Chapter 5 – Important Stuff Mechanics of compounding / discounting PV, FV, PMT – lump sums and annuities Relationships – time, interest rates, etc Calculations: PV’s, FV’s, loan payments, interest rates For those of you who don’t like numbers it may be hard. But there’s help – notes are (more complete) in Word on my Blackboard page. BUT, the number examples are different in the Word version from those in the PP version. There’s a big typo (at least in my version of the text). First one who catches it gets a $S. Understanding everything in this chapter is fundamental to passing FinCoach, the final, the exams and the quizzes. Just a word to the wise.

Time Value of Money (TVM) Time Value of Money – relationship between value at two points in time Today versus tomorrow; today versus yesterday Because an invested dollar can earn interest, its future value is greater than today’s value Problem types: monthly loan payments, growth of savings account; time to goal On the other hand, today’s value of $1 received in the future is less than $1.00 because we have to wait to receive that dollar and its benefits. Think of it as “deferred gratification” and you want to be paid for this.

Financial Calculator Keys PV - Present value FV - Future value PMT - Amount of the payment N - Number of periods (years?) I/Y - Interest rate per period Must have financial calculator. TI BA II + only one supported Know any four variables, can solve for the fifth Not a 20 min job to learn but I’ll try to make easier. Don’t think of N as years. Sometimes it is be we often encounter loans paid monthly or savings accounts that are compounded quarterly. Think in terms of “periods”, not just “years”. More later.

TI Calculator Manual Strongly Suggested Readings Getting Started – page 6 and 7 Overview – page 1-4, 1-10 and 1-20 Worksheets – pages 2-14 and 2-15 TVM – 3-1 to 3-9 Cash Flow - All These are the very minimum needed (plus attention to this lecture) but it is strongly recommended you read everything through Chapter 4 in the manual. Also, instructions are in the text’s separate Appendix A but only for the TI

Decimals and Compounding Periods Calculator Tips Decimals and Compounding Periods 2nd (gray), Format (bottom row), 4, enter, CE/C (lower left) - hit twice Compounding: 2nd , I/Y, 1, enter, CE/C – extremely important !! Right arrow key fixes “misteaks” One cash flow must be negative or error Follow with black; demonstrate calculator Watch for BGN’s – you don’t want it. Go 2nd END. When calculator manufactured or if you reset, I/Y automatically goes to 12 which you definitely don’t want. If you don’t see the answer during an exam, recheck 2 nd I/Y. I won’t help you with your calculator during an exam.

Compound Interest @ 6% Year Begin Interest FV 1 $100.00 $6.00 $106.00 1 $100.00 $6.00 $106.00 2 106.00 6.36 112.36 3 112.36 6.74 119.10 Our deposit goes in the bank today and we want to know how much we will have in three years if it earns 6%. Why earn more in year 2? Interest on interest Shows how lump sum grows, no intervening payments FV of $100 is $119.10 @ 6% in 3 years FV is sum of PV and all compounded interest payments

Future Value (FV) Algebraically FVn = PV (1 + i)n Underlies all TVM calculations Keystrokes: 100 +/- PV; 3 N; 0 PMT; 6 I/Y; CPT FV = 119.10 One cash flow must be negative Error 5 means you forgot a negative sign Which says: the future value in period n is the present value times (1 + interest rate) raised to the nth power where n equals the number of compounding periods. Often N is not the number of years Don’t ignore the algebra. The homework requires you to use them.

Future Value Interest Factor Year @2% @6% @10% 1.020 1.060 1.100 1.040 1.124 1.210 3 1.104 1.191 1.611 10 1.219 1.791 2.594 This is from Table 5-2. I don’t want you using tables but they show some interesting things. Board $469 for 10 years at 6%, FV is 469 * 1.791 = $839.98

Reading the Formulas and Tables FVn = PV* (1 + i)n Plain English = The future value in period n is the present value (PV) times the quantity (i plus the interest rate) raised to the nth power where n equals the number of compounding periods. Future value of $500 invested 3 yrs @ 6% From table: FV6%, 3 yr. = 500 *1.191 = 595.50

Future Value of $100 Notice that FV increases as you go farther out in time or if you use a higher interest factor.

FV Can Be Increased By 1. Increasing the length of time it is compounded 2. Compounding at a higher rate And/or 3. Compounding more frequently Note the relationship is positive – if one increases, the other increases.

FV – Other Keystrokes How long for an investment to grow from $15,444 to $20,000 if earn 9% when compounded annually? Must solve for N. 15444 +/- PV; 20000 FV; 0 PMT; I/Y 9; CPT N = 3 years What rate earned if start at $15,444 and reach $20,000 in 3 years? Solve for I/Y. 15444 +/- PV; 20000 FV; 0 PMT; 3 N; CPT I/Y = 9%

Time to Double Your Money “Rule of 72” Enter 100 PV; 200 FV, I/Y, solve for N or Use Rule of 72 – says number of years to double is approximately equal to 72 divided by the interest rate. Doubling time ≈ 72 Interest Rate This is not in the text but is a useful shortcut (based on prior exam questions) At 10%, it will take 7.2 years to double your money. If you start with $1,000 you will have $2,000 after 7.2 years, $4,000 after 14.4 years and $8,000 after 21.6 years. If the rate is only 5%, it will take 14.4 years to double.

Present Value (PV) If I earn 10%, how much must I deposit today to have $100 in three years? $75.10 This is “inverse compounding” Discount rate – interest rate used to bring (discount) future money back to present For lump sums (only) PV and FV are reciprocals Future value was going forward. Now with PV we are moving back to the present. PV is the current value of a future payment. Remember: the discount rate is the interest rate used to bring future values back to present value.

Present Value Formula PV = FVn [ (1 + i) n ] [ 1 ] [ 1 ] PV = FVn [ (1 + i) n ] PVIF and FVIF for lump sums only are reciprocals. For 5% over ten years FVIF = 1.629 = 1 / .614 PVIF = .614 = 1 / 1.629

Present Value Interest Factor @2% @5% @10% Year 1 .980 .952 .909 Year 2 .961 .907 .826 Year 3 .942 .864 .751 Year 10 .820 .614 .386 Nick name- pivit This table shows how much a dollar received at some future time point is worth today. For example … If I want $100 in ten years, I need to deposit $82.00 if I earn 2% but only $38.60 if I earn 10%. Highlight.

Present Value of $100 The farther out we go in time or the greater the discount factor, the lower the present value.

Keystrokes $100 @5% for ten years For PV +/-100 FV; 0 PMT; 5 I/Y; 10 N; CPT PV = 61.39 For I/Y 100 FV; 0 PMT; +/-61.39 PV; 10 N; CPT I/Y = 5 For N 100 FV; +/-61.39 PV; 0 PMT; 5 I/Y; CPT N = 10 years

PV Decreases If Number of compounding periods (time) increases, The discount rate increases, And/or 3. Compounding frequency increases Here, there’s an inverse relation.

Annuities Series of equal dollar payments Usually at the end of the year/period If I deposit $100 in the bank each year starting a year from now, how much will I have at the end of three years if I earn 6%? $318.36 We are solving for the FV of the series by summing FV of each payment. This gets a little more complicated. Actually these are compound annuities – we are investing an equal sum of money at the end of each year for a certain number of years and letting it grow.

FV of $100 Annuity @ 6% End of PMT FVIF $ Year 3 $100 1.0000 * $100.00 $318.36 * The payment at end Year 3 earns nothing Here I am introducing a series of payments. Most important: we assume the payment occurs at the end of the period. Make sure BGN does not show in calculator’s display. It’s nothing more than the sum of each payment’s future value. Remember, the $100 deposited at the end of year 2 only earns interest for one year. The one at the end of Year 3 earns nothing.

Annuity Keystrokes What will I have if deposit $100 per year starting at the end of the year for three years and earn 6%? 0 PV; 100+/- PMT; 3 N; 6 I/Y; CPT FV = 318.36 PV is zero - nothing in the bank today I call this the “savings account problem”. Algebraic instructions have been handed out and are in my Blackboard pages. You will need them to do your homework problem.

Present Value of an Annuity Amount we must put in bank today to withdraw $500 at end of next three years with nothing left at the end? Present valuing each of three payments Keystrokes: 500+/- PMT; 0 FV; 3 N; 6 I/Y; CPT PV = 1,336.51 Just looked at FV, now it’s time for PV. Recall that the withdrawal occurs at the end of the year even though the money goes in today. Only have to put in $1,337 (to get back total of $1,500 because the deposit earns interest until withdrawn.) By solving for N, I could find how long my $1,337 would last.

PV of 5 Year $500 Annuity

Nonannual Compounding Invest for ten years at 12% compounded quarterly. What are we really doing? Investing for 40 periods (10 * 4) at 3% (12%/4) Make sure 2nd I/Y is set to 1. Need to adjust rate per period downward which is offset by increase in N Poorly addressed in text. Hope this is clearer.

Nonannual Compounding FVn = PV ( 1 + i/m) m * n m = number of compounding periods per year so per period rate is i/m And m * n is the number of years times the compounding frequency which adjusts to the rate per period There is an inverse relationship between the length of the compounding period and the effective annual interest rate – the shorter the compounding period, the greater the effective annual rate. Another way to say it:For a given nominal rate, the greater the compounding frequency, the greater the future value.

Compounding $100 @10% Compounding One Year 10 Years Annually $110.00 $259.37 Semiannually 110.25 265.33 Quarterly 110.38 268.51 Monthly 110.47 270.70 These are lump sums, not annuities. The more frequent the compounding and the longer the time, the greater the FV. 47 cents over one year might not sound like much but look at the difference over ten years - $11.63 on just $100.

Amortizing Loans Paid off in equal installments Makes it an annuity Payment pays interest first, remainder goes to principal (which declines) $600 loan at 15% over four years with equal annual payments of $210.16 Where’s the money go? The $210?

$600 Loan Amortization Total To Int To Prin End Bal Year 1 210.16 90.00 120.16 479.84 Year 2 210.16 71.98 138.18 341.66 Year 3 210.16 51.25 158.91 182.75 Year 4 210.16 27.41 182.75 0 Total is the total annual payment. As the loan ages, more and more of the payment goes to amortize principal. Because principal is going down, interest goes down and more goes to principal each year.

Calculate a Loan Payment $8,000 car loan payable monthly over three years at 12%. What is your payment? How many monthly periods in 3 yrs? 36 N Monthly rate? 12%/12 = 1%/mo = I/Y What is FV? Zero because loan paid out 8000+/- PV; 0 FV; 1.0 I/Y; 36 N; CPT PMT=265.71

Perpetuities Equal payments that continue forever Like Energizer Bunny and preferred stock Present Value = Payment Amount Interest Rate Preferred stock pays $8/yr, int rate- 10% Payment fixed at $8/ .10 = $80 market price The present value is the market price (An $8 annual payment discounted at 10%). The $8 represents an $8 payment (on a security originally valued at $100). The market price is lower than $100, because the payment amount is fixed and interest rates (the denominator) have increased. We will get to this again when we get to the bond chapter.

NPV & IRR Uneven Cash Flows Occur frequently in business problems All we are doing is present valuing each cash flow, positive or negative Need to switch to CF mode in calculator Keystrokes in handout and on web page Question on final but not FinCoach Be sure to read 4-12 to 4-14 in manual and/or Table X in Appendix A of text Reason not on FinCoach is this is addressed in Chapter 9.

Cash Flow Time Line Understanding time – big problem Remember number line from algebra Visualize in picture form when each cash flow occurs by time period, amount and sign. Time 0____1____2____3____4____5___→ Flows +200 +300 +50 → -100 0 0 Understanding time is one of the biggest problems in this course. No, this does not refer to your ability to read a digital watch but rather how time affects the solutions to problems. If there are periods when there are no inflows or outflows, how does this affect subsequent cash flows and the answer you are seeking?

Present Value Irregular Flow

Keystrokes You Should Know Future value of a single payment Present value for the same Future value of an annuity Annuity’s present value Loans including monthly payments, effective rates and time to repay Present value of a perpetuity There are virtually no TVM calculations on Exam 1 but the quizzes, FinCoach and the final are loaded with them.