Post on 15-Dec-2015
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?