Goals. The Time Value of Money. First example. Compounding. Economics 71a Spring 2007 Mayo, Chapter 7 Lecture notes 3.1

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1 Goals The Time Value of Money Economics 7a Spring 2007 Mayo, Chapter 7 Lecture notes 3. More applications Compounding PV = present or starting value FV = future value R = interest rate n = number of periods First example PV = 000 R = 0% n = FV =? FV = 000*(.0) =,00

2 PV = 000 R = 0% n = 3 FV =? Example 2 Compound Interest FV = 000*(.)*(.)*(.) =,33 FV = PV*(+R)^n Example 3: The magic of compounding PV = R = 6% n = 50 FV =? > FV = PV*(+R)^n = 8 > n = 00, FV = 339 > n = 200, FV = 5,000 Example 4: Doubling times Doubling time = time for funds to double FV = 2 = ( + R)n PV log(2) = nlog( + R) n = log(2) log( + R) Example 5 Retirement Saving PV = 000, age = 20, n =45 R = 0.05 > FV = PV*(+0.05)^45 = 8985 > Doubling 4 R = 0.07 > FV=PV*(+0.07)^45 = 2,002 > Doubling = 0 Small change in R, big impact 2

3 Retirement Savings at 5% interest Goals More applications Present Value Go in the other direction Know FV Get PV PV = FV ( + R) n Example Given a zero coupon bond paying \$000 in 5 years How much is it worth today? R = 0.05 PV = 000/(.05)^5 = \$784 This is the amount that could be stashed away to give 000 in 5 years time Answer basic questions like what is \$00 tomorrow worth today 3

4 Goals More applications Annuity Equal payments over several years > Usually annual Types: Ordinary/Annuity due > Beginning versus end of period Present Value of an Annuity Annuity pays \$00 a year for the next 0 years (starting in year) What is the present value of this? R = PV = = 772 ( + R) i Future Value of An Annuity Annuity pays \$00 a year for the next 0 years (starting in year) What is the future value of this at year 0? R = 0.05 FV = 9 i=0 00(.05) i 4

5 Why the Funny Summation? Period 0 value for each > Period 0: 00 > Period 9: 00(.05) > Period 8: 00(.05)(.05) > > Period : 00(.05)^9 Be careful Application: Lotteries Choices > \$6 million today > \$33 million over 33 years ( per year) R = 7% PV = 33 ( ) i PV=\$2.75 million, take the \$6 million today Another Way to View An Annuity Annuity of \$00 > Paid year, 2 year, 3 years from now Interest = 5% PV = 00/(.05) + 00/(.05)^2 + 00/(.05)^3 = Cost to Generate From Today Think about putting money in the bank in 3 bundles One way to generate each of the three \$00 payments How much should each amount be? > 00 = FV = PV*(.05)^n (n =, 2, 3) > PV = 00/(.05)^n (n =, 2, 3) The sum of these values is how much money you would have to put into bank accounts today to generate the annuity Since this is the same thing as the annuity it should have the same price (value) 5

6 Perpetuity Discounting to infinity This is an annuity with an infinite life Math review: s = a i " as = " a i+ = " a i i=2 s # as = a s = a # a Present Value of a Constant Stream a = + R y PV = "( + R) i a = + R i = PV = y a i " i = PV = a # a y = ( + R) (y) = # ( + R) ( + R) + R + R # ( + R) = y R Perpetuity Examples and Interest Rate Sensitivity Interest rate sensitivity > y=00 > R = 0.05, PV = 2000 > R = 0.03, PV =

7 Goals More applications Mixed Stream Apartment Building Pays \$500 rent in year Pays \$000 rent 2 years from now Then sell for 00,000 3 years from now R = 0.05 PV = (.05) = 87, (.05) Mixed Stream Investment Project Pays -000 today Then 00 per year for 5 years R = PV = " = 38 (.05) i Implement project since PV>0 Technique = Net present value (NPV) 5 Goals More applications 7

8 Term Structure We have assumed that R is constant over time In real life it may be different over different horizons (maturities) Remember: Term structure Use correct R to discount different horizons Term Structure y PV = ( + R ) + y 2 ( + R 2 ) + y 3 2 ( + R 3 ) 3 Discounting payments, 2, 3 years from now Goals More examples Frequency and compounding APR=Annual percentage rate Usual quote: > 6% APR with monthly compounding What does this mean? > R = (/2)6% every month That comes out to be > (+.06/2)^2- > 6.7% Effective annual rate 8

9 General Formulas Effective annual rate (EFF) formula Limit as m goes to infinity APREFFe= EFF = e APR For APR = 0.06 limit EFF = EFF = ( + APR m )m Goals More examples More Examples Home mortgage Car loans College Calculating present values Specifications: Home Mortgage Amortization > \$00,000 mortgage > 9% interest > 3 years (equal payments) pmt Find pmt > PV(pmt) = \$00,000 9

10 Find PMT so that Solve for PMT > PMT = 39,504 Mortgage PV 3 PMT PV = =00000 (.09) i 3 PMT =00000 (.09) i Amount = \$,000 Year Car Loan > Payments in months -2 2% APR (monthly compounding) 2%/2=% per month PMT? Car Loan Again solve, for PMT 2 PMT PV = =000 (.0) i 2 PMT =000 (.0) i Total Payment 2*88.85 =, Looks like 6.6% interest Why? > Paying loan off over time PMT =

11 Payments and Principal How much principal remains after month? > You owe (+0.0)000 = 00 > Payment = > Remaining = = 92.5 How much principal remains after 2 months? > (+0.0)*92.5 = > Remaining = = 84.5 College Should you go?. Compare PV(wage with college)-pv(tuition) PV(wage without college) 2. What about student loans? 3. Replace PV(tuition) with PV(student loan payments) Note: Some of these things are hard to estimate Second note: Most studies show that the answer to this question is yes Calculating Present Values Sometimes difficult Methods > Tables (see textbook) > Financial calculator (see book again) > Excel spreadsheets (see book web page) > Java tools (we ll use these sometimes) > Other software (matlab ) Discounting and Time: Summary Powerful tool Useful for day to day problems > Loans/mortgages > Retirement We will use it for > Stock pricing > Bond pricing

12 Goals More examples 2

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