Management 3
Quantitative Methods
The Time Value of Money
Part 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
Annuities
and are found on the FVFA Table 4
The PV of an Annuity
We 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 Annuity
We 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)^t
Turn a present value into a FV
Present Value Factors “PVF” = 1/(1+r)^t
Turn a future value into a PV
Future Value Annuity Factors “FVFA” = (FVF-1)/r
Turn an Annuity into a FV
Present Value Annuity Factors “PVFA” = (-1PVF)/r
Turn 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?