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Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning What is the Time Value of Money (TVM)? • Because money earns money, the value of a dollar today is greater than the value of a dollar in the future. • Investors demand that not only do they get a return of the money invested, but that they get a return as well. • Sound financial decisions depend on an understanding of the basic mathematics of compound interest or return. • To make financial decisions, compare the value of two investments at the same point in time. Copyright 2007, The National Underwriter Company 1 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning TVM is Essential to Understanding: • The effect of time on the profitability of an investment. • How the value of an investment’s future returns affects the price that should be paid for it. • How to compute the value of an investment’s future return. • How to determine appropriate financial goals for future needs such as retirement planning, education funding and insurance, allowing for the effects of inflation and taxes. Copyright 2007, The National Underwriter Company 2 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning How Does Time Value of Money Analysis Work? • Three basic underlying rate of return principles that should govern every investment: – Timing, – Quality, and – Quantity. • In addition, the individual or family will consider personal risk preferences and financial resources. Copyright 2007, The National Underwriter Company 3 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Timing • Money now is better than money later. • Since money now can be invested to earn a return, the investor will have more money later. • Example: – Timing affects investment in the treatment of depreciation in the tax law. – Since accelerated depreciation reduces taxes faster than straight depreciation, thus putting more money in the hands of the investor sooner, the better of two otherwise financially equal investments will be the one that qualifies for accelerated depreciation. Copyright 2007, The National Underwriter Company 4 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Quality • Quality is another way of describing lower risk. • An investment that has a lower chance of losing money is a higher quality investment. • If two investments have an equal investment potential, but one is of higher quality, it is the better investment. Copyright 2007, The National Underwriter Company 5 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Quantity • Higher rate of return is better, all other things being equal. • Choosing an investment becomes like shopping: – Weigh the relative merits of two investments on the basis of all three factors. – Using the client’s risk preferences, make the choice offsetting higher risk against higher return. Copyright 2007, The National Underwriter Company 6 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning When Do Financial Planners Use Time Value of Money Analysis? • When weighing potential investments against the client’s risk preferences and timing needs. • When planning for future financial needs such as: – Education . – Retirement. – Estate planning. • While taking into consideration future income sources such as: – – – – Investments. Insurance. Social security. Retirement benefits. Copyright 2007, The National Underwriter Company 7 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Advantages Using TVM Analysis • Permits a quantitative comparison of alternative investments that have different rates of return and investment maturities. • To determine whether a particular investment is affordable. • Identifies situations in which current savings and investment will not be enough to fund future needs such as retirement, and can be used to determine how much additional savings is needed. Copyright 2007, The National Underwriter Company 8 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning PROBLEM • If I have a lump sum of $______ today, how do I calculate the value of that lump sum _____ years in the future assuming I earn ___% on my investment? • This is called calculating the Future Value of a Lump Sum, and is the most fundamental TVM calculation. Copyright 2007, The National Underwriter Company 9 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Choice of Tools to Solve the Problem • Use the Compound Interest Table in Appendix D. • Use a scientific calculator and the formula. • Use a financial calculator. • Use a computer spreadsheet program such as Excel. People who are preparing for the certification examinations need to practice using the financial calculator, since it is the only method that makes sense to use on the examination. Using the PC and spreadsheet software is often the best solution in the office, but skill with the financial calculator is useful in outside appointments. Copyright 2007, The National Underwriter Company 10 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Notation Used in the TVM Formulas • PV = Present Value • PMT = Payment (as in a loan or annuity calculation.) • FV = Future Value • N or n = number of periods • I, I/Y, or I/P = interest rate or yield • Begin or End – denotes whether the payment is made at the beginning or end of the time period. Copyright 2007, The National Underwriter Company 11 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Future Value of a Lump Sum (FVLS) • Start with a lump sum: That is the Present Value or PV. • If i = the interest or return earned per period, then the Future Value (FV) at the end of 1 period is FV= PV (1+i) – after two periods it is FV = PV (1+i)(1+i) or FV = (1+i)2 – After three periods it is FV = PV(1+i)(1+i)(1+i) = PV (1+i)3 – Therefore, after n periods it is FV = (1+i)n. Copyright 2007, The National Underwriter Company 12 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Example: FV of $10,000 @ 10% for 5 Years Beginning Balance Interest @ 10% Ending Balance Year 1 $10,000 $1,000 $11,000 Year 2 $11,000 $1,100 $12,100 Year 3 $12,100 $1,210 $13,310 Year 4 $13,310 $1,331 $14,641 Year 5 $14,641 $1,464 $16,105 Copyright 2007, The National Underwriter Company 13 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Using a Table • Use Appendix D, Compound Interest Table (future value of a lump sum). • This table reflects the amount $1 will be worth in a given number of years at various interest rates. • Multiply your Present Value (PV) by the number in the table representing the number of years and the interest rate to get the future value. • Drawbacks of the Table: – May have to interpolate if your interest rate falls between those in the table. – A comprehensive table is a thick book. – Rounding error can be significant. – You cannot take the Table to the CFP® Certification Exam. • The advantage of the table is that it is simple to use. Copyright 2007, The National Underwriter Company 14 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Using Excel Software • In Microsoft Excel 2003, move the cursor to the cell where you want to have your answer. • Click on the fx just above the worksheet. • The Insert Function window will appear. Type FV in the search box and press Go. Click OK. • The Function Arguments window will appear. Fill in the numbers for each argument. For example, if your interest rate is 6%, enter .06. Note that as you move the cursor to each argument that the software puts an explanation of the argument below. When you have finished entering the arguments, click OK. • Your answer will appear in the cell. • If you have a different version of Excel, or use Lotus, the process will be similar, but you may need to consult Help in that version of the software you are using. Copyright 2007, The National Underwriter Company 15 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Advantages and Disadvantages to Using Spreadsheet Software • Advantages: – Simple to use: Easy to do “What if” calculations. – Very accurate. – Can check the numbers in the arguments to make sure the calculation was done correctly. • Disadvantages: – Must have a PC and the software to use it. – Even a small laptop is sometimes unwieldy to take to a client meeting. – You cannot use a computer on the CFP® certification examination. Copyright 2007, The National Underwriter Company 16 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Using a Financial Calculator • Refer to your calculator manual for detailed instructions. • Clear all entries from the calculator. Put it in TVM mode. • Enter the quantity for the lump sum, then press PV. • Enter the number of periods (n) and press n. • Enter the interest rate or return rate and press I/Y or I/P (depending on make of calculator). • On HP calculators, press FV. On TI calculators, press compute then FV. Copyright 2007, The National Underwriter Company 17 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning About Financial Calculators • HP manuals are available at http://h20180.www2.hp.com/apps/Lookup?h_lang=en&h_cc=us&cc =us&h_page=hpcom&h_tool=prodhomes&h_query=calculator. • TI manuals are available at http://education.ti.com/us/global/guides.html#finance. • Both companies have tutorials online that show how to use their calculators to solve financial problems. Copyright 2007, The National Underwriter Company 18 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Computing the Present Value of a Future Lump Sum • PROBLEM: If I will have a lump sum of $______ in ___ years, how do I calculate the present value of my investment, assuming it will earn interest at the rate of ___? In other words, what is the equivalent today of $______ payable as a lump sum ___ years in the future? • You can use Appendix A – Present Value Table, or • Use the PV function in Excel, or • Use a Financial Calculator. Copyright 2007, The National Underwriter Company 19 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Development of the Formula for Present Value of a Future Lump Sum • Given the formula for computing the Future Value of a Lump sum, we can derive the formula for the Present Value: FV PV (1 i ) n • Solve for PV: FV PV (1 i ) n Copyright 2007, The National Underwriter Company 20 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Variables Affecting TVM Calculations • For every TVM problem, there are five variables: – – – – – PV = Present Value FV = Future Value PMT = Payments N = Number of Periods I, I/Y or I/P = interest rate or return per period. In addition, note whether payments occur at the beginning or end of the period, and how often the interest is compounded. Copyright 2007, The National Underwriter Company 21 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Tips on Solving TVM Statement Problems • Write down your five variables: FV, PV, PMT, N and I. As you read the problem, fill in the quantities next to the appropriate variable. • If the return is expressed as an annual return, but it is compounded, you will have to divide the annual return and multiply the number of years by the appropriate number (4 for quarterly, 12 for monthly) to solve the problem. • If you are doing a lump sum problem on a calculator, the PMT is 0. It is recommended that you put in the 0 to be safe, since prior problem’s quantities will be saved in the calculator. Copyright 2007, The National Underwriter Company 22 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Setting Up Your Calculator • Many calculators come with the number of payments per year set up as 12. If you were doing only one kind of problem that always had 12 payments per year, that would be fine. However, for financial planning, where you do varied calculations, set up your calculator for P/Y = 1. • Also set up your calculator for 4 decimal places. You can always round at the end. • Your calculator manual will have directions on how to do both of these setup functions so that they become the default for you. • It is definitely worth your time to take the calculator manufacturer's online tutorials. Copyright 2007, The National Underwriter Company 23 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Sign and Flow on the Financial Calculator • Remember that a minus sign designates an outflow and a positive number represents an inflow. If your Future Value is an inflow (positive) then your Present Value must be an outflow (negative). When you invest, your outlay (PV) is the negative number. This inflow and outflow convention is also used in the Excel software. • The flow of PMT is also signed. If you are receiving money, it is plus: If you are paying money, it is minus. • If you get Error 5 on your calculator, it is almost always due to an improperly signed inflow or outflow. Copyright 2007, The National Underwriter Company 24 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Problems Using Future Value of a Series of Payments • If, beginning today, I invest $______ a year for ___ years, how do I calculate what the value of that series of investments would be ___ years from now assuming I earn a compounded interest rate of ___% on my investments? • This type of problem requires the calculation of the future value of a regular series of payments. Where each payment is made at the beginning of a compounding period (for example, at the beginning of each year), the process is known as an “annuity due” or an “annuity in advance.” • If the first payment in the series of investments is not made until one year from now, the process is known as an “ordinary annuity” or an “annuity in arrears.” Copyright 2007, The National Underwriter Company 25 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Future Value of an Annuity Due • Remember that “Annuity Due” means payments are made at the beginning of each period. On your financial calculator and in Excel, you will choose “Begin.” In Excel, “Begin” is coded as a 1 in the function argument TYPE. • This calculation only works when the payments are made every period and are for the same amount. Otherwise, a more complex method must be used. • To use the tables, use Appendix E, Compounded Annual Annuity (In Advance) Table (Future Value of an Annuity Due). Copyright 2007, The National Underwriter Company 26 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Future Value of a Regular Series of Payments The Future Value of a Regular series of equal payments is the Sum of N lump sum calculations. If the payment is at the beginning of each period, then: FV ( PMT1 ) PMT (1 i) n FV(PMT2 ) PMT (1 i) n -1 And so on to the nth payment. You then add up the Future Value of each payment to get the Future Value of the Series. Using the formula for the sum of a series, the generalized result is: (1 i) n 1 FV PMT (1 i) i Copyright 2007, The National Underwriter Company In your calculator manual this may be called the Future Value of an Annuity Due (FVAD). The equation is different for payments at the end of each period. That is a Future Value of an Ordinary Annuity (FVOA). 27 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Setting up a Problem to solve Using a Calculator or Software • Write down your five variables, the compounding period and whether it is begin or end, thus: • FV = • PMT = • PV = • I/P = • N= • Begin? Compounding: Copyright 2007, The National Underwriter Company 28 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Setting Up the Problem for Your Calculator or Spreadsheet • Fill in the quantities from your problem. As an example, consider “What is the future value of monthly payments of $200 for 10 years, at 6% interest, compounded monthly, that start today?” • FV = ? • PMT = 200 (Press the +/- key to make it negative, since it is an outflow.) • PV = 0 • I/P = 6% per year/12 months per year = .5 per period • N = 10 years x 12 payments per year = 120 periods • Begin? Yes Compounding: Monthly Copyright 2007, The National Underwriter Company 29 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Solving the Problem • Enter the quantities into your calculator that you recorded in the worksheet. • Follow your calculator instructions to compute the answer, OR • Use the quantities that you recorded to fill in the function arguments on your spreadsheet software. Note: If you are studying for the CFP® certification examination, practicing by working many problems each week will help you become skillful at using your calculator and remove one source of anxiety on the examination. Some successful students make up problems, work them on the calculator and then check their answer using the Excel software, giving themselves double practice. Copyright 2007, The National Underwriter Company 30 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Computing the Present Value of a Regular Series of Receipts • This is the Present Value of an Annuity. If the payments start immediately and continue at the beginning of each period, it is the Present Value of an Annuity Due (PVAD). • If the first payment is at the end of the first period and every period thereafter, then you are computing the Present Value of an Ordinary Annuity (PVOA). Copyright 2007, The National Underwriter Company 31 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Computing the Present Value of a Regular Series of Receipts • PROBLEM: If, beginning one year from today, I receive $______ a year for ___ years, how do I calculate the present value of that series of payments, assuming a ___% discount rate? • SOLUTION: – Use Appendix C, Compound Discount Table (present value of an ordinary annuity), or – The PV function in Excel or – A financial calculator. Copyright 2007, The National Underwriter Company 32 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Equation for the Present Value of an Ordinary Annuity (PVOA) 1 PVOA PMT (1 i ) 1 n ( 1 i ) An ordinary scientific calculator can be used to solve a TVM problem using this formula, or any of the previous formulas. Choose End for a financial calculator, or insert a 0 into or leave blank the TYPE function argument in Excel. Copyright 2007, The National Underwriter Company 33 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Formula for the Present Value of an Annuity Due 1 PVAD PMT (1 i ) 1 (1 i ) n (1 i ) Choose Begin on a Financial Calculator or insert 1 into the TYPE function argument for PV in Excel. Copyright 2007, The National Underwriter Company 34 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Practical Examples, Problem 1: • PROBLEM: Rich Stevens, age 53, has just inherited $100,000 which he would like to use as part of his retirement nest egg. Rich would like to know just how much the $100,000 will be worth in 12 years, when he will reach age 65, assuming the funds can be invested for the entire period at a 12% annual rate. He would also like to know what the future value of the $100,000 would be in only 7 years, when he reaches age 60, in case he decides to retire early. Copyright 2007, The National Underwriter Company 35 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Problem 1 • Using the mini-worksheet below, write down your five variables, the compounding period and whether it is begin or end, thus: • FV = • PMT = • PV = • I/P = • N= • Begin? Compounding: Copyright 2007, The National Underwriter Company 36 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Problem 1 Worksheet • • • • • • FV = ? PMT = 0 PV =100,000 I/P = 12 N = 12, then redo for 7 years Begin? N/A* Compounding: Annual Begin or End only matters when there is a series of payments. When it is a lump sum calculation, Begin or End is irrelevant. Note that Rich’s age is also irrelevant, and that using this mini-worksheet makes it clear that we do not have to use his age for anything other than the years until he will retire, avoiding confusion. Copyright 2007, The National Underwriter Company 37 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Problem 1 (continued) • Since this is the Future Value of a Lump Sum, we could use Appendix D, Compound Interest Table (future value of a lump sum) to solve the problem. • The FV function in Excel could be used, filling in the function arguments with the numbers from the worksheet. • An ordinary calculator could be used with the FV formula. • The variables can be entered into a financial calculator. • Whatever method is used, the answers are approximately $389,600 in 12 years or $221,070 in 7 years. Copyright 2007, The National Underwriter Company 38 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Problem 2 • PROBLEM: Now that Rich knows how much the $100,000 inheritance will be worth in both cases, he would like to know how much he could withdraw from the fund in equal installments at the end of each year from the year he retires until he reaches age 70½, the year he must start withdrawing funds from his individual retirement account (IRA). Rich assumes the funds will continue to earn at a 12% annual rate. • In other words, Rich would like to know the annual year-end payment from (1) a 6-year annuity (from age 65 to the year he will be 70½), earning 12% annually on a principal sum of $389,600, and (2) an 11-year annuity (from age 60 to the year he will be 70½), earning 12% annually on a principal sum of $221,070. Copyright 2007, The National Underwriter Company 39 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Problem 2 Worksheet • • • • • • Scenario A (Retire at 65) FV = 0 PMT = ? PV = $389,600 I/P = 12 N=6 Begin? No Scenario B (Retire at 60) FV = 0 PMT = ? PV = $221,070 I/P = 12 N = 11 Compounding: Annual Copyright 2007, The National Underwriter Company 40 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Problem 2 (continued) • Any of the four methods can be used. – The Table that should be used is Appendix C. Pmt equals PV divided by PVOA factor. • Whichever method used should yield answers of about $94,761 per year at age 65, and $37,232 at age 60. • Note that using the financial calculator, since it saves the variables unless actively cleared, all you would have to do to compute Scenario B is enter new quantities for PV and N, and then compute PMT. Since everything else is the same, you do not have to re-enter all the variables. Copyright 2007, The National Underwriter Company 41 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Problem 3 • PROBLEM: Rich has determined that he will need $60,000 per year from the inheritance fund to handle his living needs until he reaches age 70½. Assuming the fund will continue to earn 12% annually, at what age can Rich afford to retire? (Rich has already decided not to touch his IRA funds until the latest possible date, believing he can cover his living costs with the inheritance until that time. He is even willing to adjust his retirement date by a year or so if need be.) Copyright 2007, The National Underwriter Company 42 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Problem 3 • Note that we know that the answer will be greater than age 60 and less than age 65. It is important to start asking yourself if your answer is reasonable. Figuring out an approximation ahead of solving the problem can help you avoid errors. • In this problem, we are solving for N, but we also do not know the starting PV, since it will vary according to the age of retirement. Using a financial calculator or a spreadsheet, the problem is still not difficult, since we only have to change one variable at a time to check various retirement scenarios. Copyright 2007, The National Underwriter Company 43 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Problem 3 • First, compute the value of the inheritance at each age from 61 to 64. If you have more than 4 scenarios to check, you should start at the middle of the range, compute it, then determine whether you need more or less, thus halving the remaining work to be done with each trial. Copyright 2007, The National Underwriter Company 44 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Problem 3, Worksheet 1 • • • • • • • FV = ? PMT = 0 PV =100,000 I/P = 12 N = 11, 10, 9, or 8 Begin? N/A* Compounding: Annual Recomputed for each N, changing only that variable, the value of the inheritance at each year is: – Age 61 – $247,596 Age 62 – $277,307 Age 63 – $310,584 Age 64 – $347,855 • Now it is simply a matter of determining which one of these lump sums will yield an income of at least $60,000 per year until Rich reaches the year in which he turns 70½. Copyright 2007, The National Underwriter Company 45 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Problem 3, Worksheet 2 • • • • • • FV = 0 PMT = ? PV = $247,596; $277,307; $310,584; or $347,855 I/P = 12 N = 9, 8, 7, or 6 Begin? N/A* Compounding: Annual Here, once we set up the financial calculator for the first scenario, we only have to change the PV and the N to solve each scenario. Copyright 2007, The National Underwriter Company 46 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Problem 3, Results Retire at Age # of Years Annual Lump Sum to Age 70 ½ Income from Accumulated (Rounded up to age 71) Inheritance 61 $247,596 10 $43,821 62 $277,307 9 $52,045 63 $310,584 8 $62,522 64 $347,855 7 $76,221 Copyright 2007, The National Underwriter Company 47 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Problem 3, Discussion of Results • Thus, using scenario analysis and TVM calculations, it has been determined that Rich should wait until age 63 to retire to meet his stated goals. • The best tool for this problem would be a spreadsheet, since you could set up the formula once then simply copy to cells to compute the numbers for every year. • Note, however, that this analysis assumed that there was no inflation, and that $60,000 per year would buy the same goods and services then as now. Class discussion: Will the extra $2,522 offset inflation? Copyright 2007, The National Underwriter Company 48 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Problem 4 • PROBLEM: Rich has decided that he wants to retire at age 60. He would like to know how much of his other funds need be set aside with his $100,000 inheritance in order to reach his goal of a $60,000 annuity from age 60 until the year he reaches age 70½. Rich assumes the funds can continue to earn at a 12% annual rate. Copyright 2007, The National Underwriter Company 49 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Problem 4, Worksheet 1 • The first step is to determine how much he needs to have at age 60 to give him $60,000 per year until age 70½. • FV = 0 • PMT = 60,000 • PV = ? • I/P = 12 • N = 11 • Begin? No Compounding: Annual • The result is $356,262. We already know that the inheritance will be worth $221,068, so his other money must accumulate to $135,194 by age 60. Copyright 2007, The National Underwriter Company 50 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Problem 4, Worksheet 2 • • • • • • FV = $135,194 PMT = 0 PV = ? I/P = 12 N=7 Begin? N/A Compounding: Annual Rich needs $61,155 lump sum to add to his inheritance to accumulate the needed amount by age 60. (Note: The difference between the $61,137 indicated in the textbook and the $61,155 computed here is due to differences in rounding. An Excel spreadsheet was used to compute these numbers.) Copyright 2007, The National Underwriter Company 51 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Including Taxes, Inflation and Growth in TVM Analysis • Taxes and inflation reduce return on investment. • Accounting for taxes and inflation is difficult since the rates can be different at different times and in different circumstances. • However, to design reasonable strategies for personal financial planning, it is necessary to make the best possible approximation of what the effect of taxes and inflation will be. Copyright 2007, The National Underwriter Company 52 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Taxes and Investment Return • If t is the marginal tax percentage, then: – After-tax return = pre-tax return (1-t) The result can be used as the Interest Rate in any of the formulas where income is taxed annually. • When taxes are deferred, use the before-tax return, then apply the tax rate at the end of the deferral period. • When there is a combination of ordinary income and capital gains, the calculation becomes complex. Usually the best tool will be to break down the return into its component parts and use spreadsheet software. Copyright 2007, The National Underwriter Company 53 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Inflation • Usually, when calculating long-term needs, an adjustment to return is needed to account for loss of purchasing power due to inflation. • The previous formulas can be used if the interest rate is adjusted for inflation. Simple subtraction does not give an accurate answer. Instead, use the formula: where d is the interest rate discounted for inflation, r is the nominal interest rate, and i is the inflation rate. Use d for the interest rate in calculations. • You may see this formula written in the second manner in some other text books. The two formulas are mathematically equivalent. r i d 1 i Copyright 2007, The National Underwriter Company 1 r d 1 1 i 54 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Inflation Adjusted Rate of Return and Annuities • Some annuities (and the benefits on some Long Term Care policies) have a built-in growth rate meant to adjust for inflation. • The following slides give the formulas adjusting each of the previous TVM formulas for an increasing payment with a growth rate g. • The growth formula is the same as the inflation formula, but is expressed as ρ. Copyright 2007, The National Underwriter Company 55 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Calculating ρ (growth adjusted rate) rg 1 g where r is investment return and g is growth. Example: The investment return is 12% and payments are growing by 4% per year. 12% - 4% is 8%, and 1 + 4% is 104%, so the growth adjusted rate is 7.69%. Copyright 2007, The National Underwriter Company 56 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Present Value of an Ordinary Annuity, Adjusted for Growth (PVOAg) PMT PV 1 g 1 (1 ) n if 0 PMT (n) PV if 0 1 g Copyright 2007, The National Underwriter Company 57 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Present Value of an Annuity Due, Adjusted for Growth (PVADg) 1 (1 ) n PV PMT (1 ), if 0 PV n( PMT ) , if 0 Copyright 2007, The National Underwriter Company 58 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Future Value of an Ordinary Annuity, Adjusted for Growth (FVOAg) n ( 1 ) 1 n 1 FV (1 g ) , if 0 FV PMT ( n)(1 g ) n 1 , if 0 Copyright 2007, The National Underwriter Company 59 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Future Value of an Annuity Due, Adjusted for Growth (FVADg) n ( 1 ) 1 n 1 FV PMT (1 g ) (1 r ), if 0 FV PMT ( n)1 g n 1 (1 r ) , if 0 Copyright 2007, The National Underwriter Company 60 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Net Present Value and Internal Rate Of Return • Not all investments result in regular payments. For example, an investment in a business project typically involves negative cash flow in the beginning, then (hopefully) results in positive cash flow later. • Making an investing decision among several projects, each with differing cash flows, can be difficult. • The use of Net Present Value (NPV) and Internal Rate of Return (IRR) analysis allows comparison on the same basis and aids decision. Copyright 2007, The National Underwriter Company 61 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Methods for Comparing Alternative Investments • Net present value • Internal rate of return • Adjusted rate of return • Pay back period • Cash on cash Note: The most accurate is NPV, but the others are used for a quick assessment, or to adjust for the effects of taxes. Copyright 2007, The National Underwriter Company 62 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Net Present Value • “Net” present value is the difference between – (1) the present value of all future benefits to be realized from an investment and – (2) the present value of all capital contributions into the investment. • A negative net present value should result in an almost automatic rejection of the investment. • A positive net present value indicates that the investment is worth further consideration. Copyright 2007, The National Underwriter Company 63 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Evaluating Net Present Value • A discount rate of return on investment must be assumed in computing NPV. • What is usually used as the discount rate is the minimum acceptable rate of return. • A negative NPV will indicate that the investment does not meet the investor’s minimum. • In comparing investments using NPV, the risk of the two investments must be equal for the comparison to be valid. Copyright 2007, The National Underwriter Company 64 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Example Assume a beginning of the year investment opportunity requiring a lump sum outlay of $10,000, which is currently invested in a money market fund at 6% annual net after taxes. The investment proposal projects the following after-tax cash flows at the end of each year. Assume 6% is the minimum required rate. Year 1 2 3 4 5 Cash Flow $2,000 1,500 750 500 10,000 Total 14,500 Computing the PV of each cash flow and adding them, the PV is $11,720, so the NPV is $1,720, and the proposed investment deserves consideration. If, however, the investor needs 15% return to compensate for the additional risk over the 6% money market rate, then the present value is $8,624, which gives a negative NPV of -$1,376, and the investment should be rejected. Copyright 2007, The National Underwriter Company 65 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning How to Compute Net Present Value (Lump Sum Investment, Single Future Receipt) • Given: At beginning of Year 1 invest $10,000, receive $15,000 at end of year 5. The investor’s discount rate is 6%. • PV of $15,000 at 6% for 5 years is $11,210. • $11,210 - $10,000 = $1,210 – Therefore, the investment should be considered. Copyright 2007, The National Underwriter Company 66 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning How to Compute Net Present Value (Lump Sum Investment, Multiple Future Receipt) • Given: $10,000 outlay at beginning of Year 1. Future Receipts are: Year 1 2 3 4 5 Total Receipts Amount Received $2,000 1,500 750 500 $10,000 $14,750 PV @ 6% $1,887 1,335 630 396 $7,473 $11,721 • Net Present Value is positive $1,721, so investment should be considered. Copyright 2007, The National Underwriter Company 67 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning How to Compute Net Present Value (Multiple Investments, Multiple Future Receipts • Given: Extend last problem so that outlays at beginning of Year 1 of $5,000 and Year 2 of $5,000. Receipts are still the same: Year 1 2 3 4 5 Total Receipts Amount Received $2,000 1,500 750 500 $10,000 $14,750 PV @ 6% $1,887 1,335 630 396 $7,473 $11,721 • PV of Outlays is $9,717. PV of Inflows is $11,721. NPV is $2,004. Copyright 2007, The National Underwriter Company 68 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Internal Rate of Return • The Internal Rate of Return is the rate at which the NPV of the inflows and the NPV of the outflows is equal. • Determines what percentage rate of return cash inflows will provide based on a known investment (cash outflow) and estimated cash inflows. • This is still a TVM calculation. Unlike the NPV calculation, the discount rate is the variable that is being sought, rather than the present value. Copyright 2007, The National Underwriter Company 69 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning How to Compute Internal Rate of Return • Solving manually for the Internal Rate of Return is a trial and error process. One computes the NPV using the best guess of the IRR. If NPV is positive, then a higher rate is tried. If NPV is negative, then a lower rate is tried. Continue this process until the NPV is roughly equal to $0; the result is the IRR. • The simplest way to calculate is to use a financial calculator or spreadsheet software, both of which do the same iterative process, but much faster than one can do manually. Copyright 2007, The National Underwriter Company 70 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Shortcomings of the IRR Method • Assumption that the cash flows are consumed and not reinvested. • Also cannot assume that the cash flows are reinvested at the same rate. • Despite these shortcomings, this is one of the most widely used tools for evaluating investments. Copyright 2007, The National Underwriter Company 71 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Consider 5 Possible Investments Year 0 1 2 3 IRR • Project Cash Flows A B C D E ($1,000) ($1,000) ($1,000) ($1,000) ($1,000) 100 50 (200) 200 600 100 50 (200) 200 600 1,100 1,215 1,793 869 (55) 10% 10% 10% 10% 10% Each project has a 10% IRR. Yet each has different unrecovered cash flows at a given point in time. The interest rate at which the recovered cash flows could be invested could make a great difference to the investor. Copyright 2007, The National Underwriter Company 72 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Other Weaknesses of the IRR Method • The investment with the highest IRR is not necessarily the “best” investment among a mutually exclusive set. • The unmodified IRR method does not consider realistic reinvestment rates for positive cash flows or realistic borrowing rates for negative cash flows over the holding period. • An investment project may have multiple IRRs. • Solving for the IRR often requires a series of iterative calculations to successively home in on the IRR. • However, financial calculators and computer software programs, have built-in functions that are adequate in most cases. Copyright 2007, The National Underwriter Company 73 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Modified Internal Rate of Return Methods Determining the appropriate Rate of Return: – Using a “safe” rate of return such as Treasury Bills. – Using an interest rate available to the investor on another investment. – Using a rate at which money could be borrowed. – The circumstances for each investment scenario need to be analyzed to select the right rate to use. Copyright 2007, The National Underwriter Company 74 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Pay Back Period • Based on the concept that the sooner the original investment is recovered, the better is the investment proposal. • However, this method may cause an investor to reject a project with a much higher NPV that requires longer to recover the original investment. Copyright 2007, The National Underwriter Company 75 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Cash on Cash • This method ignores time value of money and just examines how much cash the investor recovers annually. • Can cause the investor to reject a project with a higher NPV than one that returns money sooner. Copyright 2007, The National Underwriter Company 76 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Risk, Probabilities, And Modeling • In evaluating investments, there are numerous risks to evaluate that are beyond the scope of this book. (See Tools & Techniques of Investment Planning.) • In addition, the financial planner must consider noninvestment risks such as risk of dying and disability. • Time Value of Money calculations are used in techniques designed to evaluate and assess risks in investments. • Monte Carlo simulations, a recognized statistical technique, take in some or all these factors. Copyright 2007, The National Underwriter Company 77 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Monte-Carlo Simulation • Monte-Carlo simulation is the process of assessing the likelihood of an expected outcome. • Although developed during World War II, the widespread use of Monte Carlo analysis required the development of computers that could run the many scenarios in a reasonable period of time. • Part of the intelligent use of Monte Carlo analysis is the selection of factors to consider. • Monte Carlo analysis is a well-known technique used in corporate financial analysis and portfolio management that is increasingly being used by financial planners to help assess probabilities. See Gambera, M. (2002). It’s a long way to Monte Carlo. Business Economics. 37:3(34). for a discussion of assumptions and reliability that must be considered when using Monte Carlo analysis. Copyright 2007, The National Underwriter Company 78 Time Value of Money and Quantitative Analysis Chapter 20 Tools & Techniques of Financial Planning Learning More about TVM • Tools & Techniques of Financial Planning includes many excellent readings about TVM. • In addition, consider: – Booth, L. (2004). Formulating retirement targets and the impact of time horizon on asset allocation. Financial Services Review. 13.1:1-17. – Opiela, N. (2004). Retirement distributions: Creating a limitless income stream for an 'unknowable longevity‘. Journal of Financial Planning. 17.2:36-41. – Elger, J. (2003). Calculating life insurance need: Don't let the tools fool you. Journal of Financial Service Professionals. 57.3:38. Copyright 2007, The National Underwriter Company 79