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Economic System Analysis. January 15, 2002 Prof. Yannis A. Korilis. Topics. Compound Interest Factors Arithmetic Series Geometric Series Discounted Cash Flows Examples: Calculating Time-Value Equivalences Continuous Compounding.

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## Economic System Analysis

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**Economic System Analysis**January 15, 2002 Prof. Yannis A. Korilis**Topics**• Compound Interest Factors • Arithmetic Series • Geometric Series • Discounted Cash Flows • Examples: Calculating Time-Value Equivalences • Continuous Compounding**An amount P is invested today and earns interest i% per**period. What will be its worth after N periods? Compound amount factor (F/P, i%, N) Formula: Proof: Compound Amount Factor F=? P 0 1 2 N**What amount P if invested today at interest i%, will worth F**after N periods? Present worth factor (P/F, i%, N) Formula: Proof: Present Worth Factor F P=? 0 1 2 N**Annuity: Uniform series of equal (end-of-period) payments A**What is the amount A of each payment so that after N periods a worth F is accumulated? Sinking fund factor(A/F, i%, N) Formula Proof: Sinking Fund Factor 0 1 2 N**Uniform series of payments A at i%. What worth F is**accumulated after N periods? Series compound amount factor(F/A, i%, N) Formula Proof: Series Compound Amount Factor 0 1 2 N**What is the amount A of each future annuity payment so that**a present loan P at i% is repaid after N periods? Capital recovery factor(A/P, i%, N) Formula Proof: Capital Recovery Factor 0 1 2 N • EZ Proof:**What is the present worth P of a series of N equal payments**A at interest i%? Series present worth factor(P/A, i%, N) Formula Proof: Series Present Worth Factor 0 1 2 N**Series increases (or decreases) by a constant amount G each**period Convert to equivalent annuity A (A/G, i%, N): Arithmetic series conversion factor Formula: G 0 1 2 3 4 N Arithmetic Series 0 1 2 3 4 N**Proof of Arithmetic Series Conversion**• Future worth • Multiplying by (i+1):**Growing Annuity: series of payments that grows at a fixed**rate g Series present value: Growing Perpetuity: infinite series of payments that grows at a fixed rate g Series present value: . . . 0 1 2 3 4 0 1 2 3 4 Growing Annuity and Perpetuity**Payment at end of period n:**PV of payment at end of period n (discounted at rate i): Growing Annuity PV: Growing Perpetuity PV: Results follow using: For g=i, proof is easy Growing Annuity and Perpetuity: Proofs**When Marilyn Monroe died, ex-husband Joe DiMaggio vowed to**place fresh flowers on her grave every Sunday as long as he lived. Assume that he lived 30 yrs. Cost of flowers in 1962, $5. Interest rate 10.4% compounded weekly. Inflation rate 3.9% compounded weekly. 1 yr = 52 weeks. What is the PV of the commitment Not accounting for inflation Considering inflation Example: Effects of Inflation**Examples on Discounted Cash Flows**Goal: • Develop skills to evaluate economic alternatives • Familiarity with interest factors • Practice the use of cash flow diagrams • Put cash-flow problems in a realistic setting**0 1 2 3 4 5**Cash Flow Diagrams • Clarify the equivalence of various payments and/or incomes made at various times • Horizontal axis: times • Vertical lines: cash flows**0 9**Ex. 2.3: Unknown Interest Rate At what annual interest rate will $1000 invested today be worth $2000 in 9 yrs?**0 (Quarters) N**Ex. 2.4: Unknown Number of Interest Periods Loan of $1000 at interest rate 8% compounded quarterly. When repaid: $1400. When was the loan repaid?**01 03 05 07 09**Ex. 2.5: More Compounding Periods than Payments Now is Feb. 1, 2001. 3 payments of $500 each are to be received every 2 yrs starting 2 yrs from now. Deposited at interest 7% annually. How large is the account on Feb 1, 2009?**Cost of a machine: $8065. It can reduce production costs**annually by $2020. It operates for 5 yrs, at which time it will have no resale value. What rate of return will be earned on the investment? Need to solve the equation for i. numerically; using tables for (P|A,i,N); using program provided with textbook. Question: What if the company could invest at interest rate 10% annually? 0 1 2 3 4 5 Ex. 2.6: Annuity with Unknown Interest**Definition: series of payments made at the beginning of each**period Treatment: First payment translated separately Remaining as an ordinary annuity 0 1 2 4 6 8 10 12 14 Annuity Due**0 1 2 4 6 8 10 12 14**Example 2.7: Annuity Due What is the present worth of a series of 15 payments, when first is due today and the interest rate is 5%?**2001 06 11**Deferred Annuity • Definition: series of payments, that begins on some date later than the end of the first period • Treatment: • Number of payment periods • Deferred period • Find present worth of the ordinary annuity, and • Discount this value through the deferred period**With interest rate 6%, what is the worth on Feb. 1, 2001 of**a series of payments of $317.70 each, made on Feb. 1, from 2007 through 2011? 2001 06 11 Ex. 2.8: Deferred Annuity**G=$1500**0 1 2 3 4 5 6 7 8 Ex. 2.9: Present Worth of Arithmetic Gradient Lease of storage facility at $20,000/yr increasing annually by $1500 for 8 yrs. EOY payments starting in 1 yr. Interest 7%. What lump sum paid today would be equivalent to this lease payment plan? [Can be compared to present cost for expanding existing facility]**Convert increasing series to a uniform**Find present value of annuity Ex. 2.9: Present Worth of Arithmetic Gradient**Boy 11 yrs old. For college education, $3000/yr, at age 19,**20, 21, 22. Gift of $4000 received at age 5 and invested in bonds bearing interest 4% compounded semiannually Gift reinvested Annual investments at ages 12 through 18 by parents If all future investments earn 6% annually, how much should the parents invest? 5 11 12 15 18 20 22 Example: Income and Outlay**5 11 12 15**18 20 22 Income and Outlay • Evaluation date: 18th birthday • P18: present worth of education annuity • F18: future worth of gift • F=P18-F18 • F will be provided by a series of 7 payments of amount A beginning on 12th birthday**5 11 12 15**18 20 22 5 11 12 15 18 20 22 5 11 12 15 18 20 22 Income and Outlay

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