3770
Summer FINA II 2001
Topic 4--The Arithmetic of Compound Interest: Single Sums Learning Objectives Satisfied: 1. Introduction to Financial Management Also cover the major foundations of Finance such as Time Value of Money Cash Flow and Taxes and their implications for financial managers 2. Financial Markets and Interests Rates Objectives: Understand the following topics Inflation and interest rates and their relationship; theories of interest rates 3. Mathematics of Finance Objectives: Understand the following concepts Present and future value of lump sums Annualized percentage rate (APR) Effective interest rate Purpose: These notes cover the basics of compound interest, with inflation and deflation. Lump sums: -- Present value and future value, without inflation: Compounding for n discrete time periods FV = PV(1 + R )n and, PV = FV (1 + R)n
Example 1: If the interest rate is 15% compounded annually, then the future value of 3 $100 invested for 3 years is $100 1.15 = $152.09 For further practice, see Problem Set 1, problems 1, 5, 10. If the interest rate is 15% compounded annually, then the present value of $100 to be $100 $100 1.15-3 = $65.75 received in 3 years is 3 = 1.15 For further practice, see Problem Set 1, problems 24, 28, 33. Note: Your financial calculator is programmed so you can enter the interest rate as a percentage (for example, enter 15% as 15). The calculator will retrieve whatever you enter with the interest rate button and divide by 100 before plugging it into the formula. The calculator will always do this, and there is no way to change its stubborn mind.
Course handout
Pro. Kensinger
page 1 of 9
FINA 3770 Compounding periods shorter than one year R = APR with compounding m times per year n = total number of compounding periods m = number of compounding periods in one year
Summer II 2001
( ) -n PV = FV(1 + R m )
FV = PV 1 + R m
n
Note: APR means annual percentage rate (e.g., 1% per month is 12% APR). Example 2: If the APR is 15% compounded monthly, then the future value of $100 36 invested for 3 years is $100 1 + .1512 = $156.39
(
)
For further practice, see Problem Set 1, problems 1-3, 6-8, 11-12. If the APR is 15% compounded monthly, then the present value of $100 to be received in -36 = $63.94 3 years is $100 1 + .1512
(
)
For further practice, see Problem Set 1, problems 25-26, 29-31, 34, 39. Note: Most models in the current generation of financial calculators allow you to enter the APR using the button for the interest rate, and the number of compounding periods per year, using a button labeled "P/YR" on HP 10, 17, and 19. Some TI models have both a payments per year button and a compoundings per year button (for these examples, you will need only the compoundings per year feature). The calculator will then automatically do the R/m calculation, but you will need to enter the correct number of compounding periods (not years, but correct number of days, weeks, months, quarters, or other time period). Continuous compounding happens when the number of compounding periods per year is uncountably large. We'll lett be the number of years (by the way, t can be a fraction in this case). FV = PV e Rt PV = FV e - Rt Note: Financial calculators that do not offer variable-use keys (most TI models, and HP models 10 and 12) have a key labeled "e x". Calculators with heirarchical menus (such as the HP 17 or 19) have the necessary feature in the "math" mode--select math mode, and find the "exp" key. Then to calculate future value, just enter R as a decimal fraction, multiply times t, press this button (ex or exp), and multiply the result times present value (or, to calculate present value, enter R as a decimal fraction, multiply times t, change the sign to negative by pressing the "+/-" key, press the ex or exp key, and multiply the result times future value).
Course handout
Pro. Kensinger
page 2 of 9
FINA 3770
Summer II 2001
Example 3: If the APR is 15% compounded continuously, then the future value of $100 invested for 3 years is $100 e 0.153 = $100 e 0.45 = $156.83 For further practice, see Problem Set 1, problems 4, 9, 13. If the APR is 15% compounded continuously, then the present value of $100 to be received in 3 years is $100 e -0.153 = $100 e -0.45 = $63. 76 For further practice, see Problem Set 1, problems 27, 32, 35.
-- Effective rates are used for comparing APRs that do not have the same number of compounding periods per year: Let: R = APR with compounding m times per year Reff = effective rate Then:
(1 + R m) (
m
= 1 + Reff
Reff = 1 + R m
)
m
-1
Example 4: If the APR is 15% with monthly compounding, then the effective rate is 12 1 + .1512 - 1 = 16.08%
(
)
For further practice, see Problem Set 1, problems 14-16, 17-19, 21-22, 36. Note: Modern financial calculators have this conversion built in. On the menu models (such as HP 17 or 19), locate the "ICONV" choice. On the other models, locate the two keys labeled "Nom" and "Eff" (on HP models) or "APR" and "EFF" (on TI models). Then set the correct number of periods per year (with the P/YR button), enter the nominal APR with the "Nom" or "APR" button, and press the "Eff" button to display the answer. You can convert effective rates to APRs by reversing the procedure. Continuous compounding Reff = e R - 1 Example 5: If the APR is 15% with continuous compounding, then the effective rate is e.15 - 1 = .1618 = 16.18% . For further practice, see Problem Set 1, problems 16, 20, 23, 36. You may know both the present value and future value, and need to find either the annual interest rate or the number of years. There are five keys on your
Course handout Pro. Kensinger page 3 of 9
FINA 3770
Summer II 2001
calculator that are involved in time value of money calculations: n, %i, PV, FV, and PMT (up to this point, PMT has been zero). You can enter any four of these (one of these entries is zero for PMT, so you just need three other pieces of information), and then calculate the unknown variable. Example 6: Suppose you invested $1,000 and after seven years it grew to $2,000. If interest is compounded annually, then you earned a 10.41% rate of return. (P/YR is 1, n is 7, PV is 1000, FV is 2000, PMT is zero, compute %i). For further practice, see Problem Set 1, problems 37, 38. Suppose you invest $1,000 at 15% compounded annually, How many years will it take until you have $1,750? The answer is 4 years (P/YR is 1, %i is 15, PV is 1000, FV is 1750, PMT is zero, compute n). For further practice, see Problem Set 1, problems 40. Note: Whenever you enter both FV and PV in a calculation, you probably will need to observe the sign convention (this is true for most calculator models). Either FV or PV will have to be entered as a negative number. It is good habit to begin early to enter investments as negative, because those are outflows from your pocket. If you don't remember the sign convention, you will get an error indication from the calculator. You can find the interest rate or the required time with continuous compounding, too. Suppose you know both the present value and future value, and need to find either the annual interest rate or the number of years. Then you would simply rearrange the relationship as follows FV = PV e Rt Rt FV PV = e Now, "e" is the base of natural logarithms, so your calculator is programmed to give you a nice way of finding the interest rate from this point. Locate the key labeled "ln" on the keyboard (in math mode on the calculators with multifunction keys such as HP 17 or 19). Then to find the interest rate, divide future value by present value, press the ln key, and divide the result by t. If you know the interest rate and need to know the number of years required, divide future value by present value, press the ln key, and divide the result by R.
Course handout
Pro. Kensinger
page 4 of 9
FINA 3770
Summer II 2001
Example 7: If the future value is $156.83, the present value is $100, and the time is three years, the APR with continuous compounding is as follows: R= ln(156.83 100) 3 = .45 = .15 = 15% 3
If the future value is $156.83, the present value is $100, and the APR is 15% compounded continuously, then the number of years is as follows t= ln(156.83 100) .15 = .45 =3 .15
You won't be tested on continuous the time calculations in this example, so there are no repetitions in the practice problems.
Note: You may wonder how many decimal places to report when you calculate an interest rate. For our purposes in this class it will be sufficient to round the final result in such a calculation to the nearest basis point, as in the above examples (there are 100 basis points in 1%). Thus, if you set your calculator to display two places after the decimal, you will be ready to go for calculating either dollar amounts or interest rates using the financial functions. Only when doing calculations involving continuous compounding would you need to remember to convert the result to a percentage (in the above example the initial result is .1618, so multiply this time 100 and report 16.18% as the answer).
-- Inflation and the price level: Future Price Level = Present Price Level (1 + i )
n
Example 8: If the present price level index were 100 and inflation were 10% compounded annually, the index next year would be 110. The next year it would be 121, then 133.1, and so on. Present Price Level Purchasing Power = Future Amount Future Price Level 1 Purchasing Power = Future Amount n (1 + i ) Example 9: If you stuffed a $100 bill into your mattress and left it there during a year of 10% inflation, its purchasing power would shrink. With the price level index at 110% of what it was when you hid the money, your $100 would buy only 100/110ths of what it used to buy. In terms of purchasing power, it would have shrunk to $90.91. If it were left hidden for another year of 10% inflation, it would shrink to $82.64; another year would shrink it to $75.13, and so on. For further practice, see Problem Set 1, problems 41-46.
Course handout
Pro. Kensinger
page 5 of 9
FINA 3770 -- Deflation is just like inflation, but with negative i.
Summer II 2001
Example 10: If you had stuffed your $100 bill into the mattress during a year of 10% deflation, the price level at the end of the year would be 90% of what it was when you hid the money. Your cash would then buy 100/90ths of what it did before, and your $100 bill would have expanded to the equivalent of $111.11. If you left it tucked away for another year of 10% deflation, the price level would drop to 81% of what it was at the start, and your $100 bill would have expanded to $123.46. And so it goes. For further practice, see Problem Set 1, problems 47-51. -- The relationship of the real rate, r, the nominal rate, R, and the rate of inflation, i, is the first building block of the theory of interest. Single-year example of the Fisher Effect: Suppose you begin with $100 invested for one year when you expect inflation will be 4% and the desired real rate of return is 3%. You would therefore need to collect enough from the investment so that you could reinvest the principal (adjusted for inflation) and have enough left over to meet the spending goal. First, you would need $104 to keep the principal intact in real terms. Then, you would need to have enough left over so that you can spend the equivalent of $3 in today's purchasing power (thus you would need to be able to spent $3 * 1.04 = $3.12). In order to reinvest $104 and spend $3.12, you would need to collect $107.12. Thus, the required nominal return on the investment would be 7.12%. Illustration of the Fisher Effect over several years:
PV =
Purchasing Power (1 + r )n
FV n (1 + i ) since PV = (1 + r )n therefore, PV = = Since we already know that PV = FV
[(1 + r )(1 + i)]n
FV we can conclude that (1 + R)n 1 + R = (1 + r )(1 + i ) . This expression can be rearranged to yield R = r + i + ri.
Whenever you do time value calculations, you can compute the appropriate discount rate and plug it in. Don't round off these intermediate results, however, because a small error compounded many times becomes a big error, and you don't want that to happen.
Course handout
Pro. Kensinger
page 6 of 9
FINA 3770
Summer II 2001 Example 11: n=10, r=3%, i=4%, Future Amount = $198.93 198.93 10 Purchasing Power (1.04) PV = = 10 10 (1.03) (1.03) 198.93 10 134.39 (1.04) = 100 PV = = 10 10 (1.03) (1.03) therefore, R = 7.12%
To verify the nominal interest rate, enter PV= 100, FV=198.93, n=10, then calculate the interest rate. The correct answer is 7.12% For further practice, see Problem Set 1, problems 52-58. Example 12: If r =3.00% and i = 4.00%, then R = 7.12% If r =3.00% and i = 5.00%, then R = 8.15% If r =3.00% and i = 7.00%, then R = 10.21% For further practice, see Problem Set 1, problems 59. To compute the amount of purchasing power you will have in the future as a result of saving today, you would first calculate how many dollars you will have, and then adjust for inflation, as follows: Example of the Fisher Effect and future purchasing power: Suppose you begin with $100 invested for one year when the nominal rate of return is 15% and you expect inflation will be 12%. The question is how much purchasing power you will have next year, after restoring the principal. ...