Economic System Analysis January 15, 2002 Prof. Yannis A. Korilis

2-2 Topics Compound Interest Factors Arithmetic Series Geometric Series Discounted Cash Flows Examples: Calculating Time-Value Equivalences Continuous Compounding

2-3 Compound Amount Factor 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: N P F=?

2-4 Present Worth Factor What amount P if invested today at interest i%, will worth F after N periods? Present worth factor (P/F, i%, N) Formula: Proof: N P=? F

2-5 Sinking Fund Factor 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: N

2-6 Series Compound Amount Factor 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: N

2-7 Capital Recovery Factor 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: N EZ Proof:

2-8 Series Present Worth Factor 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: N

2-9 Arithmetic Series 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 N

2-10 Proof of Arithmetic Series Conversion Future worth Multiplying by (i+1):

2-11 Growing Annuity and Perpetuity 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:

2-12 Growing Annuity and Perpetuity: Proofs 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

2-13 Example: Effects of Inflation 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 1. Not accounting for inflation 2. Considering inflation

2-14 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

2-15 Cash Flow Diagrams Clarify the equivalence of various payments and/or incomes made at various times Horizontal axis: times Vertical lines: cash flows

2-16 Ex. 2.3: Unknown Interest Rate At what annual interest rate will $1000 invested today be worth $2000 in 9 yrs? 0 9

2-17 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? 0 (Quarters) N

2-18 Ex. 2.5: More Compounding Periods than Payments Now is Feb. 1, 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?

2-19 Ex. 2.6: Annuity with Unknown Interest 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?

2-20 Annuity Due Definition: series of payments made at the beginning of each period Treatment: 1. First payment translated separately 2. Remaining as an ordinary annuity

2-21 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%?

2-22 Deferred Annuity Definition: series of payments, that begins on some date later than the end of the first period Treatment: 1. Number of payment periods 2. Deferred period 3. Find present worth of the ordinary annuity, and 4. Discount this value through the deferred period

2-23 Ex. 2.8: Deferred Annuity With interest rate 6%, what is the worth on Feb. 1, 2001 of a series of payments of $ each, made on Feb. 1, from 2007 through 2011?

2-24 Ex. 2.9: Present Worth of Arithmetic Gradient Lease of storage facility at $20,000/yr increasing annually by $1500 for 8 yrs. EOY payments starting in 1 yr. Interest 7%. What lump sum paid today would be equivalent to this lease payment plan? [Can be compared to present cost for expanding existing facility] G=$

2-25 Ex. 2.9: Present Worth of Arithmetic Gradient Convert increasing series to a uniform Find present value of annuity

2-26 Example: Income and Outlay Boy 11 yrs old. For college education, $3000/yr, at age 19, 20, 21, Gift of $4000 received at age 5 and invested in bonds bearing interest 4% compounded semiannually 2. Gift reinvested 3. Annual investments at ages 12 through 18 by parents If all future investments earn 6% annually, how much should the parents invest?

2-27 Income and Outlay Evaluation date: 18 th birthday P 18 : present worth of education annuity F 18 : future worth of gift F=P 18 -F 18 F will be provided by a series of 7 payments of amount A beginning on 12 th birthday

2-28 Income and Outlay