1 Mathematics of Finance Interest is the rental fee charged by a lender to a business or individual for the use of money. charged is determined by Principle, rate and time Interest Formula I = Prt $100 At 5% Interest Compound Annually I = Prt Interest Amount (P+I) First year: I = 100x0.05x1 $5.00 $ Second year: I = 105x0.05x1 $5.25 $ Third year: I = x0.05x1 $5.51 $ Forth year: I = x0.05x1 $5.79 $ Compound Interest is not always calculated per year, it could be per quarter, per month, etc. Interest Compounded Compounding Periods per Year Annually Every year 1 Semiannually Every 6 months 2 Quarterly Every 3 months 4 Monthly Every month 12 Daily Everyday 365 Compound Interest Formula S = P(1+r) n by S = Future Value, n = total periods, r = periodic rate per period Interest Earned = S P The interest rate is usually quoted as an annual rate, called the nominal rate or the annual percentage rate (A.P.R). The periodic rate is obtained by dividing the nominal rate by the number of conversion periods per year. Periodic Rate = Example 1 Find the periodic rate and the number of compounding periods in 5 years when 12% is compounded (a) semiannually, (b) quarterly, and (c) monthly. Compounding Periods / Year Periodic Rate (r) Interest Periods (n) a. 2 12% 2 = 6% 2x5years = 10 periods b. 4 12% 4 = 3% 4x5years = 20 periods c % 12 = 1% 12x5years = 60 periods Note: 12% compounded monthly does not mean 12% per month; it means 1% per month.
2 Example 2 If money can be invested at 8% compounded quarterly, which is larger: $1,000 now or $1,210 in 5 years? $1,000 $X now 5 yrs $1,210 now 5 yrs r = % = 0.02, n = 20 $1,000 in 5 years : S = 1,000(1+.02) 20 = $1, $1,000 now is larger than $1210 in 5 years The principal P which must be invested at the periodic rate of r for n interest periods so that compound amount is S is given by P=S(1+r) -n and is called the present value of S. Example 3 Jim Ronowski needs $18,000 in 6 years. a. What amount can Ronowski deposit today at 6% compounded semiannually, so that he will have the needed money? b. Suppose he can deposit only $12,000 today. How much would he be short of the $18,000 in 6 years? r = % = 0.03, n = 12 a. P = 18,000(1+.03) -12 = $12, b. S = 12,000(1+.03) 12 = $17, He will be short of $18,000 - $17, = $
3 Effective rate The effective rate reflects the real rate of return on an investment. If money is invested at an annual rate r, compounded n times per year, the effective rate is r e = (1+ ) n -1 Example 1 The First National Bank is offering 1 6-year certificate of deposit (CD) at 6.4% interest compounded quarterly; Second National Bank is offering a 6-year CD at 6.2% compounded monthly and the Third National Bank had a 6-year CD at 6.5% compounded annually, which bank has the best rate? first bank : r e = (1+ ) 4-1 = = 6.55% BEST RATE! second bank : r e = (1+ ) 12-1 =.0638 = 6.38% third bank : r e = (1+ ) 1-1 =.065 = 6.5% Net Present Value (NPV) Net Present Value(NPV) is a way of comparing the value of money now with the value of money in the future. If an initial investment will bring in payments at future times, the payments are called cash flows. The net present value of cash flows is defined to be the sum of the present values of cash flows, minus the initial investment. If NPV > 0, then the investment is acceptable since it promises a return greater than the required rate of return. If NPV = 0, then the investment is acceptable since it promises a return equal to the required rate of return. If NPV < 0, then the investment is not acceptable since it promises a return less than the required rate of return.
4 Example 1 A project requires an initial investment of $12,000. It has a guaranteed return of $8,000 at the end of year 1 and a return of $2,000 each year at the end of years 2, 3 and 4. Would you recommend that someone invest in this project if the prevailing market rate is at 8% compounded semiannually? r = % = 0.04 NPV = 8,000(1+.04) ,000(1+.04) ,000(1+.04) ,000(1+.04) -8 12,000 NPV = NPV > 0, the investment is profitable Equation of Value An equation of value illustrates that when one is considering two methods of paying a debt (or making someone other transaction), at anytime the value of all payments under one method must equal the value of all payments under the other method. This is used when there is a need to replace a set of debts by another set of different amounts due at different times. Example 1 A debt of 20,000 which is due 8 years from now, is instead to be paid off by three payments: $4,000 in 1 year, $8,000 in 4 years and the final payment at the end of 6 years. What would this payment be if an interest rate of 6% compounded monthly is assumed? r = % $4,000 $8,000 X $20,000 4,000(1+.005) ,000(1+.005) 24 + X = 20,000(1+.005) -24 X = $3,331.03
5 Annuities Annuity is a series of equal payments made or received at fixed period for specified amount of time. Example of annuities: insurance premiums, retirement saving, saving plans for future events, car loan payments, mortgage payment. The future value of annuity refers to the total of the annuity payments plus the accumulated compound interest on these payments. The present value of annuity refers to the amount that must be deposited now at compound interest to yield a series of equal periodic payment. Ordinary annuity Future Value future value(s) is the sum of all payments and interest those payment earn n-2 n-1 n R R R R R R(1+r) R(1+r) 2 R(1+r) n-1 Formula : Interest earned : I = S - nr Example 1 A woman deposited $500 per quarter for the first ten years of an annuity but changed to quarterly payments of $1,500 for the last 10 years. Assuming 6% interest compounded quarterly, what is her accumulated value? First ten years : R = 500, n = 40, r = = next ten years 1. Find the amount of the first annuity in next ten years S = P(1+r) n S = 27,133.95( ) 40 = $49,221.48
6 2. Change deposit money : R = 1,500, n = 10(years)(4 periods per year) = 40 Her accumulated value is ($49, $81,401.84) = $130, Ordinary Annuity Present Value Present value of annuity can be thought of as the amount (lump sum) you must invest today at a specific interest rate so that when you withdraw an equal amount each period, the original principal and all accumulated interest will be completely exhausted at the end of the annuity. The present value of an annuity (A) is the sum of the present value of each payment. It represents the one deposit now that gives the same final results as an annuity n-2 n-1 n... R R R... R R R R(1+r) -1 R(1+r) -2 : R(1+r) -n The present value of annuity (A) is R(1+r) -1 + R(1+r) R(1+r) -n OR Example 1 What is the cash value of a car that can be bought for $2500 down and $450 a month for 4 years if money is worth 12% compounded monthly? Down = 2,500, R = 450, r =, n = 48 Cash value of this car = 2, , = $19,588.28
7 Example 2 A person pays $150 per month for 48 months for a car, making no down payment. If the loan costs 1.5% interest per month on the unpaid balance, what was the original cost of the car? How much total interest will be paid? R = 150, r =, n = 48 payments Interest will be paid I = (150)(48) 5, = $2, Many annuities are paid with the first payment beginning immediately. An excelltn example is the lottery which pays out equal payments over a large number of years, often 20. In order to generate publicity, the first payments an annuity due. Lease payments are also good example of an annuity due. Lease arrangements are a series of regular payments at the beginning of each period occurring at equal time intervals (usually monthly) over a specified time (usually until the end of the lease arrangement). Annuity Due Present Value n-1 n R R R R R OR Annuity Due Future Value n-1 n R R R R R R OR Example 1 Find the present value of the following annuity due; payments of $400 are made semiannually at 7% compounded semiannually for 7 years. (Present Value) R = 400, r =, n = 15
8 Example 2 Kevin rents out a building to an accounting practice. The lease arrangement is for 10 years. Every three months Jake receives $10000 in rent. Payments are made upfront. This money goes into an account earning 6% compounded quarterly. What will be the value of the account at the end of the 10 year lease arrangement if Kevin never withdraws from it? (Future Value) R = 10,000, r =, n = 40 Ordinary Annuity Size of Annuity (R), Knowing Future Value of Annuity (S) Sinking Fund Sinking funds are accounts used to set aside equal amounts of money normally at the end of each period, at compound interest, for the purpose of saving for a future obligation. Example 1 A 20 years old woman wants to retire as a millionaire by the time she turns 70. How much will she have to save at the end of each month if she can earn 6% compounded monthly, tax-free, to have $1,000,000 by the time she is 70? S = 1,000,000, n = 600, r = Ordinary Annuity Size of Annuity (R), Knowing Present Value of Annuity (A) Example 1 Justin has inherited $25,000 and is planning to invest this amount at 6% interest. At the same time he wishes to make equal monthly withdrawals to use up the entire sum in 5 years. How much can he withdraw each month? A = 25,000, n = 60, r =, a = R = = $483.32
9 Amortization Amortization is a method for repaying a loan in equal installment for a specified amount of time. Part of each payment goes toward interest due for the period and the remainder is used to reduce the principal (the loan balance). As the balance of the loan is gradually reduced, a progressively larger portion of each payment goes toward reducing principal. Amortization is especially relevant for larger purchases made over longer periods of time. Loans to pay for homes and automobiles are usually amortized. There may, or may not be a down payment. Example 1 Jay borrowed $225,000 from the bank for his new house at an interest rate of 9%. He will make equal monthly payments for the next 30 years a. How much is his monthly payment? A = 225,000, r = 0.75%, n = 360, a = R = = $1, b. How much money will he end up paying the bank over the life of the loan? What will be the finance charge of this loan? Amount paying to the bank = (360)(1,810.40) = $651,744 Finance Charge = nr A = 651, ,000 = $426,744 Amortization Schedule An amortization schedule is a table that shows each loan payment over the life of a loan, and a breakdown of the amount of interest and principle paid. Typically, it will also show the remaining balance after each payment has been made. Example 1 Bryan has borrowed $13,100 from a friend in order to invest in the stock market. The debt is to be amortized over 2-year period at 7% compounded semiannually. His semiannual payment is calculated to be $3, Construct the amortization schedule for Bryan. A = 13,100, r = 3.5%, n = 4, R = = 3, Period Principal Outstanding at the beginning of period Interest for period (3.5% of principle outstanding) Payment at the end of period Principal repaid at the end of period 1 13, , , , , , , , , , , , Total 1, , ,100
10 Example 2 Nick bought his house in He had his loan $108,830 financed for 20 years at an interest rate of 10.8% resulting in a monthly payment of $1,020. This year, 18 years later, he would like to know the following information. a. The balanced owed after 18 years b. How much of the loan had he paid off in 18 years? c. How much in interest had he paid in 18 years ,020 1,020 1,020 1,020 a. There were 144 payments left : Present Value of the remaining = 1, = 1,020( ) = $82, b. He paid off in 18 years = 108,830 82, = $26,687.7 c. How much interest had he paid = nr A = (216)(1,020) 26, =$193,632.30
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