10 years @ 10%

= $ 1,000 x (1.10)t ) for t=0 to 10-1

= $ 1,000 x FVFA (r=10%, t=10)= $ 1,000 x 15.937 from Table 4. = $ 15,937

So, if you put $ 1,000 in the bank @ 10%, each year starting in one-year, for 10 years – you should have $ 15,937 ready ten years from now.

Summary of the Factor Tables and their Functions

Future Value Factors “FVF” = (1+r)^tTurn a present value into a FV

Present Value Factors “PVF” = 1/(1+r)^tTurn a future value into a PV

Future Value Annuity Factors “FVFA” = (FVF-1)/rTurn an Annuity into a FV

Present Value Annuity Factors “PVFA” = (-1PVF)/rTurn an Annuity into a PV

Five Fundamental Practical Problems

1. Do I make “this” Investment today, i.e. does it offer a good return?

2. When do I take my Pension?

3. What will my payments be on this Loan

4. When and how much do I need to save for

something – a house, a car, or my

retirement?

5. Should I Lease or Buy this equipment?