Chapter 5 – Important StuffTime Value of Money (TVM)Financial Calculator KeysTI Calculator Manual Strongly Suggested ReadingsCalculator TipsCompound Interest @ 6%Future Value (FV)Future Value Interest FactorReading the Formulas and Tables FVn = PV* (1 + i)nFuture Value of $100FV Can Be Increased ByFV – Other KeystrokesTime to Double Your Money “Rule of 72”Present Value (PV)Present Value FormulaPresent Value Interest FactorPresent Value of $100Keystrokes $100 @5% for ten yearsPV Decreases IfAnnuitiesFV of $100 Annuity @ 6%Annuity KeystrokesPresent Value of an AnnuityPV of 5 Year $500 AnnuityNonannual CompoundingSlide 26Compounding $100 @10%Amortizing Loans$600 Loan AmortizationCalculate a Loan PaymentPerpetuitiesNPV & IRR Uneven Cash FlowsCash Flow Time LinePresent Value Irregular FlowKeystrokes You Should Know1Chapter 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 rates2Time 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 goal3Financial Calculator Keys•PV - Present value•FV - Future value•PMT - Amount of the payment•N - Number of periods (years?)•I/Y - Interest rate per period4TI Calculator ManualStrongly 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 - All5Calculator TipsDecimals 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 error6Compound Interest @ 6%Year Begin Interest FV1 $100.00 $6.00 $106.002 106.00 6.36 112.363 112.36 6.74 119.107Future Value (FV)Algebraically FVn = PV (1 + i)nUnderlies all TVM calculationsKeystrokes: 100 +/- PV; 3 N; 0 PMT;6 I/Y; CPT FV = 119.10One cash flow must be negativeError 5 means you forgot a negative sign8Future Value Interest FactorYear @2% @6% @10%1 1.020 1.060 1.1002 1.040 1.124 1.2103 1.104 1.191 1.611 10 1.219 1.791 2.5949Reading the Formulas and TablesFVn = 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 = 595.5010Future Value of $10011FV Can Be Increased By 1. Increasing the length of time it is compounded2. Compounding at a higher rateAnd/or3. Compounding more frequently12FV – 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%13Time to Double Your Money“Rule of 72”•Enter 100 PV; 200 FV, 10 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 Rate14Present Value (PV)If I earn 10%, how much must I deposittoday to have $100 in three years? $75.10This is “inverse compounding”Discount rate – interest rate used to bring (discount) future money back to presentFor lump sums (only) PV and FV are reciprocals15Present Value Formula[ 1 ]PV = FVn [ (1 + i) n ]PVIF and FVIF for lump sums only are reciprocals. For 5% over ten yearsFVIF = 1.629 = 1 / .614PVIF = .614 = 1 / 1.62916Present Value Interest Factor @2% @5% @10% Year 1 .980 .952 .909 Year 2 .961 .907 .826Year 3 .942 .864 .751Year 10 .820 .614 .38617Present Value of $10018Keystrokes$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 years19PV Decreases If1. Number of compounding periods (time) increases,2. The discount rate increases,And/or3. Compounding frequency increases20Annuities•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.21FV of $100 Annuity @ 6%End ofPMT FVIF $ Year 3 $100 1.0000 * $100.00Year 2 100 1.0600 106.00Year 1 100 1.1236 112.36$318.36* The payment at end Year 3 earns nothing22Annuity KeystrokesWhat 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 = 318.36PV is zero - nothing in the bank today23Present Value of an AnnuityAmount we must put in bank today towithdraw $500 at end of next three years, earn 6% and have nothing left at the end?Present valuing each of three paymentsKeystrokes: 500+/- PMT; 0 FV; 3 N; 6 I/Y; CPT PV = 1,336.5124PV of 5 Year $500 Annuity25Nonannual 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 N26Nonannual 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 period27Compounding $100 @10%Compounding One Year 10 YearsAnnually $110.00 $259.37Semiannually 110.25 265.33Quarterly 110.38 268.51Monthly 110.47 270.7028Amortizing 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 $210.1629$600 Loan Amortization Total To Int To Prin End BalYear 1 210.16 90.00 120.16 479.84Year 2 210.16