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The Time Value of Money Economics 71a Spring 2007 Mayo, Chapter 7 Lecture notes 3.1

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Goals Compounding and Future Values Present Value Valuing an income stream >Annuities >Perpetuities Mixed streams Term structure again Compounding More applications

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Compounding PV = present or starting value FV = future value R = interest rate n = number of periods

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First example PV = 1000 R = 10% n = 1 FV = ? FV = 1000*(1.10) = 1,100

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Example 2 Compound Interest PV = 1000 R = 10% n = 3 FV = ? FV = 1000*(1.1)*(1.1)*(1.1) = 1,331 FV = PV*(1+R)^n

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Example 3: The magic of compounding PV = 1 R = 6% n = 50 FV = ? >FV = PV*(1+R)^n = 18 >n = 100, FV = 339 >n = 200, FV = 115,000

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Example 4: Doubling times Doubling time = time for funds to double

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Example 5 Retirement Saving PV = 1000, age = 20, n =45 R = 0.05 >FV = PV*(1+0.05)^45 = 8985 >Doubling 14 R = 0.07 >FV=PV*(1+0.07)^45 = 21,002 >Doubling = 10 Small change in R, big impact

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Retirement Savings at 5% interest

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Goals Compounding and Future Values Present Value Valuing an income stream >Annuities >Perpetuities Mixed streams Term structure again Compounding More applications

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Present Value Go in the other direction Know FV Get PV Answer basic questions like what is $100 tomorrow worth today

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Example Given a zero coupon bond paying $1000 in 5 years How much is it worth today? R = 0.05 PV = 1000/(1.05)^5 = $784 This is the amount that could be stashed away to give 1000 in 5 years time

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Goals Compounding and Future Values Present Value Valuing an income stream >Annuities >Perpetuities Mixed streams Term structure again Compounding More applications

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Annuity Equal payments over several years >Usually annual Types: Ordinary/Annuity due >Beginning versus end of period

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Present Value of an Annuity Annuity pays $100 a year for the next 10 years (starting in 1 year) What is the present value of this? R = 0.05

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Future Value of An Annuity Annuity pays $100 a year for the next 10 years (starting in 1 year) What is the future value of this at year 10? R = 0.05

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Why the Funny Summation? Period 10 value for each >Period 10: 100 >Period 9: 100(1.05) >Period 8: 100(1.05)(1.05) >… >Period 1: 100(1.05)^9 Be careful!

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Application: Lotteries Choices >$16 million today >$33 million over 33 years (1 per year) R = 7% PV=$12.75 million, take the $16 million today

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Another Way to View An Annuity Annuity of $100 >Paid 1 year, 2 year, 3 years from now Interest = 5% PV = 100/(1.05) + 100/(1.05)^2 + 100/(1.05)^3 = 272.32

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Cost to Generate From Today Think about putting money in the bank in 3 bundles One way to generate each of the three $100 payments How much should each amount be? >100 = FV = PV*(1.05)^n (n = 1, 2, 3) >PV = 100/(1.05)^n (n = 1, 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)

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Perpetuity This is an annuity with an infinite life

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Discounting to infinity Math review:

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Present Value of a Constant Stream

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Perpetuity Examples and Interest Rate Sensitivity Interest rate sensitivity >y=100 >R = 0.05, PV = 2000 >R = 0.03, PV = 3333

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Goals Compounding and Future Values Present Value Valuing an income stream >Annuities >Perpetuities Mixed streams Term structure again Compounding More applications

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Mixed Stream Apartment Building Pays $500 rent in 1 year Pays $1000 rent 2 years from now Then sell for 100,000 3 years from now R = 0.05

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Mixed Stream Investment Project Pays -1000 today Then 100 per year for 15 years R = 0.05 Implement project since PV>0 Technique = Net present value (NPV)

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Goals Compounding and Future Values Present Value Valuing an income stream >Annuities >Perpetuities Mixed streams Term structure again Compounding More applications

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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

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Term Structure Discounting payments 1, 2, 3 years from now

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Goals Compounding and Future Values Present Value Valuing an income stream >Annuities >Perpetuities Mixed streams Term structure again Compounding More examples

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Frequency and compounding APR=Annual percentage rate Usual quote: >6% APR with monthly compounding What does this mean? >R = (1/12)6% every month That comes out to be >(1+.06/12)^12-1 >6.17% Effective annual rate

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General Formulas Effective annual rate (EFF) formula Limit as m goes to infinity For APR = 0.06 limit EFF = 0.0618

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Goals Compounding and Future Values Present Value Valuing an income stream >Annuities >Perpetuities Mixed streams Term structure again Compounding More examples

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More Examples Home mortgage Car loans College Calculating present values

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Home Mortgage Amortization Specifications: >$100,000 mortgage >9% interest >3 years (equal payments) pmt Find pmt >PV(pmt) = $100,000

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Mortgage PV Find PMT so that Solve for PMT >PMT = 39,504

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Car Loan Amount = $1,000 1 Year >Payments in months 1-12 12% APR (monthly compounding) 12%/12=1% per month PMT?

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Car Loan Again solve, for PMT PMT = 88.85

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Total Payment 12*88.85 = 1,066.20 Looks like 6.6% interest Why? >Paying loan off over time

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Payments and Principal How much principal remains after 1 month? >You owe (1+0.01)1000 = 1010 >Payment = 88.85 >Remaining = 1010 – 88.85 = 921.15 How much principal remains after 2 months? >(1+0.01)*921.15 = 930.36 >Remaining = 930.36 – 88.85 = 841.51

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College Should you go? 1. 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

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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…)

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Discounting and Time: Summary Powerful tool Useful for day to day problems >Loans/mortgages >Retirement We will use it for >Stock pricing >Bond pricing

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Goals Compounding and Future Values Present Value Valuing an income stream >Annuities >Perpetuities Mixed streams Term structure again Compounding More examples

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