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Economics 434 Theory of Financial Markets

Economics 434 Theory of Financial Markets. Professor Edwin T Burton Economics Department The University of Virginia. Imagine the following. Interest rates are 5 percent Someone offers to give you $ 1 every year forever (the rights to which you can leave your heirs)

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Economics 434 Theory of Financial Markets

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  1. Economics 434Theory of Financial Markets Professor Edwin T Burton Economics Department The University of Virginia

  2. Imagine the following • Interest rates are 5 percent • Someone offers to give you $ 1 every year forever (the rights to which you can leave your heirs) • What is this worth? What would you be willing to pay for this? If you owned it, at what price would you sell it?

  3. Present Value (and “Future Value” • Present value means “today’s value of a future stream of income” • Future value means “the value on a specific future today of a specific amount of money today”

  4. Present Value (money one year from today) Suppose the default free interest rate for one year is R Then M dollars received one year from today is currently worth: Present Value =

  5. Present Value (money two years from today) Suppose the default free interest rate for two years is R per year Then M dollars received two years from today is currently worth: Present Value =

  6. Present Value (money two years from today) Suppose the default free interest rate for three years is R per year Then M dollars received three years from today is currently worth: Present Value =

  7. Present Value (money six months from today) Suppose the default free interest rate for one year is R per year Then M dollars received six months from today is currently worth: Present Value =

  8. Present Value (one day from today – i.e., tomorrow..assume 365 days in a year) Suppose the default free interest rate for one year is R per year Then M dollars received tomorrow is currently worth: Present Value =

  9. Critical assumption – that R is constant….not true, obviously • R never changes and is the same for all periods of time • This is, of course, not true • The future R’s are not known exactly, though their markets already exist

  10. We will simplify matters: • RT (R with a subscript) will mean the one year default free rate starting at the beginning of year T from now) • For example: • R2 will mean the one year rate starting one year from today. • R1 will mean the one year rate today. • The future value of $ 100 two years from today will be …… FV = $ 100(1+R1)(1+R2)

  11. So, assume we know R1, R2, ….RN.. • Then Future Value of $ 100 is: • $ 100 times (1 + R1) times (1 + R2) times…..(1 + RN) • Therefore, Present Value of $ 100 three years from now is • PV =

  12. Useful Fact 1 = r

  13. What does this formula tell us? • $ 1 every year forever starting one year from now is worth: • $1 • Divided by r • If r is 5 %, then $ 20 • If r is 10% then $ 10 • Excellent shortcut

  14. Present Value is the most Crucial Concept in Finance • Value of future stream of payments • As valued today • Emphasis on “discounting” future revenue streams • Common practice to use higher rates to reflect higher uncertainty of receipt of future payments

  15. Time Value of Money From last time… BOY Balance Int. Rate EOY Balance Year Interest $x¢(1+r)0 0 (now) $x¢(1+r)1 1 $x r $x¢r $x¢(1+r)2 2 $x¢(1+r) r $x(1+r)¢r $x¢(1+r)3 3 $x¢(1+r)2 r $x(1+r)2¢r $x¢(1+r)4 4 $x¢(1+r)3 r $x(1+r)3¢r $x¢(1+r)5 5 $x¢(1+r)4 r $x(1+r)4¢r

  16. Time Value of Money So far, we’ve assumed the interest rate is constant over time. But this may not be true – future year’s interest rates can be different than this year’s. What happens if interest rates vary over time?

  17. Time Value of Money Call the interest rate in year t: rt BOY Balance Int. Rate EOY Balance Year Interest $x 0 (now) $x¢(1+r1) 1 $x r1 $x¢r1 $x¢(1+r1)¢(1+r2) 2 $x¢(1+r1) r2 $x¢(1+r1)¢r2 $x¢(1+r1)¢(1+r2)¢(1+r3) 3 $x¢(1+r1)¢(1+r2) r3 $x¢(1+r1)¢(1+r2)¢r3 $x¢(1+r1) ¢(1+r2)¢(1+r3)¢(1+r4) $x¢(1+r1) ¢(1+r2)¢(1+r3) $x¢(1+r1) ¢(1+r2)¢(1+r3)¢r4 4 r4

  18. Time Value of Money In four years, the FV of $x will be We can also use the chart to infer the PV of a fixed amount of money in the future. The PV of getting $x in 4 years is ) $x ¢ (1+r1) ¢ (1+r2) ¢ (1+r3) ¢ (1+r4) $x (1+r1)¢(1+r2)¢(1+r3)¢(1+r4) )

  19. Time Value of Money Or, more generally… The FV of getting $x today will be, in t years: And the PV of getting $x in t years is: $x ¢ (1+r1) ¢ (1+r2) ¢ (1+r3) ¢ … ¢ (1+rt-2)¢ (1+rt-1) ¢ (1+rt) $x (1+r1)¢ (1+r2)¢ (1+r3)¢ … ¢ (1+rt-2)¢ (1+rt-1)¢ (1+rt)

  20. Time Value of Money We can use these formulae to calculate the PV of a fixed stream of cash flows just as we did before. Example: You own an asset that pays $150 after 1 year and $250 after 3 years. The 1st year’s interest rate is 8%, the second year’s is 6%, and the third year’s is 7%. What is the PV of this asset’s future cash flows?

  21. Time Value of Money Present value of cash flows for each year: $150 (1+.08) Year 1’s payment: ¼ $138.89 $250 (1+.08)(1+.06)(1+.07) Year 3’s payment: ¼ $204.09 ¼ $342.98 The PV of this asset’s cash flows is $342.98

  22. Time Value of Money The PV formula works over fractions of a year as well. Example: What is the most you should pay for an asset that pays $100 in 6 months, $300 in 1 year, and $500 in 2 years, if the interest rate is a constant 10% per year?

  23. Time Value of Money Present value of cash flows for each year: $100 (1+.10)0.5 Year 0.5’s payment: ¼ $95.35 $300 (1+.10)1 $500 (1+.10)2 Year 1’s payment: ¼ $272.73 Year 2’s payment: ¼ $413.22 ¼ $781.30 total The most you should be willing to pay is $781.30

  24. Time Value of Money Note that the units for t come from the units for the interest rate. In each example so far, that has been years, but it does not have to be. Example: The interest rate is 5% every six months. What is the PV of an asset that pays $150 in one year, $250 in 18 months, and $400 in 3 years?

  25. Time Value of Money Present value of cash flows for each year: $150 (1+.05)2 Year 1’s payment: ¼ $136.05 $250 (1+.05)3 $400 (1+.05)6 Year 1.5’s payment: ¼ $215.96 Year 3’s payment: ¼ $298.49 ¼ $650.50 total The asset’s PV is $650.50

  26. Time Value of Money The Effect of Compounding • So far, we’ve assumed interest compounds annually. • However, interest can compound at any rate. • For example, 12% interest compounded… • Annually ! Balance increases by 12% after 1 year • Semiannually ! Balance increases by 6% at 6 months and another 6% at month 12 • Quarterly ! Balance increases by 3% at 3 months, 6 months, 9 months, and at month 12 • And so on….

  27. Time Value of Money If we get a 12% rate on $100 compounded annually, after one year we will have $112. How much would we have if it were instead compounded semiannually? BOY Balance Int. Rate EOY Balance Year (t) Interest $100.00 0 (now) $106.00 0.5 $100 6% $6 $112.36 1 $106 6% $6.36

  28. Time Value of Money What would happen if it were compounded quarterly? BOY Balance Int. Rate EOY Balance Year (t) Interest $100.00 0 (now) $103.00 0.25 $100 3% $3 $106.09 0.5 $103 3% $3.09 $109.27 0.75 $106.09 3% $3.18 $112.55 1 $109.27 3% $3.28 The EOY balance increases with the compounding rate.

  29. Time Value of Money How large can this increase get? Consider investing $x at an interest rate r, compounded over n periods. BOY Balance Int. Rate EOY Balance Year (t) Interest $x 0 (now) r n r n r n r n r n r n r n r n 1 n r n $x ¢(1+ )¢ $x ¢(1+ )2 $x ¢(1+ )1 $x ¢(1+ ) $x ¢(1+ )n-1¢ $x ¢(1+ )n-1 $x ¢(1+ )n $x ¢ $x r n r n r n 2 n … ……………… … ………………...… …………........ r n n n

  30. Time Value of Money We’ve seen the EOY balance increases with n… but how big can it get? This is the same as asking what is the: where e is a mathematical constant (e¼2.71828). This is called “continuous compounding.” r n = er lim n!1 (1+ )n

  31. Time Value of Money Returning to our example, $100 invested at 12% compounded continuously for one year becomes: Note that: • n = 1 ! $100 becomes $112 • n = 2 ! $100 becomes $112.36 • n = 4 ! $100 becomes $112.55 • n = 10 ! $100 becomes $112.67 • n = 25 ! $100 becomes $112.72 • n = 100 ! $100 becomes $112.74 • n = 250 ! $100 becomes $112.75 $100¢e0.12¼$112.75

  32. Time Value of Money More generally, the future value t years from now of receiving $x now at a continuously compounded interest rate r is: And the present value of receiving $x on a date t years from now, discounting at a continuously compounded interest rate r is: x ¢ ert x ert = x ¢ e-rt

  33. Time Value of Money We use continuous compounding/discounting to compute PV of future cash flows in the same way we did before. Example: What is the most you should pay for an asset that pays $100 in 6 months, $300 in 1 year, and $500 in 2 years, if the interest rate is 10% per year compounded continuously?

  34. Time Value of Money Present value of cash flows for each year: $100 e0.10*0.50 Year 0.5’s payment: ¼ $95.12 $300 e0.10*1.0 $500 e0.10*2 Year 1’s payment: ¼ $271.45 Year 2’s payment: ¼ $409.37 ¼ $776.04 total The most you should be willing to pay is $776.04

  35. Time Value of Money Earlier, the same future stream of cash flows with annual compounding gave a present value of $781.30. Changing only the compounding rate, the present value decreased to $776.04. Why is this?

  36. Time Value of Money So far, we have used the interest rate to compute present value. However, we could instead do the opposite – calculate the interest rate implied by the present value of a future stream of cash flows – and, with fixed income securities, we often will. Example: What is the interest rate implied by an asset that pays $500 in 3 years, $750 in 5 years, and $1,000 in 10 years, and has a PV = $1,100?

  37. Time Value of Money We calculate the implied interest rate as follows: $500 (1+r)3 $750 (1+r)5 $1,000 (1+r)10 $1,000 (1+r)10 $750 (1+r)5 $500 (1+r)3 PV = + + + + ) $1,100 = ) (via trial and error) that r ¼7.14% The implied interest rate with bonds is called the “yield” – very important concept in fixed income.

  38. Time Value of Money One more example… Example: What is the 6-month interest rate implied by an asset that pays $50 in 6 months, $50 in 1 year, and $50 in 18 months, and both $50 and $1,000 in 2 years, and has a PV = $950?

  39. Time Value of Money Here, r represents the 6-month rate: $50 (1+r)2 $1,050 (1+r)4 $50 (1+r)1 $50 (1+r)4 $50 (1+r)3 $50 (1+r)3 $1,000 (1+r)4 $50 (1+r)2 $50 (1+r)1 PV = + + + + + + + ) $950 = ) (via trial and error) that r ¼6.46% What would the implied 6-month interest rate be if the present value increased to $1,000?

  40. Fixed Income Securities What is fixed income? • A method of borrowing money where the return to the lender is fixed. • Return is made up of two components: • Principal = Original amount of loan • Interest = Additional compensation to lender • This exchange is a liability to the borrower and an asset to the lender. • Risks can exist along many dimensions – one main one is “default risk”

  41. Fixed Income Securities Default-Free Securities • May have other risk but default risk is absent… lender 100% assured of receiving payment as specified in the contract • No real world examples of pure default-free securities • Closest example – US government debt • Never defaulted on any debt in 220 years of borrowing • Lenders operate assuming repayment is guaranteed

  42. Fixed Income Securities Main types of U.S. Treasury Securities • U.S. Treasury Bills – orig. maturity of under one year • U.S. Treasury Notes – orig. maturity of 1 to 10 years • U.S. Treasury Bonds – orig. maturity of over 10 years U.S. Treasury Bills • Represent about 20% of all debt held by public • Discount Securities – one payment to owner

  43. Fixed Income Securities U.S. Treasury Notes and Bonds • Represent about 70% of all debt held by public • Coupon Securities – interest payments made every 6 months until maturity, on which interest + principal is paid

  44. The End

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