Chapters 5 and 6 (Cont.) 4. Semi-Annual and Other Compounding Periods All calculations are performed as explained above. You must adjust N and I in your calculations. Periodic Rate = State Annual Rate / Number of Periods per year Number of Periods = (Numbers of Years) x (Periods per Year) Example: What is the future value of $100 in 2 years at 8% (Annual Compounding) N=2 I%=8 PV= -100 PMT =0 Now assume Semi-annual compounding: N=2*2 = 4 I/%=8/2 = 4 PV = -100 PMT=0 Assume Quarterly compounding: N=2*4 = 8 I/%=8/4 = 2 PV = -100 PMT=0 Assume Monthly Compounding N=2*12 = 24 I/%=8/12 = .6667 PV = -100 PMT=0 Notice that the FV is greater as the compounding period increases. 5. Uneven Cash Flows An annuity includes constant payments. But many investments and financing decision include uneven cash flows where the dollar amounts of cash flows vary from period to period. There are two classes of uneven cash flows: (1) cash flows with amounts that may vary from period-to-period and (2) a stream of similar cash flows with an additional lump sum at the end (similar to a bond). The Present Value of an uneven stream of income is simply the sum of the Present Values of each individual cash flow. Similarly, the Future Values of an uneven stream of income is the sum of the Future Values of the individual cash flow components. Present Value of Uneven Cash Flows PV of uneven cash flows can use calculator functions to simplify calculations Timeline 0 1 2 3 4 5 6 | | | | | | | 100 100 100 200 300 500 r=8% L1 CF(1) =100 TI-83 Keys CF(2) =100 Stat Button CF(3) = 100 1 key or Enter CF(4) =200 Type in list CF(5) =300 2nd and Mode Keys CF(6) =500 APPS key npv ( 8, 0, L1) 1 key or Enter = 923.98 7 key for NPV Future Value of each cash Future Value of uneven cash flows --- unfortunately there is no automated FV function on calculator. You would need to calculate each FV calculation separately and add them together. 6. Nominal and Effective Rates Nominal rate is the contractual rate quoted by borrowers and lenders. Nominal rates can only be compared with one another if the instruments being compared use the same number of compounding periods per year. Effective Rate is the annual rate of interest actually being earned, vs. the contractual rate. Also called the Equivalent Annual Rate. (EAR) If annual compounding used, Nominal Rate = Effective Rate. If compounding occurs more than once a year, Effective Rate > Nominal Rate. EAR = (1 + INOM/M)M 1 Where INOM = Nominal Rate M = # of compounding periods per year Semi = 2, Quarterly = 4, Monthly = 12 , Daily = 365 Example: An investor has the choice to invest in: (1) Bond A that pays a nominal rate of 10%, compounded annually or (2) Bond B that pays a nominal rate of 10%, compounded semi-annually. First, you have to convert both rates to Effective Rates. Bond A Effective Rate = (1+.10/1)1 -1= 10% Bank B Effective Rate = (1 + .10/2)2 1 = 10.25% If the investor is seeking the highest return, they would select Bond B because the EAR is higher vs. Bond A. 7. Amortization Schedule Amount of payment applied to principal increases each period Amount of payment applied to interest decrease each period. PMT = Principal +Interest Calculate PMT using PVIFA Example-- The house you always loved is for sale. After taking a brief tour you decide to make an offer on the home for $250,000. The local bank approves a 30-year mortgage at an annual interest rate of 6.5%. What would be your MONTHLY payment? (**a couple of hints-- the number of periods will be months, not years. And your interest rate should also be monthly, not the annual rate) Beginning Amount Payment Interest Repayment of Principal Ending Balance % of Principal Retired (Paid) Month 1 $250,000.00 $1,580.18 $1,354.17 $226.01 $249,773.99 0.09% 2 249,773.99 1,580.18 $1,352.94 $227.24 $249,546.75 0.18% 3 249,546.75 1,580.18 $1,351.71 $228.47 $249,318.28 0.27% 4 249,318.28 1,580.18 $1,350.47 $229.71 $249,088.57 0.36% 5 249,088.57 1,580.18 $1,349.23 $230.95 $248,857.62 0.46% 6 248,857.62 1,580.18 $1,347.98 $232.20 $248,625.42 0.55% 7 248,625.42 1,580.18 $1,346.72 $233.46 $248,391.96 0.64% 8 248,391.96 1,580.18 $1,345.46 $234.72 $248,157.24 0.74% 9 248,157.24 1,580.18 $1,344.19 $235.99 $247,921.25 0.83% 10 247,921.25 1,580.18 $1,342.91 $237.27 $247,683.97 0.93% 11 247,683.97 1,580.18 $1,341.62 $238.56 $247,445.41 1.02% 12 247,445.41 1,580.18 $1,340.33 $239.85 $247,205.56 1.12% 13 247,205.56 1,580.18 $1,339.03 $241.15 $246,964.41 1.21% 14 246,964.41 1,580.18 $1,337.72 $242.46 $246,721.96 1.31% 15 246,721.96 1,580.18 $1,336.41 $243.77 $246,478.19 1.41% 16 246,478.19 1,580.18 $1,335.09 $245.09 $246,233.10 1.51% 17 246,233.10 1,580.18 $1,333.76 $246.42 $245,986.68 1.61% 18 245,986.68 1,580.18 $1,332.43 $247.75 $245,738.93 1.70% 19 245,738.93 1,580.18 $1,331.09 $249.09 $245,489.83 1.80% 20 245,489.83 1,580.18 $1,329.74 $250.44 $245,239.39 1.90% 21 245,239.39 1,580.18 $1,328.38 $251.80 $244,987.59 2.00% 22 244,987.59 1,580.18 $1,327.02 $253.16 $244,734.43 2.11% 23 244,734.43 1,580.18 $1,325.64 $254.54 $244,479.89 2.21% 24 244,479.89 1,580.18 $1,324.27 $255.91 $244,223.98 2.31% After 2-full years of payments, only 2.31% of the loan has been repaid. 232 146,398.48 1,580.18 $792.99 $787.19 $145,611.29 41.76% 233 145,611.29 1,580.18 $788.73 $791.45 $144,819.84 42.07% It take over 19 years (of your 30 year mortgage) before the majority of your monthly payment is being allocated to repay the loan and not simply going to interest on your loan. 256 126,281.01 1,580.18 $684.02 $896.16 $125,384.86 49.85% 257 125,384.86 1,580.18 $679.17 $901.01 $124,483.84 50.21% It take over 21years (of your 30 year mortgage) before you have repaid half of your mortgage.