Thomas D. Le
Financial Mathematics
Part II  Problems
Chapter 3 The Time Value of Money The previous chapters gave us basic understanding of the fundamental facts and concepts of the financial markets. This chapter presents financial mathematics to quantify some of these concepts with a very practical purpose of applying the concepts to daily life. In the most basic scenario, when an investor invests a sum of money in an instrument and holds it over time, it accumulates interest. The longer the term of the investment, the greater the accumulation. This is the simple meaning of time value of money. We will discuss a number of ways money generates money over time. Interest Rates The first topic of discussion must be the interest rate since most everyone borrows money at some time to buy a car or home, or invest surplus cash. To the borrower the interest rate is the cost of money, or the price paid to postpone payment until a future time. To the investor it is the earnings she can expect to make. If you borrow P dollars to be repaid in three years, the interest rate i charged is the cost of postponing the repayment of P. Your creditor expects to get back P in addition to the return on investment (ROI) equal to the interest earned (P * i * 3) after three years. Conversely, if you invest P dollars for three years at the interest rate of i, the rate of return of your investment is i, and you want to recuperate your principal P and earn an additional (P * i * 3) after three years. There are all sorts of interest rates for: mortgages, car loans, home equity loans, consumer loans (charged by retail stores), credit card loans, business loans. There are discount rate (charged by the Fed to commercial banks), Fed funds rate (for overnight loans among commercial banks), prime rate (charged by commercial banks to their most valued customers), CD rate, Treasury Tbill rate, Tbond rate, and more. Just about the only thing they have in common is that they are all different. Yet the interest rate is at the heart of all the calculations. Viewed from different perspectives, the interest rate is the cost of money for the debtor, the income for the creditor, the opportunity cost or return on investment for the investor, the hurdle rate or opportunity cost of capital for the finance manager of a firm, and so on. Two basic financial principles underlie the investor’s decision. First, a dollar today is worth more than a dollar tomorrow because she can invest it now to start earning interest immediately. Second, a safe dollar is worth more than a risky one. In all the discussion below, we assume that inflation has no effect on the calculations. In reality, everyone should worry about inflation. For simplicity, we assume that the prevailing interest rate already includes the expected inflation rate. 3.1 The Compound Value (CV) or Future Value (FV) The concept of compounding, i.e., accumulating, is very important. The compound value or future value of a sum of money invested (called principal) is its value plus accrued interest over time. When an amount is invested at a given interest rate, the interest starts accruing. How interest accrues makes a difference in the total return on investment. Simple Interest In an investment that earns simple interest, the interest earned is calculated on the original principal for each time period. A $10,000 certificate of deposit (CD) placed at a simple interest rate of 5% per year for three years will yield: Year Principal Interest earned 1 10,000 10,000 * 0.05 = 500
After three years, the total interest income is: $500 * 3 = $1,500. The total investment at the end of three years is: $10,000 + $1,500 = $11,500. Compound Interest – Frequency of Compounding With the compound interest the interest earned after each period is added to the principal to earn more interest in the following period. In the following example the compounding frequency is annual, i.e., interest accumulates only once a year. If the interest is compounded annually, the CD will earn: Year Principal Interest earned 1 10,000 10,000 * 0.05 = 500
The total compound interest earned after three years is: $500 + $525 + $551.25 = $1,576.25. And the compound value of the CD at the end of three years is: $10,000 + $1,576.25 = $11,576.25, which is a nontrivial improvement over the simple interest. And the gains become larger as the period becomes longer for the same annual interest rate. They also become larger as the principal gets larger. In the latter scenario, the interest after each year is added to the principal to earn more interest. This is the secret of compounding. Clearly, the more often the compounding takes place, the greater income it generates. A quarterly compounding interest produces greater income than a semiannual one. The compound or future value after one year is summarized as follows: V_{1} = P_{0} + P_{0} r = P_{0} (1 + r) where V_{1} is the compound value at the end of 1 period, P_{0} the principal or beginning amount at time 0, and r the annual interest rate. The compound value after the second year becomes: V_{2} = V_{1 }(1 + r) = P_{0} (1 + r) (1 + r) = P_{0 }(1 + r)^{2} The compound value after the third year is: V_{3 }= V_{2} (1+ r) = P_{0 }(1+ r)^{2} (1+ r) = P_{0 }(1+ r)^{3 } In general, the compound amount at the end of any year t is given by the fundamental compound interest equation: V_{t }=_{ }P_{0 }(1+ r)^{t} (1)^{ } Note that formula (1) may be easily applied by using an Excel spreadsheet. The following table is a spreadsheet, and contains five periods. The table may be expanded to cover as many periods as desired.
Note that the ending balance of each period becomes the beginning balance of the following period. Thus the interest earned is added to the principal to earn more interest. The quantity (1+ r)^{t} in formula (1) is called compound value interest factor (CVIF). Because the CVIF may be calculated mechanically, tables have been constructed for wide ranges of r and t. Appendix 1 contains a partial table of CVIF that gives the compound (future) value of $1 for various interest rates r and number of periods t. Note that the values of CVIF are approximate only, because they are arbitrarily shown with 4 decimal places. This affects the final result, which varies slightly from that obtained with a spreadsheet formula, which holds up to 15 decimals. Substituting the CVIF for (1+ r)^{t} in formula (1) yields: V_{t }=_{ }P_{0 }(CVIF) (2)^{ } In our example above, the CD value at the end of the third year is: V_{t }=_{ }P_{0 }(CVIF) V_{t }=_{ } 10,000 * 1.1576 ≈ $11,576.00 (rounded) Our Excel spreadsheet gives the value of $11,576.25 (rounded to two decimal places). We use the CVIF table to avoid having to calculate the values of (1+ r)^{t}. First locate the annual interest rate at the top of each column. In our case, it is 5%. Go down the column. Each row in this column gives the compound or future value of $1 at the end of the period (indicated in the leftmost column) at 5% per year. For example, the CVIF for period 5 is 1.2763 (due to rounding). That is, for every dollar invested at 5%, its future value is $1.2763 at the end of 5 years. Linear Interpolation Interest rates in the CVIF table are integers. For interest rates that fall between two consecutive values in the table, we use linear interpolation to determine the interest factor (IF). Linear interpolation is based on the assumption that intermediate values are uniformly distributed between given values. The general formula for the IF is: IF for intermediate interest rate = [ (r – r_{L}) / (r_{H} – r_{L}) ] (IF_{H } – IF_{L}) + IF_{ L} (3) where r is the interest rate in question, r_{H} the interest rate just higher than r in the table, r_{L} the interest rate just lower than r in the table,_{ }IF_{H } and IF_{L} the interest factors for r_{H} and r_{L} CVIF for 7.25% at year 10 = [ (7.25 – 7) / (8 – 7) ] (2.1589 – 1.9672) + 1.9672 = (0.25 / 1) (0.1917) + 1.9672 = 2.0151 If the original investment amount is $5,000, its compound amount after 10 years at 7.25% per annum is: V_{10} = P_{0 }(CVIF) V_{10} = $5,000 * 2.0151 = $10,075.63 This investment doubled in value after 10 years. Total Compound Interest By comparing the compound amount and the principal at time zero, we can derive the compound interest: Compound interest = V_{t }_{ }P_{0} (4) Compound interest = $10,075.63  $5,000 = $5,075.63 Using a Formula in an Excel Spreadsheet to Calculate the Compound Value Formula (1) may be written in an Excel spreadsheet as follows: V_{t }=_{ }P_{0 }(1+ r)^{t} (1) In an empty cell in an Excel spreadsheet, enter the formula for the compound value: = P_{0 }* POWER((1+ r),t) or = P_{0 }* (1+ r)^t (5) where P_{0} is the principal at time zero, r the annual interest rate in decimals, t the number of periods, or years in annual compounding. Note that (1+ r)^t is the CVIF. Plug it into your spreadsheet, and you don’t need the CVIF table. Using the same example as above, we enter in an empty cell: = 5000*(1+ 0.0725)^10 and get the result as $10,068.00, which differs slightly from the result obtained by using the linear interpolation above. Frequency of Compounding Other Than Annually Interest may compound more frequently than annually. Most corporate bonds, and Treasury bonds, compound semiannually. There are quarterly and monthly frequencies as well. When calculating the compound value of an investment with a frequency other than annual, the interest rate must be divided by the compounding frequency to get the periodic rate, and the length of time in periods must be multiplied by the compounding frequency. Problem: What is the compound (future) value of $6,000 invested at 5.5% compounded quarterly (4 times a year) after 5 years? Hint: Use the spreadsheet formula. Applying the compound value formula (1), we get: V_{t }=_{ }P_{0 }(1+ r)^{t} V_{t }= $6,000 (1+ (0.055 / 4))^{5*4} = $7,884.40 Using the spreadsheet formula (5) above, we enter in an empty cell of the spreadsheet: = P_{0 }* (1+ (r / f))^(t * f) (6) where P_{0} is the principal at time zero, r the annual interest rate in decimals, t the number of periods, and f the compounding frequency. The value of f is annual = 1, semiannual = 2, quarterly = 4, monthly = 12, daily = 365. Using the same example as above, we enter in an empty cell: = 6000 * (1+(0.055 / 4))^(5 * 4) and get the compound value of $7,884.40. The use of formulas (5) and (6) obviates the need for linear interpolation, and even the CVIF table itself. Continuous Compounding As seen above, the more frequently an investment is compounded, the higher the return it will generate. However, there is a limit on the interest earned regardless of how frequent the compounding is. Think of the scenario where $1 is invested at an annual interest rate of 100% compounded f times a year. Applying the formula (6) for calculating the compound amount: V_{t} = P_{0 }* (1+ (r / f))^(t * f) (6) where V_{t} is the compound amount, P_{0 } the amount of the investment, r the annual interest rate, f the compounding frequency, and t the number of years, we derive the compound amount of $1 as follows: 2.71828 = 1 * (1+ (1 /31536000))^(1 * 31536000) The investment of $1 at 100 % compounding every second of the year (31536000 seconds) grows to $2.71828 after one year. This is as close to continuous compounding as it gets. If you look closely, the number 2.71828 is the irrational number denoted e, so frequently found in mathematics, that is the base of the natural logarithm, and is generated by the expression: (1 + (1 / n))^{n} as n grows very large, and to which the formula (6) reduces since P_{0} = 1, r = 1, t = 1, and f = n in the above expression. Substituting e in the formula (6), we obtain the formula for continuous compounding as follows: V_{t} = P_{0}*e^{rt} Continuous compounding formula Problem: Calculate the compound amount of $6,000 invested at 5.5% compounded continuously for 5 years. Using the continuous compounding formula: V_{t} = P_{0}*e^{rt} or its spreadsheet equivalent, we obtain the compound amount: = P_{0}*exp(1)^(r*t) = 6000*exp(1)^(0.055*5) = $7,899.18 Note that exp(1) is Excel's way of generating e. Nominal Interest Rate vs. Effective Interest Rate In our CD example above, the stated annual interest rate is 5%. Yet the compound values vary depending on whether the interest is simple or compound. Not all CD’s are created equal. If you are quoted a rate of 5% with an annual yield of 5.25%, for example, then your interest is compound interest. The stated rate is then known as the nominal rate, and the annual yield of 5.25% is the effective rate. To illustrate the effective rate, let us use formula (1). A $1,000 investment earns an annual interest rate of 8%. After one year, the compound amount becomes: V_{t }=_{ }P_{0 }(1+ r)^{t} V_{t }= $1,000 (1.08) = $1.080.00 Now suppose this same investment has an interest that compounds quarterly, at an annual rate of 8% or 2% every quarter. The compound amount after one year is: V_{t }= $1,000 (1.02)^{4} = $1.082.43 Because of compounding, the effective rate is greater than the nominal rate of 8%. The formula for the effective interest rate is: Effective rate = (1 + (r / f))^{f }– 1 (7) where r is the nominal interest rate, and f the frequency of compounding. Applying formula (7) yields an effective rate of: Effective rate = (1 + (0.08 / 4))^{4}  1 = 1.082432 – 1 = 0.082432 The effective rate of 0.082432 is higher than the nominal rate. The equivalent spreadsheet formula is: Effective rate = [= ((1+ (r/4))^4) – 1] (8) Note that the square brackets around the spreadsheet formula, which begins with an equals sign, are for notational purposes only. The Excel spreadsheet formula should always begin with an equals sign. The nominal rate is derived from formula (1): V_{t }=_{ }P_{0 }(1+ r)^{t} as follows: V_{t }/ P_{0 }= (1+ r)^{t} ^{t}√ (V_{t }/ P_{0}) = 1+ r r_{p} = ( ^{t}√ (V_{t }/ P_{0}) ) – 1 (9) The spreadsheet formula is: = ((V_{t }/ P_{0})^(1/ t) – 1) (10) where r_{p} is the nominal periodic rate, i.e., per frequency period, t the time periods, V_{t } the compound value, P_{0} the principal at time zero. The compounding frequency f is annual = 1, semiannual = 2, quarterly = 4, monthly = 12, daily = 365. The time periods t is equal to the frequency times the number of years or life of the investment. For example, if the investment is placed at an interest rate that compounds quarterly for 5 years, the time periods are 4 * 5 = 20. Recall that all interest rates are quoted as annual rates. However, in cases of compounding more frequently than annually, r represents the periodic rate. To get the annual rate, we must multiply the periodic rate times the frequency as shown: r = (( ^{t}√ (V_{t }/ P_{0})) – 1) * f Using the example for the effective periodic rate above, we obtain: r_{p} = ( ^{4}√ (1082.43_{ }/ 1000) ) – 1 The spreadsheet formula should show: = (1082.43_{ }/ 1000)^(1 / 4) – 1 The annual rate is then: r = (( ^{4}√ (1082.43_{ }/ 1000)) – 1) * 4 In the spreadsheet, it looks like: = ((1082.43_{ }/ 1000)^(1 / 4) – 1) * 4 = 0.079998 ≈ 0.08 Problem: Compare the effective interest rates of three CD's offered by three different banks, and indicate which bank offers the best rate for the investor. (1) Bank 1: 6.125% compounded monthly; (2) Bank 2: 6.155% compounded quarterly; (3) Bank 3: 6.2% compounded semiannually. Using the effective rate formula (7), we obtain: (1) Effective rate offered by Bank 1: (1 + (0.06125 / 12))^{12}  1 ≈ 6.2999% (2) Effective rate offered by Bank 2: (1 + (0.06155 / 4))^{4}  1 ≈ 6.2985% (3) Effective rate offered by Bank 3: (1 + (0.062 / 2))^{2}  1 ≈ 6.2961% Bank 1, which offers the lowest nominal rate of 6.125%, actually yields the best return of all. Bank 3's highest nominal rate of 6.2% turns out to be the least attractive to the investor. The reason for this discrepancy is compounding frequency. Calculating the Annual Nominal Rate Given the Effective Rate From the effective rate r_{eff} formula, r_{eff} = (1 + (r / f))^{f }– 1 (7) we derive the nominal rate r as follows: (r_{eff} + 1)^{1/f} = (1 + (r / f)) (r_{eff} + 1)^{1/f}  1 = r / f r = ((r_{eff} + 1)^{1/f}  1 ) f Annual nominal rate formula or its spreadsheet equivalent: r = ((r_{eff} + 1)^(1/f)  1) * f Problem: A bank CD yields an annual effective rate of 6.532% compounded monthly. What is it annual nominal rate? Using the nominal rate formula above, we get: r = ((0.06532 + 1)^(1/12)  1 ) * 12 r = 6.344% Calculating the Effective Rate Given the Original Principal, the Compound Amount and the Number of Years From the compound amount formula: V_{t} = P_{0} * (1+ (r / f))^{(tf)} we get: V_{t} / P_{0} = (1 + (r / f))^{tf} Substituting the nominal rate formula for r yields: V_{t} / P_{0} = (1 + (((r_{eff} + 1)^{1/f}  1 ) * f ) / f )^{tf} V_{t} / P_{0} = (1 + ((r_{eff} + 1)^{1/f}  1 ))^{tf} V_{t} / P_{0} = ((r_{eff} + 1)^{1/f} )^{tf} V_{t} / P_{0} = (r_{eff} + 1)^{t} (V_{t} / P_{0})^{1/t} = r_{eff} + 1 The effective rate can now be calculated with: r_{eff} = (V_{t} / P_{0})^{1/t}  1 Effective rate formula 2 or with its spreadsheet counterpart: r_{eff} = ((V_{t} / P_{0})^(1/t))  1 where V_{t} is the compound amount, P_{0} the original principal invested, and t the number of years. Problem: Find the effective rate of an investment of $6,000 in an instrument that returns $7,884.40 after 5 years. Using the preceding effective rate formula, we obtain: = ((7884.4/6000)^(1/5))  1 = 5.6144836% Compare this with the result obtained by the first effective rate formula through use of nominal rate and compounding frequency: = ((1+ (0.055/4))^4)  1 = 5.6144809% Number of Years You want to save up for a future purpose, perhaps for a down payment on a house, a trip to Europe, or a digital camera, and know ahead of time how much money you will need. You also know what you can afford to invest, and the interest rate your investment earns. Now you want to know how many years you need to stay invested to reach your financial goal. Problem: How long does it take for an investment of $600 to grow to $900 at annual rate of 8% compounded quarterly? From formula (1) we get: V_{t }=_{ }P_{0 }(1+ r)^{t} $900_{ }= $600 (1.02)^{ t} (1.02)^{ t } = $900 / $600 (1.02)^{ t } = 1.5 Taking the natural logarithms on both sides of the above equation yields: t ln (1.02) = ln (1.5) t = ln (1.5) / ln (1.02) t = 20.47532 in quarters, or 20.47532 / 4 = 5.1188 years. More generally, t in years is found in: t = (ln( 1 + (r / f )) / ln( V_{t }_{ }P_{0 })) / f (11) The equivalent spreadsheet formula is: =(LN(1 + (r / f )) / LN(V_{t }_{ }P_{0})) / f (12) 3.2 The Present Value (PV) The concept of the Present Value (PV) is very important because it is used to set prices in all financial markets. The prices of financial assets, be they bonds, stocks, loans, or projects, are the discounted values of an expected stream of future earnings. For example, the price of a coupon bond is the stream of interest payments generated by the bond over its life plus its face value due at maturity, at the specified discount rate, i.e., the annual interest rate. Problem: You have $10,000 to invest and want to earn an interest rate of 7% compounded annually. A bank offers $19,672 at the end of 10 years in exchange for your $10,000 now. Would you take up the offer? What is asked of you is simply this: How much is the stream of future earnings discounted at the discount rate of 7% for 10 years worth now? If the present value were $10,000, would you want to invest? If you work out the formula (1) or spreadsheet formula (6), you will find that investing $10,000 at 7% for 10 years yields $19,671.51. You could also consult the CVIF table, and find virtually the same answer. You would be indifferent as to the outcome. The latter sum is the compound (future) amount discussed in Section 3.1 above. And the 7% rate is called the opportunity cost, i.e., the cost of having to forego a chance of earning by choosing a different alternative. The $10,000 is the present value of a stream of future income or cash flows. The opportunity cost in this case is also referred to as the discount rate. Finding the present value, otherwise known as discounting, is the reverse of compounding. The term discounting indicates that the present value is less than the future value of the same instrument. Formula (1) contains the compound or future value and the present value. Knowing one, we can solve for the other. We use formula (1) to solve for the present value: V_{t }=_{ }P_{0 }(1+ r)^{t} (1) P_{0} = V_{t} / _{ }(1+ r)^{t} P_{0} = V_{t } (1 / (1+ r)^{t} ) (13) P_{0} is the present value, denoted as PV: PV = P_{0} = V_{t } (1 / (1+ r)^{t} ) (14) Or PV = P_{0} = V_{t }(1+ r)^{t} (15) Its spreadsheet formula is: = V_{t } * (1+ r)^t (16) When we examine the formula for the present value, we quickly conclude its relationship to the interest rate r in the denominator. A lower denominator results in a higher present value. In other words, the lower the interest rate, the higher the present value (or price). Like (1+ r)^{t}, its inverse 1 / (1+ r)^{t}, known as Present Value Interest Factor (PVIF), can be calculated ahead of time for wide ranges of r and t. The present value interest factor is the present value of $1 given a specified interest rate and term. The present value factors are reciprocals of the compound value factors. These values are collected in a partial table in Appendix 1, which lists the present value of $1 in the future income stream. The present value interest factor (PVIF) table is used in the same way as the future value interest factor (CVIF) table in Section 3.1. Substituting PVIF for 1 / (1+ r)^{t} in formula (14) yields: PV = P_{0} = V_{t } (PVIF) (17) The table in Appendix 1 lists approximate values of PVIF for a number of r and t. As with the table of CVIF, we can use linear interpolation to find intermediate values that are not listed. Recall that the spreadsheet formula =1/(1+r)^t or =(1+r)^t gives a more accurate PVIF than the table. Let us use the PVIF. If you have invested at 7% and expected to have $19,672 in future income 10 years from now, what is its present value? Look up the future value table, and you will find the PVIF to be 0.5083 at the intersection of the column 7% and the row for period 10. PV = P_{0} = $19,672 (0.5083) = $9,999.28 ≈ $10,000 The spreadsheet formula gives a comparable answer: = V_{t }* (1+ 0.7)^10 PV ≈ $10,000.25 The present value is useful in making decisions about setting up a trust fund for a child’s education, or for some other future needs by leaving the invested amount in place to earn a steady income stream for an extended period of time. Problem: You are setting up a trust fund for your child’s education with one single payment that needs to grow to $60,000 in 12 years. If the investment earns 6% compounded semiannually, how much money should this single payment be? Applying the present value formula (15), and noting that the investment earns r/2 (half the annual rate) or 3% for 12*2 or 24 semiannual periods, we obtain the initial outlay as: PV = P_{0} = V_{t }(1+ r)^{t} PV = $60,000 * (1.03)^{24} ≈ $29,516.02 Present Values of Two Loans with Different Maturities and Uneven Cash Flows Problem: You owe two sums of money to the bank: $5,000 due in 3 years, and $4,000 due in 6 years. Calculate the single payment now to pay off both debts if the interest rate is 8% compounded quarterly. The single payment P now is the sum of the present values of both amounts: P = 5000(1.02)^{(3*4)} + 4000(1.02)^{(6*4)} = 5000(1.02)^{12} + 4000(1.02)^{24} (18) = $3,942.46 + $2,486.89 = $6,429.35 Another method of payment consists of two payments in the future. Multiplying both sides of equation (18) by (1.02)^{24}: P(1.02)^{24} = 5000(1.02)^{12 }+ 4000 (19) The equations (18) and (19) are equivalent, and are called equations of value.Equation (19) is useful when P is known, but one of the future payments is not, as illustrated in the following problem. Problem: You owe $18,000, due 5 years from now, to be paid off in three payments: $2,000 now, $3,000 in three years, and the final payment at the end of five years. Assuming an interest rate of 6% compounded semiannually, what is the amount of the last payment? Let x be the final payment due in 5 years. At year 5 we compute the future values of $2,000 and $3,000, and the present value of $18,000: 2000(1.03)^{10} + 3000(1.03)^{4} + x = 18000(1.03)^{2} Knowing P the present value, the last payment x derives in the same way as equation of value (19): 18000(1.03)^{2} = 2000(1.03)^{10} + 3000(1.03)^{4} + x x = 18000(0.942596 )  2000(1.343916)  3000(1.125509) x = $16,966.73  $2,727.83  $3,376.53 = $10,902.37 Present Value of a Coupon Bond A coupon bond pays interest every six months at a fixed interest rate (known as coupon) until maturity, at which time the bondholder gets the entire face amount plus the last coupon payment. The present value of a bond is its price. Problem: Calculate the present value (PV), i.e., current bond price, of a $100,000 coupon bond paying an annual rate of 5% compounded semiannually for 10 years. This bond pays $5,000 in interest a year, or $2,500 every six months. At maturity the bondholder gets the bond’s face amount of $100,000 back in addition to the interest due for the last period. The present value or price of the bond is. _{ t }
PV = ($2,500 / (1.025))+ ($2,500 / (1.025)^{2 }) + …+($2,500 / (1.025)^{19}) + ($102,500 / (1.025)^{20}) where t is every payment period except the last and 20^{th} period. This is the period of maturity, when the bond’s face amount is due along with the last coupon. The bond price (PV) is exactly its face amount of $100,000. You can verify this for yourself by using the spreadsheet formula (16) for each period, and summing up the results. Now take the scenario that at the time of trade the market yield, i.e., current interest rate prevailing in the market, rises to 7%. The bond still pays $2,500 every 6 months. To find the bond price at this time, we discount at the current market yield, not at the original interest rate. PV = ($2,500 / (1.035))+ ($2,500 / (1.035)^{2}) + …+($2,500 / (1.035)^{19})+ ($102,500 / (1.035)^{20}) PV = $85,787.60 In the above scenario the current price of the bond is lower than the original $100,000 price at 5%. Clearly, a higher interest rate depresses the bond price. Consider a lower prevailing interest rate due to market conditions, say 4%. What effect does it have on the bond’s price? Because the interest rate is in the denominator, the present value, i.e., current price, rises. PV = ($2,500 / (1.02)) + ($2,500 / (1.02)^{2}) + …+($2,500 / (1.02)^{19})^{ } + ($102,500 / (1.02)^{20}) PV = $108,175.72 We can generalize this fact in mathematical terms by saying that bond prices vary in inverse relation to interest rates or yields. 3.3 The Net Present Value (NPV) The Net Present Value (NPV) is used often in business to help make financial and investment decisions. It follows the discounted cash flows procedure of the present value in Section 3.2. Essentially, the net present value is the present value of the investment minus the cost of the investment.The NPV allows you to compare alternative investment opportunities. Problem: A doctor in private practice wants to purchase an expensive piece of medical equipment. Before making a decision, she calculates the net present value. If the net present value is positive, the project is worthwhile. A negative net present value militates against the project’s acceptance. She has $50,000 and wants to know whether to buy the medical equipment or to invest it in a CD. The proposed equipment costs $50,000 and is expected to generate a cash flow stream consisting of a revenue of $12,000 after the second year, $15,000 after the third year, $18,000 after the fourth year, and $20,000 after the fifth year. For simplicity, assume no salvage value: after the fifth year the equipment is considered worth nothing and needs replacement. The market interest rate is 8% per year compounded annually. Again for simplicity, we disregard depreciation of the equipment as well as other associated costs such as installation cost, service contract, and training of personnel. The present value of the expected cash flow stream may be written as follows: PV = ($12,000 / (1.08)^{2}) + ($15,000 / (1.08)^{3}) + ($18,000 / (1.08)^{ –4}) + ($20,000 / (1.08)^{5}) PV = $12,000(0.8573388) + $15,000(0.7938322) + $18,000(0.7350299) + $20,000(0.6805832) PV = $10,288.07 + $11,907.48 + $13,230.54 + $13,611.66 PV = $49,037.75 The net present value is the difference between the present value of the revenue stream and the initial investment: NPV = PV – Cost of initial investment (20) NPV = $49,037.75  $50,000 =  $962.25 Since the NPV is negative, the proposed investment should be rejected. She would do better by buying a $50,000 CD paying 8% since her business proposal is equivalent to investing only $49,037.75. 3.4 Perpetuities Some financial assets do not have a maturity date but keep making payments indefinitely. Such are perpetual bonds, which have been issued in a number of countries, including the United Kingdom, where they are called Consols. Equity share prices are another example of perpetuities. A Consol is a promise by the British government to pay a fixed coupon to the bondholder forever. If a Consol that pays a coupon of $25 is sold for $250, its yield is $25 / $250 = 10%, as given by: k_{c} = Q_{c} / P_{c} (21) where k_{c} the interest rate or yield of the Consol, Q_{c} the Consol’s coupon, P _{c} the Consol’s price. From the formula, it is clear that the bond yield varies inversely with its price, which is consistent with what we have seen about bonds. Using the present value formula, in which k_{c} is the yield, and N is the number of periods, we get: PV = Q_{c}[1/(1+ k_{c}) + 1/(1+ k_{c})^{2} +…+ 1/(1+ k_{c})^{N}] (22)^{ } Multiplying both sides of the equation by (1+ k_{c}) gives: PV * (1+ k_{c}) = Q_{c}[1/(1+ k_{c}) + 1/(1+ k_{c})^{2} +…+ 1/(1+ k_{c})^{N1}] (23)^{ } Subtracting equation (23) from equation (22), we obtain: PV  PV * (1+ k_{c}) = Q_{c}[1 + 1/(1+ k_{c})^{N}] (24) In the case of a perpetuity as N approaches infinity, Q_{c}*1/(1+ k_{c}) approaches zero. Equation (24) becomes: PV* k_{c = }Q_{c } PV = P_{c} = Q_{c }/ k_{c } (25) Equations (21) and (25) show that given the bond price and its coupon, we can derive the yield; and given the yield and its coupon, we can derive the price. And as mentioned above, the bond price is in inverse relation to its yield 3.5 Annuities An annuity is a series of fixed payments made at fixed periods (payment periods) over a length of time called term. This is the situation where you invest a fixed amount on a periodic basis over a number of years. Or it may be a mortgage loan or a car loan, for which you pay a fixed amount every month until the loan has been paid off. The Compound or Future Value of an (Ordinary) Annuity The compound or future value of an annuity is the value of the compound amounts of all payments at the end of term. Since most annuities in consideration are ordinary annuities, i.e., with payments coming at the end of each period, we will henceforth refer to them simply as annuities. Consider an annuity of n payments of P dollars at the end of each period, where the periodic interest rate is r. The compound amount of the last payment P does not accrue interest. The (n – 1) ^{th} payment earns interest for one period, and the first payment earns interest for (n – 1) periods. The future value s of the annuity is expressed as a series: s = P + P(1+ r) + P(1+ r)^{2} +…+ P(1+ r)^{n1} (26) This is a geometric series of n terms with the first term P and common ratio 1 + r. Multiplying both sides by (1+ r) gives: s(1+ r) = P(1+ r) + P(1+ r)^{2} +…+ P(1+ r)^{n1} + P(1+ r)^{n} (27) Subtracting (27) from (26) gives: s  s(1+ r) = P  P(1+ r)^{n} s(1  (1+ r)) = P(1  (1+ r)^{n}) s = P(1  (1+ r)^{n}) / (1  (1+ r)) s = P((1  (1+ r)^{n}) /  r) Hence, the compound value of an annuity is: s = P(((1+ r)^{n} – 1) / r ) (28) Remember that every parameter in equation (28) refers to payment periods. Thus r is the periodic, not annual, rate. If the interest rate is compounded quarterly, r must be the annual rate divided by 4. By the same token, n is the number of payment periods, not years. Thus, if the security matures in three years and pays every quarter, the number of periods n is 3*4 =12. Translating equation (28) in a spreadsheet, we get: =P*(((1+r)^n)  1) / r (29) The interest factor (((1+r)^n)  1) / r, called compound value interest factor for an annuity (CVIFA), has been calculated for wide ranges of r and n. Appendix 1 contains a partial table of CVIFA for annuities of $1at the end of n periods. Again, note that the spreadsheet formula (((1+r)^n)  1) / r calculates the CVIFA. Problem: What is the future amount of an annuity consisting of a payment of $60 at the end of every quarter for four years at 6% compounded quarterly? Also find the compound interest. P = 60, r = 0.06 / 4 = 0.015, and n = 4 * 4 = 16. Applying spreadsheet formula (29), we obtain: =60*(((1+0.015)^16) – 1) / 0.015 s = $1,075.94 The compound interest is the difference between the value of the compound amount s, and the sum of all payments, $60*16 = $960: Compound interest = $1,075.94  $960 = $115.94 Problem:In Section 3.2 we had a problem on setting up a trust fund for a child’s college education by finding out what one initial payment must be in order to obtain the desired amount in the future. In this problem, we will find the future amount of the trust fund by investing a series of payments into the trust fund. If the amount invested each period is $1,200 and earns 6% compounded semiannually, what is the future value of the trust fund at the end of 12 years? Applying the compound value formula (28) or its spreadsheet equivalent (29), and noting that the investment earns r/2 (half the annual rate) or 3% for 12*2 or 24 semiannual periods, we apply: s = P(((1+ r)^{n} – 1) / r ) using the spreadsheet formula: =1200*(((1.03)^24)  1) / 0.03 s = $41,311.76 Now suppose you could get an investment that earns 6% compounded monthly for 12 years, and your monthly payment into the fund is $200, what is its future amount? P = 200, n = 12*12 = 144, and r = 0.06 /12 = 0.005. =200*(((1.005)^144)  1) / 0.005 s = $ 42,030.03 By merely switching from the semiannual compound frequency to monthly, and keeping the yearly investment constant at $1,200 every six months, you gain: $42,030.03  $41,311.76 = $718.27. That’s another illustration of the time value of money, and the power of compounding. Payments at the Beginning of the Period So far all the payments into the annuity occur at the end of each period, which is the normal way interest accrues. Such an annuity is known as an ordinary annuity. What if the payments are made at the beginning of each period? Such is the case with paying the monthly rent for an apartment, or commercial real property, for example. Problem: You pay $500 at the beginning of each quarter into a savings account that pays 6% compounded quarterly. Find the balance in the account after three years. Since the payment occurs at the beginning of the period, you can use the equation (28) for a total of 13 quarterly payments of $500 each minus the final payment, which earns no interest. = (500*(((1.015)^13)  1) / 0.015) – 500 s = $7,118.41  $500 = $6,618.41 An annuity that pays at the beginning of a period is referred to as annuity due. Sinking Fund A sinking fund is a fund into which periodic payments are deposited to meet some future obligations. A loan may have a sinking fund provision to make sure funds are available to pay off the debt at maturity. Other reasons to set up a sinking fund may be to save up enough money for a college education, a trip, a piece of equipment, or some other purpose. The future amount is known, we want to find the periodic payment. Problem:You have a piece of machinery that costs $8,000 to replace 5 years from now. You can invest a certain amount every three months in a fund that pays 5.5% compounded quarterly. How much must your quarterly payment be? s = P(((1+ r)^{n} – 1) / r ) Using the equation (28) for the future amount of an annuity, we solve for the periodic payment P: P = s / (((1+ r)^{n} – 1) / r ) And for the spreadsheet: 8000 = P*((((1.(055/4)^20)  1) / (0.055 / 4)) P = 8000 / ((((1 + (0.055 / 4))^20)  1) / (0.055 / 4)) P = $350.24 When you think about it, if this were a loan, P would be the principal to be repaid at each period. Present Value of an Annuity The present value of an annuity is the sum of the present values of all the periodic payments. It is the amount to be invested now to earn the future stream of income. This is the situation of a car loan or a mortgage loan. The finance company or mortgage company essentially purchases the stream of future income (the debtor’s monthly payments) with the loan extended. Unless otherwise stated, we assume that the payment occurs at the end of each period, which is the normal way interest accrues. The present value of an annuity, PVA, is a series of n payments consisting of P periodic payments discounted at the current interest rate. In a car loan or mortgage loan, PVA is the loan amount: PVA = P(1+ r)^{1} + P(1+ r)^{2} +…+ P(1+ r)^{n} This is a geometric series consisting of n terms with the first term P(1+ r)^{1} and common ratio (1+ r)^{1}: PVA = (P(1+ r)^{1}((1  (1+ r)^{n})) / ((1  (1+ r)^{1}))) = P(1  (1+ r)^{n}) / (1+ r)(1  (1+ r)^{1}) = P(1  (1+ r)^{n}) / ((1+ r) – 1) Finally, PVA = P((1  (1+ r)^{n}) / r) (30) Its spreadsheet equivalent would be: = P((1  (1+ r)^n) / r) (31) If PVA is known, the periodic payment P may be derived as follows: P = PVA / ((1  (1+ r)^{n}) / r) (32) Its spreadsheet counterpart is: = PVA / ((1  (1+ r)^n) / r) (33) The (1  (1+ r )^{n}) / r is called the Present Value Interest Factor for an Annuity (PVIFA), which is the present value of an annuity of $1 per period for n periods. It is found in the table of PVIFA in Appendix 1 for limited ranges of r and n. The spreadsheet formula ((1  (1+ r)^n) / r) calculates the PVIFA. Recall that r is the periodic (not annual) rate, n is the number of payments, and P is the periodic payment of the annuity. Problem: What is the present value of an annuity in which a monthly amount of $1,352.25 is paid for 360 months, if the interest rate is 6.75% compounded monthly? You can easily recognize that this is a 30year mortgage loan. You want to know how much you can borrow, given a monthly payment you feel comfortable with and a known interest rate. We use the spreadsheet formula (31): = P((1 (1+ r)^n) / r) (31) = 1352.25*((1(1+(0.0675/12))^360) / (0.0675/12)) PVA ≈ $208,488.1234 Problem: John wanted to borrow money to purchase a car at an interest rate of 9.25% to be repaid in 4 years. He can afford a monthly payment is $399.17. How much could he borrow? The spreadsheet formula (31) gives: =399.17*((1(1+(0.0925/12))^48) / (0.0925/12)) The amount of the loan John could borrow is: PVA = $15,964.30 Problem: Now, suppose John wanted to borrow no more than $14,000 at the same interest rate of 9.25% for 4 years. How much is his monthly payment going to be? Apply spreadsheet formula (33): = PVA / ((1  (1+ r)^n) / r) (33) =14000 / ((1(1+(0.0925/12))^48) / (0.0925/12)) His monthly payment will be reduced to: P = $350.05 Problem: John wanted to accelerate his payments. He would still borrow $14,000 at the same interest rate of 9.25% but for a shorter term of 3 years. How much would his monthly payment be? =14000 / ((1(1+(0.0925/12))^36) / (0.0925/12)) His monthly payment would be higher: P = $446.83 What the Equation for the Present Value of an Annuity Tells Us. The equation for the present value of an annuity is given by: PVA = P((1  (1+ r)^{n}) / r) (30) Equation (30) shows that the present value of an annuity varies in direct relation to the periodic payment P. In practical terms, the greater the periodic (monthly) payment you can afford, the larger loan amount you can borrow for given interest rate and number of payments Since the interest rate r is in the denominator, the present value of an annuity varies in inverse relation to the interest rate. That means for the borrower if the interest rate drops, she can afford to borrow a larger sum for the same periodic payment and loan term. The number of payments n being a negative exponent reduces the effect of the interest rate in (1+ r) ^{n}, which tends to decrease the present value. For the investor this implies that for a given periodic payment of income, the investment produces less as the cash flow stream shortens. Keeping the present value constant, if the number of payments increases, the periodic payment decreases, and vice versa. For the borrower, if she chooses a smaller monthly payment, she will have to pay longer, which may not be a good idea. 3.6 Loan Amortization Automobile loans and mortgage loans are typical of the loans that require payment of interest and principal in each monthly payment to retire the debt at the end of the term. The calculations are such that at the end of the last payment period all interest and principal have been fully paid. Although the monthly payment remains constant through the term of the loan, the interest portion decreases and the principal portion increases as time goes by. Periodic Payment The periodic payment is the payment for each period of the term, in this case, at the end of each monthly period. The periodic payment comprises the payment of interest and principal due for that period. The principal thus paid reduces the principal balance on which interest of the following period is computed. Therefore, the interest paid diminishes with each succeeding period while the principal payment increases. Recall that the present value of an annuity in a mortgage or car loan is the principal amount of the loan. Problem: Calculate the monthly payment of a $15,000 3year car loan at 8.25% annual percentage rate. All car loan payments are due monthly, and the interest rate compounds monthly. See Appendix 2 for a loan amortization table of this problem. Since a car loan is an annuity, the payment is derived from the present value for an annuity, by formula (32), P = PVA / ((1  (1+ r)^{n}) / r) or the spreadsheet formula (33), we obtain: = PVA / ((1  (1+ r)^n) / r) (33) P = 15000 / ((1  (1+ (0.825/12))^36) / (0.0825/12)) P = $471.78 Problem: Calculate the monthly payment of a $180,000 30year mortgage at 7.75%. Recall that mortgage payments are due monthly, and the interest rate compounds monthly. See Appendix 3 for a loan amortization schedule of this problem. Since a mortgage loan is an annuity, the payment is derived from the present value for an annuity, by formula (32), P = PVA / ((1  (1+ r)^{n}) / r) or the spreadsheet formula (33). We obtain the monthly payment P: = PVA / ((1  (1+ r)^n) / r) (33) P = 180000 / ((1  (1+ (0.0775/12))^360) / (0.0775/12)) P = $1,289.54 Interest Portion in the First Periodic Payment The interest payment is the product of the loan principal outstanding and the periodic interest rate. I_{1} = PVA * r (34) where I_{1} is the interest portion in the first payment, PVA the original loan principal, r is the interest rate per period. Problem: Calculate the interest portion in the first monthly payment of a $120,000 30year mortgage at 7.75%. I_{1}= PVA * r =120000 * (0.0775 / 12) = $775.00 Total Interest Paid The total interest paid can be calculated by subtracting the loan principal from the total payments for all periods. Total interest paid = (P * n) – PVA (35) where P is the periodic payment, n the number of periods (or term), and PVA the loan principal. Problem: Find the total interest paid for a 30year mortgage loan of $120,000 at 7.75%, the loan for which P was calculated in the previous problem. Total interest paid = ($859.69 * 360)  $120,000 ≈ $309,490.09  $120,000 = $189,490.09 In this case, the interest paid is more than 1.5 times the loan amount. Principal Outstanding at the Beginning of k^{th} Period When the current market interest rate drops below the original interest rate, mortgage holders may prepay their current loans, and refinance their loans by taking advantage of the new lower rate. In which case, the borrower will want to know how much of the old mortgage principal is outstanding. We use the formula (30) for the present value of an annuity but slightly modify the period: PVA = P((1  (1+ r)^{n}) / r) (30) PVA_{k} = P((1  (1+ r)^{n + k1}) / r) (36) where PVA_{k} is the principal outstanding at the beginning of k^{th} period, k is the period in question. Its spreadsheet equivalent would be: = P((1  (1+ r)^(n+k1)) / r) (37) Problem: What is the principal outstanding at the beginning of the 43^{rd} period of a 30year $150,000 mortgage issued at 8.125%? First we find the periodic payment: P = 150000 / ((1  (1+ (0.08125/12) )^360) / (0.08125/12)) = $1,113.75 We then apply the formula (37) for the principal outstanding at the beginning of the 43^{rd} period: PVA_{k} = 1113.75*((1  (1+ (0.08125/12))^(360+43 1)) / (0.08125/12)) = $145,252.23 Interest in the k^{th} Payment The mortgage borrower may want to know at the time of prepayment how much of the k^{th} payment is the interest portion. Basically, the interest due in k^{th} period is the product of the outstanding principal balance at the beginning of k^{th} period and the periodic interest rate. Hence, we have: I_{k} = PVA_{k} * r_{ } I_{k} =_{ }P((1  (1+ r)^{n + k1}) / r) * r (38) And its spreadsheet counterpart: =_{ }P*(((1  (1+ r))^(n+k1)) / r) * r (39) Problem: Find the interest portion of the 50^{th} payment of a 30year $120,000 mortgage issued at 7.75%? P = 120000/((1(1+(0.0775/12))^360)/(0.0775/12)) = $859.69 I_{k} = 859.69*(((1(1+(0.0775/12))^(360+501))/(0.0775/12))*( 0.0775/12)) = $743.59 Principal Portion in the k^{th} Periodic Payment In case of prepayment, it may be useful to also know what portion of the k^{th} payment is the principal. The principal portion in the k^{th} payment is the difference between the periodic payment and the interest portion in the k^{th} payment. We have: PP_{k} = PVA_{k}  I_{k} PP_{k} = [P((1  (1+ r)^{n + k1}) / r)] _{ }[P((1  (1+ r)^{n + k1}) / r) * r] PP_{k} = P [1  ((1  (1+ r)^{n + k1}) / r) * r] (40) Or its spreadsheet equivalent: = P*(1  ((1  (1+ r)^(n+k1)) / r) * r) (41) Problem: Find the principal portion of the 50^{th} payment of a 30year $120,000 mortgage issued at 7.75%? PP_{k} =859.69*(1  ((1  (1+ (0.0775/12))^(360+501)) / (0.0775/12)) * (0.0775/12)) = $116.10 Number of Periods At times we want to know, for a given payment, loan principal and interest rate, the number of payments (or term) needed to pay off the debt. This is particularly useful when we plan to prepay or accelerate the debt retirement. Given the formula for the present value of an annuity, we can derive the term of the loan. PVA = P((1  (1+ r)^{n}) / r) (30) (PVA*r) / P = 1  (1+ r)^{n} (1+ r)^{n} = 1  ((PVA*r) / P) (1+ r)^{n} = (P  (PVA*r)) / P Taking the natural logarithm of both sides: n ln(1+ r) = ln((P  (PVA*r)) / P) n =  (ln((P  (PVA*r)) / P)) / ln(1+ r) (42) Or its spreadsheet formula: =  (ln((P  (PVA*r)) / P)) / ln(1+ r) (43) Problem: How long does it take to pay off a $30,000 loan at an interest rate of 8.765% compounded monthly if you want to make a monthly payment of $512.98? Applying formula (43), we get: n = (LN((512.98(30000*(0.08765/12)))/512.98))/LN(1+(0.08765/12)) ≈ 76.56 months If you want to pay off the same loan faster, and can afford a monthly payment of $700.00, how many months will it take? n = (LN((700(30000*(0.08765/12)))/700))/LN(1+(0.08765/12)) ≈ 51.59 months Credit Card Debt Credit card debt is a type of loan, called revolving loan, in which the loan amount is the amount of all purchases made by the cardholder during any monthly billing period plus any account balance remaining from the previous period minus any payment made for the previous period. As the balance is paid down each month, the cardholder may make additional purchases as long as the overall charges do not exceed a preset limit. This balance, if paid in full by the payment due date (called the grace period), incurs no finance charges. Also included in this balance are cash advances and balance transfers from other credit card accounts, which generally enjoy no grace period. The periodic finance charges are typically computed each day on the average daily balance at the daily periodic rate, which when annualized is expressed as the annual percentage rate (APR) that the credit card company charges the cardholder. The entire amount due is called the new balance for a given billing period. However, the cardholder has the option of paying the minimum payment due (which could be 1% or less of the new balance). The finance charges for purchase transactions are typically lower (e.g., 12% per year) than the finance charges for cash advance transactions (close to 20%). Problem: Assuming no new purchases are made, how long does it take you to pay off a $3,000 credit card balance at an annual percentage rate of 12% compounded monthly if you pay (1) $31.00 each month? (2) the minimum payment due of $20.00 each month? For simplicity, we assume the finance charges accrue monthly. (1) Payment of $31.00 each month. Applying formula (43), we get: n =(LN((31(3000*(0.12/12)))/31))/LN(1+(0.12/12)) ≈ 345.11 months (2) Payment of $20.00 each month. Applying formula (43), we get: n = (LN((20(3000*(0.12/12)))/20))/LN(1+(0.12/12)) ≈ ? months The Excel spreadsheet formula above does not give any result. Note that $20.00 is far less than the monthly finance charges incurred, which is about $30.00. In fact, if you just pay $30.00 a month, you will only pay the interest, and will never pay off the amount of the debt. The Loan Amortization Table All the calculations seen so far about loan amortization can be applied in an amortization table for convenient use. Two such tables are provided in Appendix 2 for a car loan, and Appendix 3 for a mortgage loan. The utility of these schedules is that they combine all information needed about the loan and answer many questions you may have about the loans. The table gives the amount of interest and principal paid for each period, the cumulative interest paid, the cumulative principal paid, the number of periods, the periodic payment, the outstanding principal balance at the beginning of any period. An added feature is the yeartodate interest paid. This column comes in handy when you want to verify the loan interest paid for any year for tax purposes. Note the following facts about the loan amortization of a shortterm car loan in Appendix 2 and the longterm mortgage loan in Appendix 3: In the early stages of the loan, the interest portion is smaller than the principal portion in a shortterm loan. Just the opposite is true of the longterm mortgage. In our mortgage example, the interest portion ($1,162.50) is nine times as large as the principal portion ($127.04) for the first period. As time goes by, the monthly interest payment diminishes. Still the borrower ends up paying $284,235.14 on a $180,000 loan. This points to two conclusions: (1) the shorter loan term drastically reduces the interest payment, and (2) prepayment, whether partial or total, is most profitable in the early stages of the loan, when the interest payment is highest. For the past year and a half or so, interest rates have fallen to record lows, triggering a frenzy of home mortgage refinancing in the years 20022003. Although home mortgage prepayment generally incurs no penalty, borrowers must factor in the cost of refinancing as it is paid up front. Since any money paid now has a future value if invested (cf. Chapter 3, Section 3.1), they should do a little homework using the calculations presented above before refinancing. Unlike mortgages, car loans may have a prepayment penalty. From the discussion above it appears that, all risks and other things being equal, a debtor is better off borrowing as little as possible for the shortest possible term at the least possible interest rate. And from the investor’s point of view, the opposite is true: invest for the highest return as long as possible.
Chapter 4 Practical Problems 4.1 The Compound Value or Future Value 4.1.1 Find the (a) compound amount, (b) the compound interest at the given rate, and the (c) annual effective interest rate:
 $5,000 for 5 years at 7% compounded semiannually.
4.1.2 Find the effective interest rate corresponding to:
 nominal rate = 6% compounded quarterly.
4.1.3 What are the annual nominal rates of interest compounded quarterly that correspond to the effective rates of 5%, 6,25%, 8,75%, and 10.5%? Find the annual nominal rates for the same effective rates, if they compound (a) semiannually, and (b) monthly. 4.1.4. How many years would it take to double a principal at a given effective rate?
 4 percent.
4.1.5 If you have a choice of investing a sum of money at 8% compounded semiannually, and 7.8% compounded quarterly, which is the better of the two rates? 4.1.6 A commercial bank offers CDs at a rate of 3.75% compounded daily. Find the effective rate, if a year is assumed to be (1) 360 days, and (2) 365 days. Hint: Divide the rate by the number of days (frequency) assumed to be in the year. 4.1.7 An investment of $1,500 after three years yields $1,800. If the interest rate compounds semiannually, what is the annual interest rate earned by the investment? 4.1.8 Calculate the effective rate of
 an investment of $1,500 that returns $1,800 after three years.
4.1.9 Compare the effective interest rates of three CD's offered by three different banks, and indicate which bank offers the best rate for the investor.
 (1) Bank 1: 6.12% compounded monthly; (2)
Bank 2: 6.15% compounded quarterly; (3) Bank 3: 6.2% compounded semiannually.
4.1.10 What are the compound amounts of the following investments, all of which compound continuously?
 an investment of $3,500 at 3.75% for 5 years.
4.1.11 Find the nominal rates of bank CD's that yield
 an annual effective rate of 4.32% compounded monthly.
4.2 The Present Value 4.2.1 Find the present value of a future payment stream at a specified interest rate:
 $8,000 due in 10 years at 6% compounded quarterly.
4.2.2 A trust fund for a child’s education is being set up with one single payment. It is expected to generate $25,000 in 15 years. The single payment can earn an interest rate of 5%, compounded semiannually. How much should be invested initially? 4.2.3 A debt of $6,000 at an interest rate of 8% compounded quarterly due in 6 years from now is to be repaid by a payment of $2,000 now and a second payment at the end of 6 years. What is the amount of the second payment? 4.2.4 A debt of $4,000 due in two years and $2,000 due in four years is to be repaid by a single payment of $1,000 now, and two equal payments which are due one year from now and three years from now. If the interest rate is 7% compounded annually, calculate the amount of each of the equal payments. 4.3 The Net Present Value 4.3.1 A business investment of $30,000 is expected to produce the following cash flows: Year Cash Flow
1
$9,000
If the interest rate is 4.5% compounded semiannually, find the net present value of the cash flow. Is the investment profitable? 4.3.2 Assume a business venture requiring an initial outlay of $45,000 that guarantees an income of $10,000 after the first year, $13,000 after the second year, $15,000 after the third year, and $18,000 after the fourth year. What is the net present value of the cash flow stream, assuming an interest rate of 3.5% compounded semiannually? Would this be a profitable venture? 4.4 The Compound Value of an Annuity 4.4.1 For an annuity of $100 paid at the end of every six months for 7.5 years at an interest rate of 5% compounded semiannually, find (1) the future value, and (2) the compound interest. 4.4.2 You have a piece of machinery that costs $10,000 to replace 6 years from now. How much must be your semiannual payment into a sinking fund that pays 4.5% compounded semiannually? 4.4.3 A trip to Europe will cost you $6,000 in 5 years. How much do you need to invest annually in a fund that earns 3% annually to have enough money for the trip? 4.4.4 A college trust fund being set up for your twoyearold child to be used at his eighteenth birthday earns 5% compounded annually. You anticipate a need for $80,000. What is the annual investment amount required? 4.5 The Present Value of an Annuity 4.5.1 Assume an interest rate of 4.75% compounded annually, what is the present value of the income stream of a sevenyear business venture consisting of $2,000 after each of the first three years, $3,000 after each of the next two years, and $4,000 after each of the remaining two years? 4.5.2 Given an interest of 5% compounded annually, find the present value of the annuity consisting of $1,500 each year for three years, and $2,000 each year for the next 5 years. 4.5.3 What is the maximum cost of the home you could purchase if you were able to pay a monthly payment of $2,123 on a 15year mortgage at 7.125%? Recall that mortgage interest rates compound monthly, and the monthly payments are to pay interest and principal, exclusive of insurance, tax escrow, and the like. 4.5.4 The home you want to purchase costs $250,000. You can obtain 95% of the financing from a mortgage company at a 6.75% interest rate. If you can pay $2,500 a month, how many months will it take to pay off your mortgage? 4.5.5 You borrow $5,000 to be repaid by a sum of $1,200 at the end of each year for three years, and a final payment at the end of the fourth year. If the interest rate is 5.5% compounded annually, how much is the final payment? 4.5.6 A doctor sets up a sinking fund of $25,000 to replace a piece of medical equipment six years from now, with a regular yearend payment into the fund, earning 5% compounded annually. After three years, the investment earns 6% compounded annually.Due to higher earnings, she can reduce her annual payment and still reach her goal. What is the amount of the new payment? 4.5.7 You borrow $3,000 and will pay off the loan by equal payments at the end of each month for 3 years. Assuming the bank charges 7.78% compounded monthly, how much is each monthly payment? 4.5.8 A person accumulates a credit card debt of $4,000, stops charging, and pays a minimum of $25 each month. If the credit card company charges an annual 19% finance charge on the remaining balance, how many years will it take him to pay off his debt? Hint: The finance charge compounds monthly. Good luck. 4.6 Loan Amortization 4.6.1 The new Lexus you are buying costs $38,000. You trade in your old jalopy for $2,000. The dealer offers two options: (1) $3,000 rebate and an interest rate of 6.75%, or (2) no rebate but an interest rate of 5.5%. Assuming you want to pay off the loan in 5 years, which option will you prefer? Why? Show your calculations. 4.6.2 You want to pay off your fiveyear car loan of $21,000 at an interest rate of 2.9% after the 34^{th} payment. What is the payoff principal balance? 4.6.3 Assume the same scenario as in problem 4.6.2 above. Also assume that the extra cash you have at the time of the proposed prepayment could be invested at the interest rate is 3.5% compounded annually, would you go ahead and prepay? Why or why not? Show your calculations. 4.6.4 Suppose your 30year $150,000 mortgage at 7.45% is at the beginning of its eleventh year, and you could obtain a new mortgage rate of 5.5%. If the cost of refinancing were $3,000, would you go ahead and refinance? Show your calculations. 4.6.5 What is the outstanding principal balance of a 5year $20,000 car loan issued at 8.875%, at the beginning of the 24^{th} month? 4.6.6 What are the amounts of the interest and principal paid at the end of the 60^{th} month on a 20year $175,000 mortgage at 8.755%? 4.6.7 You have a 30year $200,000 mortgage at 6.75%. How much would you save in interest payment if the loan were for (a) 10 years, (b) 15 years, and (c) 20 years? 4.6.8 You can afford a monthly payment of no more than $300 on a $15,000 car loan at 9.75% annual percentage rate. How many months will it take you to pay off the loan? 4.6.9 If a pension fund offers you (a) a lump sum of $100,000 now, or (b) a monthly payment of $500 for the rest of your life, expected to be another 30 years. Assuming you can invest your money and expect to earn 4% compounded annually for the next 30 years, which option would you take? Why? Show your calculations. 4.6.10 What is the total interest payment of a 30year $500,000 mortgage loan at 7.125%? 4.6.11 Your 5year $20,000 car loan at 9.25% had its first payment on 6/30/2002. Calculate the interest paid for the year 2003. 4.6.12 A credit card debt of $5,000 is being repaid by a monthly payment of $70. If the finance charge is 12%, how many years will it take to pay off the debt? 4.6.13 Examine the amortization table in Appendix 2. Answer the following questions: (1) What is the monthly payment? (2) What is the total interest paid at maturity? (3) What is the total interest paid in 2003? If you want to prepay this loan in December 2003, how much money do you need? 4.6.14 Construct an amortization table for a loan of $19,530.35 with interest at 7.545% for 4 years. The first payment date is 15 December 2003. Use Appendix 2 as a model. Hint: Use a spreadsheet, and the spreadsheet formulas discussed in the text.
Appendix 1
Future Value Interest Factors – FVIF or CVIF
Present Value Interest Factors  PVIF
Future Value Interest Factors of an (Ordinary) Annuity – FVIFA or CVIFA
Present Value Interest Factors of an (Ordinary) Annuity  PVIFA
Appendix 2 Loan Amortization – 3Year Car Loan Principal: $15,000.00 Monthly payment: $471.78 Interest rate: 8.250% Term: 3 years First payment date: 15 January 2003.
Period
Interest
Principal
Total Int
Total Princ
Princ Balance
YTD Int
Date
1
103.13
368.65
103.13
368.65
14,631.35
103.13
1/15/03
2
100.59
371.19
203.72
739.84
14,260.16
203.72
2/15/03
3
98.04
373.74
301.75
1,113.58
13,886.42
301.75
3/15/03
4
95.47
376.31
397.22
1,489.89
13,510.11
397.22
4/15/03
5
92.88
378.90
490.11
1,868.78
13,131.22
490.11
5/15/03
6
90.28
381.50
580.38
2,250.28
12,749.72
580.38
6/15/03
7
87.65
384.12
668.04
2,634.40
12,365.60
668.04
7/15/03
8
85.01
386.76
753.05
3,021.17
11,978.83
753.05
8/15/03
9
82.35
389.42
835.40
3,410.59
11,589.41
835.40
9/15/03
10
79.68
392.10
915.08
3,802.69
11,197.31
915.08
10/15/03
11
76.98
394.80
992.06
4,197.49
10,802.51
992.06
11/15/03
12
74.27
397.51
1,066.33
4,595.00
10,405.00
1,066.33
12/15/03
13
71.53
400.24
1,137.87
4,995.24
10,004.76
71.53
1/15/04
14
68.78
402.99
1,206.65
5,398.24
9,601.76
140.32
2/15/04
15
66.01
405.77
1,272.66
5,804.00
9,196.00
206.33
3/15/04
16
63.22
408.55
1,335.88
6,212.56
8,787.44
269.55
4/15/04
17
60.41
411.36
1,396.30
6,623.92
8,376.08
329.97
5/15/04
18
57.59
414.19
1,453.88
7,038.11
7,961.89
387.55
6/15/04
19
54.74
417.04
1,508.62
7,455.15
7,544.85
442.29
7/15/04
20
51.87
419.91
1,560.49
7,875.06
7,124.94
494.16
8/15/04
21
48.98
422.79
1,609.47
8,297.85
6,702.15
543.14
9/15/04
22
46.08
425.70
1,655.55
8,723.55
6,276.45
589.22
10/15/04
23
43.15
428.63
1,698.70
9,152.18
5,847.82
632.37
11/15/04
24
40.20
431.57
1,738.91
9,583.75
5,416.25
672.58
12/15/04
25
37.24
434.54
1,776.14
10,018.29
4,981.71
37.24
1/15/05
26
34.25
437.53
1,810.39
10,455.82
4,544.18
71.49
2/15/05
27
31.24
440.54
1,841.63
10,896.36
4,103.64
102.73
3/15/05
28
28.21
443.56
1,869.85
11,339.92
3,660.08
130.94
4/15/05
29
25.16
446.61
1,895.01
11,786.53
3,213.47
156.10
5/15/05
30
22.09
449.68
1,917.10
12,236.22
2,763.78
178.20
6/15/05
31
19.00
452.78
1,936.10
12,689.00
2,311.00
197.20
7/15/05
32
15.89
455.89
1,951.99
13,144.88
1,855.12
213.08
8/15/05
33
12.75
459.02
1,964.74
13,603.91
1,396.09
225.84
9/15/05
34
9.60
462.18
1,974.34
14,066.09
933.91
235.44
10/15/05
35
6.42
465.36
1,980.76
14,531.44
468.56
241.86
11/15/05
36
3.22
468.56
1,983.98
15,000.00
0.00
245.08
12/15/05
Appendix 3 Loan Amortization  30Year Mortgage Principal: $180,000.00 Monthly Payment: $1.289.54 Interest Rate: 7.750% Term: 30 years First Payment Date: 30 July 2001
Period
Interest
Principal
Total Int
Total Princ
Princ Balance
YTD Int
Date
1
1,162.50
127.04
1,162.50
127.04
179,872.96
1,162.50
7/30/01
2
1,161.68
127.86
2,324.18
254.90
179,745.10
2,324.18
8/30/01
3
1,160.85
128.69
3,485.03
383.59
179,616.41
3,485.03
9/30/01
4
1,160.02
129.52
4,645.06
513.11
179,486.89
4,645.06
10/30/01
5
1,159.19
130.36
5,804.24
643.47
179,356.53
5,804.24
11/30/01
6
1,158.34
131.20
6,962.59
774.67
179,225.33
6,962.59
12/30/01
7
1,157.50
132.05
8,120.08
906.71
179,093.29
1,157.50
1/30/02
8
1,156.64
132.90
9,276.73
1,039.61
178,960.39
2,314.14
2/28/02
9
1,155.79
133.76
10,432.51
1,173.37
178,826.63
3,469.93
3/28/02
10
1,154.92
134.62
11,587.44
1,307.99
178,692.01
4,624.85
4/28/02
11
1,154.05
135.49
12,741.49
1,443.47
178,556.53
5,778.90
5/28/02
12
1,153.18
136.36
13,894.67
1,579.84
178,420.16
6,932.08
6/28/02
13
1,152.30
137.25
15,046.96
1,717.08
178,282.92
8,084.38
7/28/02
14
1,151.41
138.13
16,198.37
1,855.22
178,144.78
,235.79
8/28/02
15
1,150.52
139.02
17,348.89
1,994.24
178,005.76
10,386.30
9/28/02
16
1,149.62
139.92
18,498.51
2,134.16
177,865.84
11,535.93
10/28/02
17
1,148.72
140.83
19,647.23
2,274.99
177,725.01
12,684.64
11/28/02
18
1,147.81
141.73
20,795.04
2,416.72
177,583.28
13,832.45
12/28/02
19
1,146.89
142.65
21,941.93
2,559.37
177,440.63
1,146.89
1/28/03
20
1,145.97
143.57
23,087.90
2,702.94
177,297.06
2,292.86
2/28/03
21
1,145.04
144.50
24,232.94
2,847.44
177,152.56
3,437.91
3/28/03
22
1,144.11
145.43
25,377.05
2,992.87
177,007.13
4,582.02
4/28/03
23
1,143.17
146.37
26,520.22
3,139.24
176,860.76
5,725.19
5/28/03
24
1,142.23
147.32
27,662.45
3,286.56
176,713.44
6,867.41
6/28/03
25
1,141.27
148.27
28,803.72
3,434.83
176,565.17
8,008.69
7/28/03
26
1,140.32
149.23
29,944.04
3,584.05
176,415.95
9,149.00
8/28/03
27
1,139.35
150.19
31,083.39
3,734.24
176,265.76
10,288.36
9/28/03
28
1,138.38
151.16
32,221.78
3,885.40
176,114.60
11,426.74
10/28/03
29
1,137.41
152.14
33,359.18
4,037.54
175,962.46
12,564.15
11/28/03
30
1,136.42
153.12
34,495.61
4,190.65
175,809.35
13,700.57
12/28/03
31
1,135.44
154.11
35,631.04
4,344.76
175,655.24
1,135.44
1/28/04
32
1,134.44
155.10
36,765.48
4,499.86
175,500.14
2,269.88
2/28/04
33
1,133.44
156.10
37,898.92
4,655.97
175,344.03
3,403.31
3/28/04
34
1,132.43
157.11
39,031.35
4,813.08
175,186.92
4,535.74
4/28/04
35
1,131.42
158.13
40,162.77
4,971.20
175,028.80
5,667.16
5/28/04
36
1,130.39
159.15
41,293.16
5,130.35
174,869.65
6,797.55
6/28/04








350
88.16
1,201.38
283,788.67
167,551.05
12,448.95
918.38
8/28/30
351
80.40
1,209.14
283,869.07
168,760.19
11,239.81
998.77
9/28/30
352
72.59
1,216.95
283,941.66
169,977.14
10,022.86
1,071.37
10/28/30
353
64.73
1,224.81
284,006.39
171,201.95
8,798.05
1,136.10
11/28/30
354
56.82
1,232.72
284,063.21
172,434.67
7,565.33
1,192.92
12/28/30
355
48.86
1,240.68
284,112.07
173,675.36
6,324.64
48.86
1/28/31
356
40.85
1,248.70
284,152.92
174,924.05
5,075.95
89.71
2/28/31
357
32.78
1,256.76
284,185.70
176,180.81
3,819.19
122.49
3/28/31
358
24.67
1,264.88
284,210.36
177,445.69
2,554.31
147.15
4/28/31
359
16.50
1,273.05
284,226.86
178,718.73
1,281.27
163.65
5/28/31
360
8.27
1,281.27
284,235.14
180,000.00
0.00
171.93
6/28/31
References
Billingsley, Patrick, D. James Croft, David V. Huntsberger and Collin J. Watson, Statistical Inference for Management and Economics, Third Edition, Allyn and Bacon, Inc. Boston, 1986 Brealey, Richard and Stewart Myers, Principles of Corporate Finance, McGrawHill Company, New York, 1981. Campbell, Tim S., Financial Institutions, Markets and Economic Activity, McGrawHill Company, New York, 1982. Dornbusch, Rudiger and Stanley Fischer, Macroeconomics, Second Edition, McGrawHill Company, New York, 1981. Edmonds, Thomas P., Shannon M. McKinnon and Larry F. McCrary, Principles of Accounting, Dame Publications, Inc., Houston, 1985. Francis, Jack Clark, Investments: Analysis and Management, McGrawHill Company, New York, 1980. Haeussler, Jr., Ernest and Richard S. Paul, Introductory Mathematical Analysis for Students of Business and Economics, 4^{th} Edition, Reston Publishing Company, Inc., Reston, VA, 1983. Radcliffe, Robert C, Investment: Concepts, Analysis, and Strategy, Scott, Foresman and Company, Glenview, IL, 1982. Sullivan, Michael, College Algebra, Fourth Edition, PrenticeHall, Inc., Upper Saddle River, NJ, 1996. Weber, Jean E. Mathematical Analysis: Business and Economic Applications, Third Edition, Harper & Row, Publishers, New York, 1976. Weston, J. Fred and Eugene E. Brigham, Managerial Finance, Seventh Edition, The Dryden Press, Hillsdale, IL, 1981.
Internet Resources Annuities
http://www.prenhall.com/divisions/bp/app/cfldemo/TVM/Annuities.html
Bond Valuation http://teachmefinance.com/bondvaluation.html Bureau of the Public Debt http://www.publicdebt.treas.gov/ Commercial Paper http://www.ny.frb.org/pihome/fedpoint/fed29.html
Compound Value, Future Value
http://www.prenhall.com/divisions/bp/app/cfldemo/TVM/FutureValue.html
Credit Unions http://www.ncua.gov/ Employee Benefits Security Administration (EBSA) http://www.dol.gov/ebsa/ Eurodollars http://wfhummel.cnchost.com/eurodollars.html
Federal Deposit Insurance Corporation http://www.fdic.gov/ The Federal Financing Bank (FFB) http://www.treasury.gov/ffb/ Federal Home Loan Bank, Federal Housing Finance Board http://www.fhfb.gov/ Federal Reserve System http://www.federalreserve.gov/general.htm Federal Reserve’s Federal Open Market Committee http://www.federalreserve.gov/FOMC/ Federal Reserve Requirements http://www.federalreserve.gov/monetarypolicy/reservereq.htm Frequency of Compounding http://www.prenhall.com/divisions/bp/app/cfldemo/TVM/OtherCompounding.html Interest Rates http://teachmefinance.com/kindsofinterestrates.html Loan Amortization http://www.math.utah.edu/~alfeld/Loan/ Money and Monetary Policy
http://www.amosweb.com/cgibin/gls.pl?fcd=dsp&key=money+supply
Office of Thrift Supervision http://www.ots.treas.gov/ Pension Benefit Guarantee Corporation http://www.pbgc.gov/ Perpetuities http://teachmefinance.com/perpetuities.html
Present Value, Net Present Value http://www.investopedia.com/terms/n/npv.asp
Risk http://www.defaultrisk.com/papers.htm Securities and Exchange Commission, SEC http://www.sec.gov/index.htm
Standard and Poor’s http://www2.standardandpoors.com/NASApp/cs/ContentServer?pagename=sp/Page/HomePg Stock Valuation http://teachmefinance.com/stockvaluation.html Time Value of Money
http://www.prenhall.com/divisions/bp/app/cfldemo/TVM/TimeValueOfMoney.html
Treasuries http://www.investorwords.com/cgibin/getword.cgi?5057

Mathematics of Finance, Financial System  The Universe, Part I  The Universe, Part II  Quantum Mechanics  The Multiverse  Mars  Mars Exploration Rover Spirit  Mars Exploration Rover Opportunity 