FINA 351 Managerial Finance, Ch.4-5, Time-Value-of-Money (TVM), Notes

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1 FINA 351 Managerial Finance, Ch.4-5, Time-Value-of-Money (TVM), Notes The concept of time-value-of-money is important to know, not only for this class, but for your own financial planning. It is a critical in understanding car loans, student loans, mortgages, investments and retirement planning. You want to make the concept of time-value-money work for you rather than against you. Example 1:

2 Example 2:

3 Example 3: Suppose that you just graduated from Western Wedding University with a spouse in one hand and a diploma in the other. When you start your career, you save $100 per month for the next 40 years at a rate of 12%. You have would have $1,176,477 dollars when you retire. Example 4: If you had invested $100 in Berkshire Hathaway (Warren Buffet s company) in 1965, it would be worth nearly $1 million today. If you had invested $100 in 1965 in the 500 stocks in the S&P500, your investment would have growth to only about $30,000. Buffet s compound annual return has averaged nearly 22%, while the S&P500 averaged about 10% over the same period. Example 5: In order to buy a house, you get a $200,000, 8%, 30-year mortgage and you put 5% down. You would pay $1, a month for the mortgage payment, which totals $328,321 of interest over the life of the loan, considerably more than the value of your home. In addition, add to your monthly payment about $350 for taxes and insurance. However, if you paid an extra principal payment of $100 per month, you would save nearly $80,000 of interest and cut 72 monthly payments off or your mortgage. Once asked, What is the most powerful force on earth? Albert Einstein replied without hesitation, Compound interest. And Benjamin Franklin defined the term as the stone that will turn all your lead into gold. METHODS: There are five methods of solving time value of money calculations. 1. Long-hand--which will be used only to demonstrate that the other methods are much faster and give the same answer. 2. Mathematical formulas--which can become cumbersome to use, so we mostly ignore this method, except to show that it works. 3. Tables--this may be the easiest method without a financial calculator. The tables are simply an extension of the formulas. The numbers on the tables are called interest or value factors. 4. Financial calculators/financial cellphone apps--these are fast and accurate methods. But you obviously have to first obtain and then learn how to use them. If you choose to purchase a calculator, either the HP10bii or TI BAiiPlus is recommended. The cost to download a phone app is usually under $ Spreadsheets--Excel has TVM functions which easily solve all of these problems. Note: regardless of the method, on your assignments you should record the main variables in the calculation (e.g. rate, term, present or future value amount, etc.) as well as the answer, so that you know how you got your answer when reviewing for an exam.

4 I. FUTURE VALUE (FV) AND SIMPLE INTEREST Simple interest is interest on principal only. Example: How much would you have in the bank in 2 years if you put $100 in the bank today and earn simple interest of 10 percent? Year 1 Prin: $100 Year 1 Int: $100 * 10% = $10 Year 1 Bal: $100 + $10 = $110 Year 2 Int: $100 * 10% = $10 Year 2 Bal: $110 + $10 = $120 II. FUTURE VALUE (FV) OR COMPOUNDING OF SINGLE SUM (SS) Compound interest is interest on principal and interest. Compound interest is compounded annually, unless stated otherwise. The abbreviation APR stands for Annual Percentage Rate, which means annual compounding. A single sum is one payment or deposit, as compared to a series of payments or deposits, which is called an annuity. Notice that when it comes to single sums, the present value and future value are at opposite ends of the same timeline. PRESENT VALUE (amount today is prin. only) grows over time with interest to equal the FUTURE VALUE (future amount is prin. and int.) PV FV Example: How much will you have in 2 years if you put $100 in the bank today and earn interest compounded annually at 10 percent? Let s solve it four ways: 1. Long-hand Year 1 Prin: $100 Year 1 Int: $100 * 110% = $10 Year 1 Bal: $100 + $10 = $110 Year 2 Int: $110 * 10% = $11 Year 2 Bal: $110 + $11 = $121 (notice the extra $1 of interest on interest) 2. Formula FVSS(n,i%) = PVSS * (1 + i)^n FVSS(2,10%) = $100 * (1.1)^2 FVSS(2,10%) = $ Tables (use tables in Appendix A.1) FVSS(n,i%) = PVSS * Factor from Table FVSS(2,10%) = $100 * 1.21 FVSS(2,10%) = $ Financial calculator n=2, i=10, PV= -100, FV=? (Note that PVs are always entered as negative) III. RULE OF 72 The rule of 72 is a simple way to approximate the effect of compounding on a single amount (calculating the future value of a single sum). In words, the formula is: The number of years it takes to double an amount, times the annual interest (or growth) rate, equals 72. Example: You bought a share of IBM stock for $100, and 8 years later you sold it for $200. What was your annual average return? Answer: 8*I=72; I=72/8; I = 9%.

5 IV. PRESENT VALUE(PV) OR DISCOUNTING OF SINGLE SUM(SS) Calculating PV of SS is exactly the reverse of calculating the FV of SS. In mathematical terms, their interest factors are the reciprocal of (or 1 divided by) each other. Example: How much must you put in the bank today in order to have $121 at the end of two years, if your money is compounding annually at 10 percent? Let s solve it three ways: 1. Long-hand (Forget it! Too cumbersome.) 2. Formula PVSS(n,i%) = FVSS * 1/((1 + i)^n) PVSS(2,10%) = $121 * 1/((1.1)^2) PVSS(2,10%) = $ Tables (see tables in Appendix A.2, p. 592 of your text) PVSS(n,i%) = FVSS * Factor from Table PVSS(2,10%) = $121 *.8264 FVSS(2,10%) = $ Financial calculator n=2, i=10, FV= 121, PV=? NOTE: the compounding period (year, quarter month, etc.) must first be determined and then the interest rate for that compounding period. Example: If the interest rate was 12% Annual Percentage Rate (APR), what interest rate would be used if the compounding period were monthly? The answer is 1% (12%/12 = 1%). V. FUTURE VALUE (FV) OF AN ORDINARY ANNUITY(OA) An annuity is a bunch of single sums of equal amounts paid in equal intervals of time (e.g. rent of $400 due first of every month.) An ordinary annuity is one where the payment occurs at the end of each period. An annuity due, which we won't deal with a whole lot, is where the payment is due at the beginning of each period. On most financial calculators, you toggle annuity due calculations off and on with the beg button. The future value of an ordinary annuity (FVOA) has a balance which gets bigger over time. Both the annuity payments and the interest add to the balance. FV PV = 0 Example: How much would you have in the bank three years from now if you deposit $100 at the end of each year and earn 10% compounded annually? 1. Long-hand Year 1 Prin: $100 Year 1 Int: $0 (deposit occurs at end of first year so no interest is earned) Year 1 Bal: $100 Year 2 Prin: $100 Year 2 Int: $100 * 10% = $10 Year 2 Bal: $100+ $100 + $10 = $210 Year 3 Prin: $100 Year 3 Int: $210 * 10% = $21 Year 3 Bal: $210+ $100 + $21 = $ Formula - can use it if you want to but the tables are easier.

6 3. Table (see tables in Appendix A.4, p. 596 of your text) FVOA(n,i%) = Annuity Amt * Factor from Table FVOA(3,10%) = $100 * 3.31 FVOA(3,10%) = $ Financial calculator n=3, i=10, PMT=100, FV=? VI. PRESENT VALUE OR DISCOUNTING OF AN ORDINARY ANNUITY (OA) While the PV and FV of a single sum are at opposite ends of the same calculation, this is not the case with annuities. FV or PV of annuities have no relation to each other at all. In a PVOA calculation, the annuity payment decreases the balance, but the interest increases the balance. PV FV = 0 There are two primary types of PVOA problems: one is a loan amortization, and the second is when you are withdrawing money periodically out a bank account. In both cases, the payment reduces the balance but the interest adds to the balance. It is practically impossible to solve PVOA problems long-hand and the formula becomes complicated. So we stick to the tables and financial calculators for these computations. Example: You are about to purchase a used car for $12,500. You do not have this kind of money sitting around so you do some shopping for a loan. The best interest rate you could find is 12% APR, available at the credit union. The credit union requires a down payment of 20% and you have the amount of cash needed for the down payment. The amount borrowed plus interest will be repaid in 6 monthly payments of the same amount. A. After paying the 20% down payment, how much has to be borrowed in order to have enough to pay for the car? Answer: $12,500 * 20% = $2,500; $12,500 - $2,500 = $10,000 B. Determine the amount of each of the 6 monthly loan payments (or annuity amount) assuming they include interest at 12% APR. Use tables in Appendix A.3, p PVOA(n,i%) = Annuity Amt * Factor from Table PVOA(6,1%) = Annuity Amt * Annuity Amt = $10,000 / = $1, C. How much will be repaid to the lender in total, including interest? (Six times the monthly payment) 6 * $1, = $10, D. There is a simple way to calculate total interest on a loan. This involves deducting the amount borrowed (letter A above) from the total amount repaid (letter C above). Determine the total interest to be paid on this loan using this method: $10, $10, = $ E. Complete the amortization table below on this loan. Don't worry if you are a few pennies off at the bottom due to rounding. A B C D (D x I / 12) (A B) (Previous D C) BALANCE BALANCE OWING MONTH PAYMENT INTEREST REDUCTION 10, , , , , , , , , , , , , , *0.09 TOTAL 10, * * off a few pennies due to rounding errors Note that amortization tables can be done very nicely using spreadsheets, as discussed at the end of Chapter 5.

7 VII. TIME VALUE OF MONEY DECISION TREE: Single-Sum, Multiple Uneven-Sums OR Annuity? Single-Sum: PV and FV at opposite ends of same time line Annuity (equal pmts & time periods): FV or PV? (balance increasing or decreasing?) Multiple Uneven-Sums (many single sums of different amounts): find PV or FV of each single sum and add them up Ordinary annuity or annuity due? (payment at end or beg. of period)? VIII. MULTIPLE CASH FLOWS (SINGLE SUMS) OF DIFFERENT AMOUNTS An annuity is when the payment each period is exactly the same amount, like rent. Multiple cash flows of different amounts are when a payment occurs at the same time each period but the amounts are different. To solve these problems, we treat each payment as a single sum; find the present or future value of each single sum, and then add them all together. Example: How much would you have in the bank at the end of 3 years if you made a deposit of $1,000 at the end of Year 1, $2,000 at the end of Year 2, and $3,000 at the end of Year 3, assuming 10% APR? 1. Long-hand Year 1 Prin: $1,000 Year 1 Int: $0 (deposit wasn t made until the end of the year) Year 1 Bal: $1,000 Year 2 Prin: $2,000 Year 2 Int: $100 ($1,000 * 10%) Year 2 Bal: $1,000 + $2,000 + $100 = $3,100 Year 3 Prin: $3,000 Year 3 Int: $310 ($3,100 * 10%) Year 3 Bal: $3,100 + $3,000 + $310 = $6, Formula (forget it too hairy) 3. Tables Find the FVSS for each deposit separately and add them up together. FVSS(2,10%) = $1,000 * 1.21 = $1,210 FVSS(1,10%) = $2,000 * 1.1 = $2,200 FVSS(0,10%) = $3,000 * 1.0 = $3,000 Total of above = $6,410

8 4. Financial Calculators Find the future values of each cash flow separately and add them together. IX. PUTTING TWO DIFFERENT TMV CALCULATIONS TOGETHER Many real-life TMV problems require that two different calculations be used in one situation. For example, suppose that you are 22 years old today. You plan to go to graduate school when you are 28 years old and get an MBA (finance concentration) from Ivy League University, which will take two years. You are worried about paying the tuition, which you estimate will be $40,000 per year. You decide to start a college savings fund by depositing an equal amount in the bank at the end of each of the next 6 years. At the end of years 7 and 8 you want to be able to withdraw $40,000 for tuition. You expect to earn 8% interest rate compounded annually. How much will you need to deposit at the end of each of the next 6 years in order to have enough to pay for the MBA program? Notice that you first need to calculate the balance in the fund needed at Point A, which is $71,332, in a PVOA calculation. This balance, then, becomes the future value in an FVOA calculation needed to determine the annual deposits (B). STEP A PVOA (2,8%) = * = 71,332 +B +B -40,000 STEP B +B FVOA (6, 8%) = ANN * IF 71,332 = ANN * B ANN = 9,724-40,000 +B +B X. EFFECTIVE ANNUAL RATES (EAR)AND THE EFFECT OF COMPOUNDING EAR is a much better rate than APR because it takes into account the number of times a year the interest in compounded. APR is a poor measure--so poor that the Gov=t now requires that financial institutions (banks, etc.) quote interest rates using the EAR, which they call annual percentage yield (APY). The formula is: EAR = (1 + NR/m) m - 1 Where: NR = nominal rate (APR) m = number of compounding periods per year Example: If both Bank A and B quote a nominal interest rate of 10% APR, but Bank A compounds annually and Bank B compounds semi-annually, which is the better rate? APR EAR or APY Bank A 10.00% 10.00% Bank B 10.00% 10.25% = ( /2)^2-1 Most financial calculators and spreadsheets can do these calculations for you. See the discussion in Ch. 5 for more details.

9 Continuous Compounding Taking it to the limit: can there be such a thing as continuous compounding? Yes. The formula for EAR under continuous compounding is: EAR = e qr - 1 Where: e = mathematical constant = qr = quoted nominal rate (APR) Moral of the story: the more often you compound the rate, the more times interest is earned on interest, and the greater the total interest earned.

10 The effect of compounding more often gets more pronounced the higher the rate;