Time Value of Money Topics to be covered 1. Future Value - Simple Interest - Compound Interest - Effective Interest Rate (EAR) - Annual Percentage Rate (APR) 2. Present Value 3. Calculating future/present value if interest is not compounded annually. 4. Present and Future Values of Multiple Cash Flows 5. Perpetuity: Indefinite Cash Flow 6. Annuity: Equal Cash Flow Ordinary Annuity Annuity Due 7. Mortgage Payment 8. Interest Rate Nominal Interest Real Interest I Future Value: Amount to which an investment will grow after earning interest. 1.1 Simple Interest: Interest is earned only on the original investment; no interest is earned on interest. 1.2 Compound Interest: Interest is earned on interest. Example1: How much your money will grow if you deposit $100 which earned 6% simple interest for 4 years? How much if it is compound interest? a) Simple Interest T=0 T=4 PV=$100 FV=? 0 1 2 3 4 Interest Earned 100*6% =6 100*6% =6 100*6% =6 100*6% =6 Value 100 106 112 118 124 Simple Interest = P * R * T b) Compound Interest 0 1 2 3 4 Interest Earned 100*6% 106*6% 112.36*6% 119.10*6% =6 =6.36 =6.74 =7.14 Value 100 106 112.36 119.10 126.24 Compound Interest = (1+r)t 1 Interest earned on Interest = Compound Interest – Simple Interest Future Value = PV * (1+r)t Example 2: What is the future value of $100 if interest is compounded annually at a rate of 6% for 4 years? 1) FV = 100 *(1+0.06)4 = 126.24 2) 100 PV 6 %I 4 N Cpt FV 126.24 3) Table FV = 100 *(1.262) = 126.24 Example 2B: Sale of Manhattan Island for $24 in 1626 (N=385) N = 385 I = 8% PV = $24 * (1+.08) ^385 = 177,156,500,000,000 = 164 trillion With R = 3.5 % PV = 13,559,570 Effective Annual Interest Rate (EAR) VS Annual Percentage Rate (APR) 1.3 EAR: Interest rate that is annualized using compound interest. 1.4 APR: Interest rate that is annualized using simple interest. Example 3A: Suppose you borrow money at a monthly rate of 1%, what is the EAR and what is the APR? APR = 1% * 12 = 12% EAR = (1+ APR/m)m - 1 = (1+0.12/12)12 –1 = 12.68% Where m number of compounding periods per year 12 2ND EFF 12 = 12.68% 12.68 2nd APR 12 = 12% Example 3B: Calculate the EAR for APR of 10% with monthly compounding, 8% with quarterly compounding and 10% with semiannual compounding. a) 10% 2nd EFF 12 = 10.47% b) 8 2nd EFF 4 = 8.24% c) 10 2nd EFF 2 = 10.25% Compounding Period Periods per year (m) 1 year 1 Semiannually 2 Quarterly 4 Monthly 12 Weekly 52 Daily 365 2 History on Compound Interest (From Wikipedia): If the Native American tribe that accepted goods worth 60 guilders for the sale of Manhattan in 1626 had invested the money in a Dutch bank at 6.5% interest, compounded annually, then in 2005 their investment would be worth over €700 billion (around USD $1,000 billion), more than the assessed value of the real estate in all five boroughs of New York City. With a 6.0% interest however, the value of their investment today would have been €100 billion (7 times less!). Compound interest was once regarded as the worst kind of usury, and was severely condemned by Roman law, as well as the common laws of many other countries. [2] Richard Witt's book Arithmeticall Questions, published in 1613, was a landmark in the history of compound interest. It was wholly devoted to the subject (previously called anatocism), whereas previous writers had usually treated compound interest briefly in just one chapter in a mathematical textbook. Witt's book gave tables based on 10% (the then maximum rate of interest allowable on loans) and on other rates for different purposes, such as the valuation of property leases. Witt was a London mathematical practitioner and his book is notable for its clarity of expression, depth of insight and accuracy of calculation, with 124 worked examples.[3][4] The Qur'an, revealed over 1400 years ago, explicitly mentions compound interest as a great sin. Interest known in Arabic as riba is considered wrong: "Oh you who believe, you shall not take riba, compounded over and over. Observe God, that you may succeed. (Qur'an 3:130)" http://en.wikipedia.org/wiki/Compound_interest 2. Present Value Present Value (PV) = FV/(1+r)t Discount Factor = 1/(1+r)t (This measures the PV of $1 received in year t) 3. FV/PV if interest is not compounded annually Example 4: Assume that you deposit $1,000 today in an account that pays 8 percent compound quarterly, how much will you have in 4 years? Compound Quarterly Compound Semiannually Compound Monthly -1,000 PV -1,000 PV -1,000 PV 8/4 %I 8/2 %I 8/12 %I 4*4 N 4*2 N 4*12 N CPT FV 1,372.78 CPT FV 1,368.56 CPT FV 1,375.66 Example 5: How long does it take your money to grow to 5,000 if you deposit 1,000 today and bank pays 6% quarterly compound interest? I/P = 6/4 = 1.5 1000(1.015) ^n = 5000 n=108 quarters Example 6: How much interest will you earn on a $100 deposit at 8% compound semiannually if you deposit for 5 years? 100(1+.04) ^10 = 148.02 3 Example 6B: K auto offering free credit on $10,000 car, pay $4000 down and balance in 2 years. T auto gives $1000 off the list price. If the interest rate is 10% who is giving better deal? K auto T-auto 4000 6000 10,000 -1000 = 9000 0 2 years PV= 4000 + 6000/(110)2 = 4000 + 4958.68 = 8958.68 Hence K auto gives a better deal Example 6C: Calculate interest rate if FV of $129 25 years from now is $1000? PV= -129 FV = 1000 N=25 Compute I/Y = 8.54% __________________________________________________________________________ 4. Multiple (a stream of) Cash Flows A) In many situations, there are more than one cash flows. Whether they are equal or unequal, they are referred to as a stream of cash flow. B) Calculating the present value (future value) of an unequal series of future cash flow is determined by summing the present value (future value) of each discounted single cash flow. C) The PV of a stream of future cash flows is the amount you would have to invest today to generate that stream. D) Calculating the present value (future value) of an equal series of future cash flow is called an annuity. E) An annuity is an equally spaced, level stream of cash flow for a fixed period of time. F) If the cash flows occur at the end (beginning) of each period it is called ordinary annuity (annuity due). F) If the multiple cash flow is the same amount and be equally spaced and lasts forever, it is perpetuity. 4 Present Value of a Multiple Cash Flows Unequal Cash flow Example 7: If you deposit $100 in one year, $200 in two years, and $300 in three years, what is the present value of this cash flow stream if the interest rate is 8%? T =0 t=1 t=2 t=3 100 200 300 PV=? (1) PV = 100/(1+0.08)1 + 200/(1+0.08)2 + 300/(1+0.08)3 = 92.59+171.53+238.15 =502.27 (2) 100 FV 200 FV 300 FV 1 N 2 N 3 N 8 %I 8 %I 8 %I CPT PV 92.59 + CPT PV 171.53 + CPT PV 238.15 = 502.27 Future Value of a Multiple Cash Flow Example 8: Using the same example, how much will you have in three years? T=0 t=1 t=2 t=3 100 200 300 (1) FV = 100(1+0.08)2 + 200(1.08)1 + 300 = 116.64+216+300 = 632.64 (2) -100 PV -200 PV 2 N 1 N 8 %I 8 %I CPT FV 116.64 + CPT FV 216 + 300 = 632.64 Example 9: You are offered an investment that will pay you $200 in one year, $400 the second year, $600 the third year, and $800 at the end of the last year. You can earn 12% on very similar investments. What is the most you should pay (PV) for this one? Solution: 200 400 600 800 0 1 2 3 4 PV= 200/ (1.12) + 400/ (1.12)2 + 600/ (1.12)3 + 800/ (1.12)4 = 178.57 + 318.88 + 427.07 + 508.41 = $1,432.93 5 Comparing two multiple cash flows Key: A future stream of cash flows associated with an investment may be compared or summed if adjusted to a common time period, usually the present (PV). When comparing two multiple cash flows of different time periods, we need to discount back and compare the present value. Example 10: Investment A pays $100 per year for three years. Investment B pays $80 per year for four years. Which investment would you choose if the discount rate is 10% and why? Solution: Investment A: Investment B: 100 100 100 80 80 80 80 0 1 2 3 0 1 2 3 4 PV for Investment A: PV for Investment B: =100/ (1.1) +100/ (1.1)2+100/ (1.1)3 =80/ (1.1) +80/ (1.1)2+80/(1.1)3+80/(1.1)4 =$248.69 =$253.60 Example 11: You have joined a company.