1. Management 3 Quantitative Methods The Time Value of Money Part 2

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.

3. 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.

4. 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 1 st payment is not due the day you borrow, but one month later.

5. 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.

6. 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.

7. 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

8. 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.

9. 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.

10. 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.

11. 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.

12. 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

13. 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?