University:
Michigan State University
Course:
EC 301 | Intermediate Microeconomics
Academic year:

2021
Views:

251

Pages:

35
Author:

Quintin Ware

Chapter 4: The Time Value of Money Simple interest 𝐼𝐼 = 𝑃𝑃 × 𝑁𝑁 × 𝑖𝑖 𝐼𝐼: Total simple interest paid or earned. 𝑃𝑃: Principal amount lent or borrowed. 𝑁𝑁: Number of interest periods (e.g., years). 𝑖𝑖: Interest rate per interest period. Example: $1,000 loan for 3 years at a simple interest rate of 10% per year. 𝑃𝑃 = Principal = $1,000. 𝑁𝑁 = Number of interest periods = 3 years. 𝑖𝑖: Interest rate per interest period = 10% per year. The total interest paid = 𝐼𝐼 = $1000 × 10% × 3 years = $300. The total amount repaid at the end of the loan period = principal (𝑃𝑃) + interest (𝐼𝐼) = $1000 + $300 = $1300. Example You borrowed $5,000 at a simple interest rate = 0.5% per month to be repaid after 4 years. How much will you pay back? or what is the future equivalent of the borrowed $5,000? 𝑃𝑃 = Principal = $5,000. 𝑁𝑁 = Number of interest periods = 4 years. 𝑖𝑖: Interest rate per interest period = 0.5% per month × 12 months/year = 6% per year. The total interest paid = 𝐼𝐼 = $5,000 × 6% × 4 years = $1,200. The total amount repaid (or future equivalent) = $5,000 + $1,200 = $6,200. Compound interest  Interest is based on the remaining principal + accumulated interest. Example: $1,000 loan for 3 years at a compound interest rate of 10% per year. = amount owed at beginning of period × interest rate = $1,000 × 10% Compounding period (e.g., years, months) Amount owed at end of previous period Period 1 2 3 = amount owed at beginning of period + interest for period = $1,000 + $100 Amount owed at Interest amount Amount owed at beginning of period for period end of period $1,000 $1,100 $1,210 $100 $110 $121 $1,100 $1,210 $1,331 Simple vs. compound interest Economic Equivalence Cash-flow diagrams Upward arrows: receipts (positive cash flow or cash inflow) 0 Downward arrows: expenses (negative cash flow or cash outflow) 1 2 N-1 N Time scale (moving from left to right) Direction of the arrows depends on the point of view (lender vs. borrower) Example An investment of $10,000 will produce a uniform annual revenue of $5,000 for 5 years and have a market (recovery) value of $2,000 at the end of year (EOY) five. Annual operating and maintenance expenses are estimated at $3,000 at the end of each year. Draw a cash-flow diagram from the corporation’s view point. Market recovery ($2,000) Annual revenue ($5,000 per year) 0 1 2 3 4 5 Annual operating and maintenance ($3,000 per year) Capital investment ($10,000) Net cash-flow? Cash flow diagrams Notation 𝑖𝑖: effective interest rate per interest period. 𝑁𝑁: number of compounding interest periods (e.g., years). 𝑃𝑃: present sum of money (or the equivalent sum of one or more cash flows at present time). 𝐹𝐹: future sum of money (or the equivalent sum of one or more cash flows at future time). 𝐴𝐴: end-of-period cash flow (or equivalent end-of-period value) in a uniform series starting at the end of first period and continuing through the last period. Beginning of period 1 (year 1, month 1, or any other compounding period) 𝐹𝐹 𝐴𝐴 0 1 𝑃𝑃 2 End of period 1 N-1 N Relating present and future equivalent values • For a single cash flow and using the compound interest rate formula. or or 𝐹𝐹 = 𝑃𝑃 1 + 𝑖𝑖 𝑁𝑁 𝐹𝐹 = 𝑃𝑃 𝐹𝐹 ⁄𝑃𝑃 , 𝑖𝑖𝑖, 𝑁𝑁 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑖𝑖𝑖𝑖 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝐶𝐶 𝑃𝑃 = 𝐹𝐹 1 + 𝑖𝑖 −𝑁𝑁 𝑃𝑃 = 𝐹𝐹 𝑃𝑃⁄𝐹𝐹 , 𝑖𝑖𝑖, 𝑁𝑁 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑖𝑖𝑖𝑖 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝐶𝐶 Example $1,000 loan for 3 years at a compound interest rate of 10% per year. How much will be repaid? 𝑃𝑃 = $1,000 𝐹𝐹 = ? 𝑁𝑁 = 3 years 𝑖𝑖 = 10% 𝐹𝐹 = 1,000 1 + 0.1 3 = $1,331 or Go to Appendix C, 𝑖𝑖 = 10% page to find 𝐹𝐹 ⁄𝑃𝑃 , 10%, 3 𝐹𝐹 = 1,000 × 1.331 = $1,331 Example You need $10,000 after five years so you decided to save money now. How much do you need to deposit now in the bank given that the interest rate is 5% per year? 𝐹𝐹 = $10,000 𝑁𝑁 = 5 years 𝑖𝑖 = 5% 𝑃𝑃 = ? 𝑃𝑃 = 10,000 1 + 0.05 −5 = $7,835.26 or Go to Appendix C, 𝑖𝑖 = 5% page to find 𝑃𝑃⁄𝐹𝐹 , 5%, 5 P = 10,000 × 0.7835 = $7,835 Finding 𝒊𝒊 given 𝐹𝐹, 𝑃𝑃, and 𝑵𝑵 𝑖𝑖 = 𝑁𝑁 𝐹𝐹/𝑃𝑃 − 1 Finding 𝑁𝑁 given 𝐹𝐹, 𝑃𝑃, and 𝑖𝑖 log(𝐹𝐹 ⁄𝑃𝑃) 𝑁𝑁 = log(1 + 𝑖𝑖) Example: How many years does it take to double my money at an interest rate of 5% per year? 𝐹𝐹/𝑃𝑃 = 2 𝑁𝑁 = ? 𝑖𝑖 = 5% log(2) 𝑁𝑁 = = 14.2 years log(1 + 0.05) (to use Appendix C tables, you need interpolation). Example: What is the interest rate that will double an investment of $50,000 in 10 years? 𝑃𝑃 = $50,000 𝑖𝑖 = ? 𝑖𝑖 = 𝐹𝐹 = $100,000 10 𝑁𝑁 = 10 years 100,000/50,000 − 1 = 0.0718 = 7.18% (to use Appendix C tables, you need interpolation). Uniform series (annuity) to present and future 𝐴𝐴: series of uniform (equal) payments occurring at the end of each period for 𝑁𝑁 periods … also called annuity. 𝐹𝐹 Present equivalent value (a single value at time 0) 𝐴𝐴 0 1 𝑃𝑃 2 Annuity starts one interest period after the present equivalent Example: repaying a loan in uniform monthly payments. N-1 N Future equivalent value (occurs at the same time as the last A, or N periods after P) Finding 𝑭𝑭 given 𝑨𝑨 Or 1 + 𝑖𝑖 𝑁𝑁 − 1 𝐹𝐹 = 𝐴𝐴 𝑖𝑖 𝐹𝐹 = 𝐴𝐴 𝐹𝐹 ⁄𝐴𝐴 , 𝑖𝑖𝑖, 𝑁𝑁 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑖𝑖𝑖𝑖 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝐶𝐶 Finding 𝑷𝑷 given 𝑨𝑨 Or 1 + 𝑖𝑖 𝑁𝑁 − 1 𝑃𝑃 = 𝐴𝐴 𝑖𝑖 1 + 𝑖𝑖 𝑁𝑁 𝑃𝑃 = 𝐴𝐴 𝑃𝑃⁄𝐴𝐴 , 𝑖𝑖𝑖, 𝑁𝑁 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑖𝑖𝑖𝑖 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝐶𝐶 • A given F Or • A given P Or Finding 𝑨𝑨 given 𝑷𝑷 or 𝑭𝑭 𝑖𝑖 𝐴𝐴 = 𝐹𝐹 1 + 𝑖𝑖 𝑁𝑁 − 1 𝐴𝐴 = 𝐹𝐹 𝐴𝐴⁄𝐹𝐹 , 𝑖𝑖𝑖, 𝑁𝑁 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑖𝑖𝑖𝑖 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝐶𝐶 𝑖𝑖 1 + 𝑖𝑖 𝑁𝑁 𝐴𝐴 = 𝑃𝑃 1 + 𝑖𝑖 𝑁𝑁 − 1 𝐴𝐴 = 𝑃𝑃 𝐴𝐴⁄𝑃𝑃 , 𝑖𝑖𝑖, 𝑁𝑁 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑖𝑖𝑖𝑖 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝐶𝐶 Example: You took a loan which is to be repaid in uniform payments over 4 years. Assuming the interest rate is 1% per month, and your monthly payment is $300. What is the principal amount (the amount of money borrowed)? 𝐴𝐴= $300 𝑖𝑖 = 1% per month 𝑁𝑁 = 4 years × 12 months/year = 48 months *** the period 𝑁𝑁 should be consistent with the interest rate (interest per month  period in months) 𝑃𝑃 = ? 1 + 0.01 48 − 1 𝑃𝑃 = 300 = $11,392.2 48 0.01 1 + 0.01 Or from Appendix C tables, 𝑃𝑃⁄𝐴𝐴 , 1%, 48 = 37.9740  𝑃𝑃 = $300 × 37.9740 = Example: How much will you have in 40 years if you invest $3,000 of your income each year in a project that earns 8% per year? 𝐴𝐴= $3,000 𝐹𝐹 = ? 𝑖𝑖 = 8% 𝑁𝑁 = 40 years 1 + 0.08 40 − 1 𝐹𝐹 = 3,000 = $777,169.6 0.08 Or from Appendix C tables, 𝐹𝐹 ⁄𝐴𝐴 , 8%, 40 = 259.0565  𝐹𝐹 = $3,000 × 259.0565 = $777,169.5 Example Calculate the compounded future value of 20 annual payments of $5,000 each into a savings account that earns 6% per year. All 20 payments are made at the beginning of each year. - Definition of annuity: occurs at the end of each compounding period. - In the example, payments are made at the beginning of each period. 𝐹𝐹 = $5,000 𝐹𝐹 ⁄𝐴𝐴 , 6%, 20 𝐹𝐹 ⁄𝑃𝑃 , 6%, 1 = $5,000 × 36.7856 × 1.06 = $194,963.68 Payments start at the beginning of each year, so the first annuity is at time 0. Hence, the present equivalent is at year -1 and the future equivalent is at year 19. We first use the (F/A) relationship to determine the future equivalent at year 19 and then we determine the future equivalent at year 20 using the (F/P) relationship. Another way to solve: 𝐹𝐹 = $5,000 𝐹𝐹 ⁄𝑃𝑃 , 6%, 20 + $5000 𝐹𝐹 ⁄𝐴𝐴 , 6%, 19 𝐹𝐹 ⁄𝑃𝑃 , 6%, 1 𝐹𝐹 = $5,000 × 3.2071 + $5000 × 33.7600 × 1.06 Example A loan of $10,000 is to be repaid in 4 equal payments (over 4 years) and the interest rate is 10% per year. Determine the interest paid and principal repayment every year.  Find the annual payment (annuity) 𝐴𝐴 = 𝑃𝑃 𝐴𝐴⁄𝑃𝑃 , 10%, 4 = $10,000 × 0.3155 = $3,155 per year.  Fill out a table Loan amount Year Amount owed at beginning of period Interest 1 2 3 4 $10,000 $7,845 $5,475 $2,867 $1,000 $785 $547 $287 = amount owed at beginning of previous year – principal repayment in previous year Annual payment Principal repayment $3,155 $3,155 $3,155 $3,155 = amount owed at beginning of period × interest rate $2,155 $2,371 $2,608 $2,868 = annual payment – interest Solving for 𝑵𝑵 • You borrowed $100,000 at an interest rate of 7% per year. If the annual payment is $8,000, how many years does it take to repay the loan? • You invested $20,000 in a project and you are expected to gain $4,000 annually. At a 10% interest rate, when will you recover your investment? Solving for 𝒊𝒊 Deferred annuities - Ordinary annuity (uniform series) appears at the end of the first period. - Deferred annuity (also uniform series) begins at later time. Annuity is deferred for J periods Deferred annuities - example How much money should be deposited each year for 12 years if you wish to withdraw $309 each year for five years, beginning at the end of the 14th year? Assume the interest rate is 8% per year. 0 1 $309 12 14 𝐴𝐴? 15 18 Uniform (arithmetic) gradient of cash flows - Cash flow that changes by a constant amount (G) each period. Examples At a 15% interest rate, determine the present equivalent for the following cash flow. At a 15% interest rate, determine the present equivalent for the following cash flow. Geometric sequence of cash flows _ - Cash flow that changes by a constant rate (𝑓𝑓 ) each period. - First payment at EOY 1. Example: Assume that a payment of $1,000 is made at EOY 1 and decreases by 20% per year after the first year for 4 years. At a 25% interest rate, Determine the present equivalent. Nominal and effective interest rates - If compounding period is less than a year. - Annual rate is called nominal interest rate or annual percentage rate (APR). - Actual or exact rate is called effective interest rate. - Example: if annual interest rate is 10% compounded annually, then the effective rate = nominal rate = 10%. Where: 𝑖𝑖: effective interest rate per year. 𝑟𝑟: nominal interest rate per year. 𝑀𝑀: number of compounding periods per year. Example A credit card company charges 1.375% per month on the unpaid balance. They claim that the annual interest rate is (12 × 1.375% = 16.5%). - What is the effective interest rate per month? Since compounding is monthly, effective monthly rate = nominal monthly rate = 1.375%. - What is the effective interest rate per year? 𝑟𝑟 = 16.5% 𝑀𝑀 = 12 compounding periods per year - Does this card provide a better deal than another card which charges 16.8% annual rate compounded bimonthly? 17.81% < 18.02% ⇒ the first card (16.5% per year compounded monthly) is better. Example A loan of $15,000 requires monthly payments of $477 over a 36month period. - What is the nominal interest rate (APR)? 𝑃𝑃 = $15,000 𝐴𝐴 = $477 𝑁𝑁 = 36 months - What is the effective interest rate per year? - What is the amount of unpaid loan principal after 20 months? Examples • A loan of $2,000 at 10% annual interest rate for 8 years is to be repaid in two equal payments, @ EOY 4 and EOY 8. What is the value of the payments? • If the monthly interest rate is 1%, what is the effective semi-annual rate? Continuous compounding - Allowing interest to compound continuously throughout the period ⇒ 𝑀𝑀 approaches ∞. 𝑖𝑖 = 𝑒𝑒 𝑟𝑟 − 1 Where 𝑖𝑖 is the effective rate and 𝑟𝑟 is the nominal rate. Continuous compounding factors Notice: 𝑟𝑟 is substituted (not 𝑖𝑖) So we can use these formulas or we can substitute the effective interest rate (𝑖𝑖) in P/F. F/A, and P/A equations presented earlier in the chapter Example • A bank offers loans at an annual interest rate of 12% compounded continuously, - What is the effective annual interest rate? - What is the effective monthly interest rate? - If you borrowed $10,000 on these terms, what is the future equivalent of this loan after 5 years? Example • A nominal interest rate of 8% is compounded continuously. - What is the uniform EOY amount for 10 years that is equivalent to $8,000 at EOY 10? - What is the present equivalent value of $1,000 per year for 12 years? - What is the future equivalent at the end of the 6th year of $243 payments made every 6 months during the 6 years (first payment occurs 6 months from the present and the last occurs at EOY 6)?

Chapter 4 The Time Value of Money

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