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?

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1 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=? Interest Earned 100*6% =6 100*6% =6 100*6% =6 100*6% =6 Value Simple Interest = P * R * T b) Compound Interest Interest Earned 100*6% =6 106*6% = *6% = *6% =7.14 Value Compound Interest = (1+r) t 1

2 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 = ) 100 PV 6 %I 4 N Cpt FV ) Table FV = 100 *(1.262) = 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 2 ND EFF 12 = 12.68% nd 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% 2 nd EFF 12 = 10.47% b) 8 2 nd EFF 4 = 8.24% c) 10 2 nd EFF 2 = 10.25% Compounding Period Periods per year (m) 1 year 1 Semiannually 2 Quarterly 4 Monthly 12 Weekly 52 Daily 365 2

3 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)" 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, CPT FV 1, CPT FV 1, 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 = (1.015) ^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 =

4 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 , = years PV= /(110) 2 = = Hence K auto gives a better deal Example 6C: Calculate interest rate if FV of $ 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

5 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= PV=? (1) PV = 100/(1+0.08) /(1+0.08) /(1+0.08) 3 = = (2) 100 FV 200 FV 300 FV 1 N 2 N 3 N 8 %I 8 %I 8 %I CPT PV CPT PV CPT PV = 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= (1) FV = 100(1+0.08) (1.08) = = (2) -100 PV -200 PV 2 N 1 N 8 %I 8 %I CPT FV CPT FV = 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: PV= 200/ (1.12) + 400/ (1.12) / (1.12) / (1.12) 4 = = $1,

6 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 $80 per year for four years. Which investment would you choose if the discount rate is 10% and why? Solution: Investment A: Investment B: PV for Investment A: PV for Investment B: =100/ (1.1) +100/ (1.1) / (1.1) 3 =80/ (1.1) +80/ (1.1) 2 +80/(1.1) 3 +80/(1.1) 4 =$ =$ Example 11: You have joined a company. They ve offered you two different salary arrangements. You can have $40,000 per year for the next two years or $20,000 per year for the next two years along with a $30,000 signing bonus today. If the interest is 16% compound annually, which do you prefer? Arrangement 1: Arrangement 2: 40K 40K 30K 20K 20K PV for Arrangement 1: PV for Arrangement 2: = 40,000/ (1.16) + 40,000/ (1.16) 2 =20,000/ (1.16) + 20,000/ (1.16) ,000 = $ 64,209 =62, Perpetuity: A stream of level of cash payments that never ends PV = C/r = Cash Payment/Interest Rate Where C represents cash flows. R represents discount rate. 2 Types of perpetuity: (1) Perpetuity that starts to make payments at the end of the first year. (2) Perpetuity that does not start to make payments for several years. 6

7 Example 12: What is the present value of a $100 cash flow forever if the interest rate is 12% PV = 100/ Example 13: With a 10% interest rate, calculate the present value of the following streams of cashflow (1) $1,000 per year forever PV=c/r = 1000/0.10 = 10,000 (2) $500 per year forever, with the first payment two years from now α α So PV 1 = 500/0.10 = $ PV o = 5000/1.1 = (3) $2,420 per year forever, with the first payment three years from now α α PV 2 =2420/0.10 = $ PV= FV/(1+r) 2 = 24200/(1.1) 2 = $20,000 7

8 6. Annuity: an equally spaced, stream of cash flow for a limited number period of time PV = C/r T=0 t=1 t=2 t=3 t=4 t=5 t=6 T= (1) Perpetuity A. $1 $1 $1 $1 $1 $1 PV = 1/r T=0 t=1 t=2 t=3 t=4 t=5 t=6 T= (2) Perpetuity B. $1 $1 $1 PV = 1/(r(1+r) 3 ) T=0 t=1 t=2 t=3 t=4 t=5 t=6 T= (3) 3 Yrs Annuity. $1 $1 $1 PV A - PV B = PV of 3-yrs $1 annuity = 1/r-1/(r(1+r) 3 ) PV of t-year annuity = C[1/r-1/(r(1+r) t )] t-year annuity factor = [1/r-1/(r(1+r) t )] 2 Types of Annuity: (1) Ordinary Annuity: Payment is made at the end of the year. (2) Annuity Due: Payment is made at the beginning of the year or started immediately. 6.1 Ordinary Annuity: PV = C[1/r 1/r(1+r) t ] Example 14: Suppose you deposit $100 at the end of each year for three years in an account paying 8%. What is the present value of the cash flow stream? (Equal Cash Flow) t=0 t=1 t=2 t=3 $100 $100 $100 (1) 100 FV 100 FV 100 FV 1 N 2 N 3 N 8 %I 8 %I 8 %I CPT PV CPT PV CPT PV =

9 (2) 100 PMT 3 N 8 %I CPT PV (3) PV = 100[1/0.08 1/0.08(1+0.08) 3 ] = Annuity Due Using the same example, what is the present value if you deposit the money at the beginning of each year? t=0 t=1 t=2 t=3 2 nd BGN 100 PMT N 8 %I CPT PV FV 100 FV 2 N 1 N 8 %I 8 %I CPT PV CPT PV = Finding Present Value of Annuity Problem 1: Suppose we were examining an investment that promised to pay $500 at the end of each year for the next three years. We want to earn 10 percent on our money, how much would we offer for this annuity? How much, if the payment is made at the beginning of each year? Solution: A) B) R=10%; N=3; Pmt Amount=500 Change to Beginning CPT PV=$ R=10%; N=3; Pmt Amount=500 CPT PV=$ Problem 2: After carefully going over your budget, you have determined you can afford to pay $632 per month towards a new sports car. You call up your local bank and find out that the going rate is 1 percent per month for 48 months. How much can you borrow? Payment = 632 I/Y= 1 N = 48 FV = 0 Compute PV = $-23,999,54 This is the maximum you can borrow. 9

10 Finding Payment Problem 3: Suppose you start a new business, and you need to borrow $100,000. You propose to pay off the loan by making 5 equal annual payments. If the interest rate is 18 percent, what will the payment be? Solution: PV=-100,000; N=5; R=18 CPT Payment = $31,978 Problem 4: Using the same example, what will the payment be if you propose to pay off by making a monthly payment for 5 years and the annual interest rate is 18%. PV = -100,000 N = 5 x 12 = 60 I/Y = 18/12 = 1.5 FV = 0 Compute Payment= $2539,34. Finding the Number of Periods Problem 5: You borrow $1,000 on your credit card. You can only afford to make the minimum payment of $20 per month. The interest rate on the credit card is 1.5 percent per month. How long will you need to pay off the $1,000? PV = -$1000 Payment= 20 I/Y = 1.5 FV = 0 Compute N = It will take you 94 payments to pay $ payments of $20 each and then in the 94 th payment of about $2.2 you will repay the entire loan. Finding the interest rate Problem 6: An insurance company offers to pay you $1,000 per year for 10 years if you will pay $6,710 up front (now). What rate is implicit in this 10-year annuity? Payment = 1000 N = 10 PV = FV = 0 Compute I/Y = 8% 10

11 Finding Future Value of Annuity Problem 7: Suppose you plan to contribute $2,000 every year into a retirement account paying 8 percent. If you retire in 30 years, how much will you have? 7. Mortgage Payment Example 15: You are purchasing a car. You are scheduled to make 3 annual installment of $4,000 per year. Given interest rate of 10%, what is the price you are paying for the car? Example 16: If you take out a $9,000 car loan that calls for 48 monthly payments at an APR of 10%. (1) What is your monthly payment? Answer: (2) What is the effective annual interest rate on the loan? Answer: 10.47% (3) If you deposit the same monthly payment at an APR of 10% for 4 years, how much will your money grow? Answer: $13,404 (4) Assume that you have to make a monthly payment as calculated from (1) for your car loan at the beginning of period, what is the price you are paying for the car? Answer: $ Mortgage Amortization So far, we have looked at the type of loan, in which the monthly payment is fixed over the life of the mortgage which is called an amortizing loan. Amortizing means that part of the monthly payment is used to pay interest on loan and part is used to reduce the amount of the loan. Example 17: If you take out a $8,000 car loan that calls for 12 monthly payments at an APR of 10%. (1) What is the monthly payment? 8,000 PV 12 N 10/12 %I CPT PMT (2) How much of the first monthly payment is interest paid? 8,000 * 0.1/12 = (3) How much of the first monthly payment is principal repayment? = (4) What is the remaining balance at the end of the first month? 8, = 7, (5) How much of the second payment is the interest paid? 7,363.34*0.1/12 = (6) How much of the second monthly payment is principal repayment? = (7) What is the remaining balance at the end of the second month? 7, = 6,

12 Periods Beginning Balance Payment Interest Paid Principle Repayment Remaining Balance Problem (8) If you take out a $10,000 car loan that calls for 36 monthly payments at an APR of 15%, calculate the answer for (1)-(7) as shown in example (17)? Saving for Retirement Problem 33 from the book. (Page 184) Solution: $100,000 save for 10 years Receive retirement money for 25 years 0 10Years 35 years 40 K Inflation = 5% PV=40K; R=5%; N=10 FV= $ 65, years money he receives for 25 years. Annuity Due: $65, years N=25; Payment= $ ; R= 8 %; FV=0 12

13 PV=$ 751, you need this much money at the end of year 10 Present value of $ equals PV 0 =$347, He has $100,000 so the Difference needed = $ 247,934.90=PV N=10; R=8 CPT Payment=$36, Inflation and the Time Value of Money Nominal Interest Rate: Real Interest Rate: Inflation: Rate at which money invested grows. Rate at which the purchasing power of an investment increases Rate at which prices as a whole are increasing. Approximate Formula (Fisher Equation) Nominal Rate Real Rate + Inflation Rate Exact Formula (1) (1+Nominal Rate) = (1+Inflation)(1+Real Rate) (2) 1+Inflation Rate = 1+Nominal Rate 1+Real Rate (3) 1+Real Rate = 1+Nominal Rate 1+ Inflation Problem (9) Suppose that you invest your fund at an interest of 8%, if the inflation rate is 3.8%. What is the real interest rate using exact formula? And what is using approximate formula? 1+RR = 1+NR/(1+Inflation) 1+RR = 1.08/1.038 = RR = 4.05% Additional Solved Problems on TVM: Problem (1): Your landscaping company can lease a truck for $8000 a year (paid at the year-end) for 6 years. It can instead buy the truck for $40,000. The truck will be value less after 6 years. If the interest rate your company can earn on its funds is 7%, is it cheaper to buy or lease? Leasing the truck means that the firm must make a series of payments in the form of an annuity. Using the financial calculator Payment = $8000 N=6 I= 7% FV=0 Compute PV = $38, Since $ < $40000 (the cost of buying a truck), it is less expensive to lease than to buy. 13

14 Problem (2): Suppose that you will receive annual payments of $10,000 for a period of 10 years. The first payment will be made four years from now. If the interest rate is 5%, what is the present value of this stream of payments? This problem can be approached in two steps. First, find the PV of the $10,000, 10-year annuity as of year3, when the first payment is exactly one year away (and it is therefore an ordinary annuity). Then discount this value back to today. Using financial calculator PMT=10,000 FV=0 N=10 I=5% Compute PV 3 =$77, PV 0 =PV 3 / (1+r)* 3 PV 0 =$77,217.35/1.05* 3 = $66, Problem 3: A retiree wants level consumption in real terms over a 30-year retirement. If the inflation rate equals the interest rate she earns on her $450,000 of savings, how much can she spend in real terms each year over the rest of her life? The real rate is zero. With a zero real rate, her spending each year is simply $450,000/30 = $15,000 per year Problem 4: A local bank advertises the following deal Pay us $100 a year for 10 years and then we will pay you (or your beneficiaries) $100 a year forever. Is this a good deal if the interest rate available on other deposits is 6%? The present value of your payments to bank equals: PMT=$100 N=10 I=6% PV= $ The present value of your receipts is the value of a $100 perpetuity deferred for 10 years: (100/.06) * 1/ (1.06) 10 = $ This is a good deal if you can earn 6% on your other investments. Problem 5: A factory costs $400,000. You forecast that it will produce cash inflows of $120,000 in year1, $180,000 in year 2 and $300,000 in Year 3. The discount rate is 12%. Is the factory a good investment? PV of cash flows = ($120,000/1.12) + ($180,000/ ) + ($300,000/ ) = $464, This exceeds the cost of the factory, hence the investment is attractive 14

15 Problem 6: You borrow $1000 and you repay the loan by making total payments of $1200: $100 a month for 12 months. The lender argues that since the loan is for 1 year and the total amount of interest paid is $200, the interest rate is 20 %. What is the APR and effective annual rate on this loan? Is either rate as low as 20%? Why is the 20% rate quoted on this loan deceptive? You borrow $1000 and repay the loan by making 12 payments of $100. Solve for the interest rate using financial calculator PMT= $100 PV= -$1000 N=12 Compute I = 2.923% Therefore APR = 2.923% * 12 = % Using the Iconv function in financial calculator For Nom= and C/Y=12 Compute EAR= % If you borrowed $1000 today and paid $1200 one year from now, the true rate would be 20%. You should have known that the true rate must be greater than 20% because $100 payments are made before the end of year, thus increasing the true rate above 20% (Time value of money!!) Problem 7: You believe you will need to have saved $500,000 by the time you retire in 40 years in order to live comfortably. If the interest rate is 6% per year how much must you save each year to meet your required goal? The future value of the payments into your savings must accumulate to $500,000 by year 40 FV= $500,000 N=40 I= 6% PV = 0 Compute PMT $