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The Time Value of Money II

The Time Value of Money II. Econ 173A. Scenario #2a. We can trade single sums of money today (PV) for multiple payments (FV’s) paid-back periodically in the future: Borrow today (a single amount) and make payments (periodically in the future) to repay the Loan. Annuities.

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The Time Value of Money II

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  1. The Time Value of Money II Econ 173A

  2. Scenario #2a We can trade single sums of money today (PV) for multiple payments (FV’s) paid-back periodically in the future: • 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, such as: • $ 1,000 per year/month for 3 years/36 months. • These payments can be made at the beginning, or at the end, of the financing period: • Annuities “Due” are made at the beginning; • “Ordinary” Annuities are made at the end. • 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 isn’t due the day you borrow, but one month later.

  4. 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.

  5. 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.

  6. 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

  7. 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.

  8. 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.

  9. Scenario #2b We deposit multiple small sums of money regularly (FV’s) to achieve a single large accumulation (FV) in the future: • Save an amount each year to achieve a future goal.

  10. 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.

  11. Five Fundamental Practical Problems • Do I make “this” Investment today, i.e. does it offer a good return? • When do I take my Pension? • What will my payments be on this Loan • When and how much do I need to save for something – a house, a car, or my retirement? • Should I Lease or Buy this equipment?

  12. The five fundamental problems each have two things in common: Saving, receiving, or paying money in the future, i.e. over a time frame, and calculating the PV of that with another PV … A value, or decision, in dollars today.

  13. #1 - Is this a good Investment? If I invest $100 today • in exchange for $ 150 in 10 years? • or for $ 8 per year for 20 years, what is my “return”? You know the PV, the FV, and t-years, so you can solve for “r”.

  14. #2 - When Do I Take My Pension? A pension is a guaranteed annual “receipt” of money for life. How long is “life”? The sooner it starts the longer the pension life, not your life. But the later it starts the greater the monthly amount. So, do you want less for longer or more for shorter?

  15. My Social Security The Question: Do I want $1,720 per month for 23 years starting 4 years from now or do I want $2,352 per month for 19 years starting 8 years from now?

  16. #3 What will my Payments be on a Loan? If you borrow $PV money at rate “r” for a period “t” to be repaid over “n” periods then what willyour payments “A” be? or If rates are = “r” and the maximum period that you can repay a loan is “t” in “n” payments, then what is the most $PV you can borrow?

  17. What is a Loan? It is an exchange of a big package of money today in exchange for many small packages on a future schedule. So the person with the big package today – the lender - is selling that big package to the borrower at interest rate “r”.

  18. You want to borrow $30,000 You Know • $30,000 is the PV • For 5 years = t • At 8 percent per year interest = r And you want to find the monthly Payments, i.e. the amount of each “Future” payment. Since you know 3 of the 4 parameters, you can calculate the 4th – you payments.

  19. Monthly payments on a $30,000 loan – approximated by doing annual discounting & dividing the result by 12 Solve this Value exchanged must be the same, so: $30,000 = P. Value sold= P. Value paid $ 30,000 = “Payment” x PV Factor for the Payment Use 5 years = t and 8 percent = r So $ 30,000 = $Payment x PVFA(8%, 5 years) Go to the Table; get the PVFA = 3.99 Solve for $A – the annual payment = $ 7,500 Divide by 12 for an approximation to the month payment.

  20. The PVFA and an Example • If someone is willing to pay you $ 625 per month for 5 years and relevant rates are 8% , then the most you should lend is … $30,000. Because the PV of $625/month for 60 months @ 8% interest rates is approximately equal to $30,000 PV = 30,000 = Payment x PVFA for t = loan periods & r = loan rate)

  21. #4 - How much do I need to Save for my Retirement? • The more you save the more you can spend. • The longer you save the more you can spend. • The sooner you start spending, the less you can spend (per year). • The later you start spending, the more you can spend (per year).

  22. #5 Should I Lease or Buy the equipment? If you “buy” you need the purchase price now and you own the equipment! If you “lease” you need a smaller amount now, then regular rental payments, but you don’t own the equipment.

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