Economics 434 Theory of Financial Markets

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

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

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?

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

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) )

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)

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?

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

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?

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

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?

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

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….

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

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.

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 r n 1 n • $x • $x ¢(1+ )n-1 • $x ¢(1+ )n-1¢ • $x ¢(1+ )2 • $x ¢(1+ )¢ • $x ¢(1+ )1 • $x ¢(1+ )n • $x ¢(1+ ) • $x ¢ r n 2 n r n r n … ……………… … ………………...… …………........ r n n n

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

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

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

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?

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

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?

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?

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.

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?

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-monthinterest rate be if the present value increased to $1,000?

Economics 434 Theory of Financial Markets