F371 Financial Manageent Part 3.2: DISCOUNTED CASH FLOW VALUATION AND MULTIPLE CASH FLOWS COMPARING RATES WITH DIFFERENT COMPOUNDING PERIODS The stated rate required on consuer loans is called the Annual Percentage Rate (A.P.R.). APR = R and R = APR where = nuber of copounding periods in a single year. Exaples: APR = 12% copounded onthly R = 1% per onth. APR = 16% copounded seiannually R = 8% per seiannual period. R = 2% per quarter APR = 8% copounded quarterly. Effective annual interest rate (E.A.R.): The rate, on an annual basis, that reflects copounding effects. Given soe rate, the EAR is the annual rate which gives the sae Future Value at the end of one year. Starting fro the sae PV, if the FV is the sae at the end of one year, then the (1+r) ters ust be the sae. In other words: (1 + EAR as a decial) 1 = (1 + given rate) F371 Page 1 of 6
Exaple: Consider a rate of 2% per quarter. What is the EAR that is equivalent to 2% per quarter? (1 + EAR) 1 = (1 +.02) 4 1 + EAR = 1.02 4 = 1.0824 EAR = 8.24% per year NOTE: The EAR is NOT necessarily the sae as the APR. This is because the APR does not include the effects of copounding. To find the EAR when starting fro an APR: Steps: 1) Divide the APR by (the nuber of copounding periods in one year). 2) Convert this percentage to (1 + r) forat (divide by 100, add 1). 3) Raise this ter to the power of (nuber of copounding periods). This gives you the EAR in (1 + r) forat. 4) Convert fro (1 + r) forat back to a percent (inus 1, ties 100). The foral equation is: EAR = (1 + = (1 + APR as a decial ) 1 Quoted rate ) 1 Exaple: Consider an APR of 10% copounded quarterly. What is the effective annual rate (EAR)? EAR = (1 + 0.10 4 ) 4 1 = 1.025 4 1 = 1.1038 1 EAR = 10.38% per year F371 Page 2 of 6
Exaple 2: Which option would you choose if you were getting a loan? A) 10% copounded onthly B) 10% copounded quarterly C) 10.25% copounded annually D) 11% copounded annually IMPLICATIONS FOR SOLVING FOR PV AND FV When we solve for present or future value, we ust either use the E.A.R. with years or the periodic rate (quoted/) and N periods (where N = * nuber of years). => FV = PV * ( 1 q t 1 FV ) ; PV = FV * [ ] ( 1 q ; PV = t q t ) ( 1 ) Exaple: What is the present value of $100 to be received in 2 years at 10% copounded quarterly? There is no periodic payent in this one (PV and FV only), so either of these two ethods will work: Method 1: Use a quarterly rate and the total nuber of quarters. Method 2: Use the EAR (yearly rate) and the nuber of years. F371 Page 3 of 6
RATE ADJUSTMENTS FOR MULTIPLE COMPOUNDING PERIODS--SUMMARY Whenever the copounding periods differ fro once per year (e.g. seiannually, quarterly, onthly, etc.) you ust ake the appropriate adjustents to all forulas! t (n) => = the # of years * # copounding periods per year. r => = stated rate / # copounding periods per year. With annuities, you ust be sure to atch the appropriate rate to your cash flows! Exaple 1: Suppose a copany has borrowed $500,000. The loan will be repaid in five annual payents starting at the end of the first year. The interest rate on the loan is 9% copounded onthly. What is the aount of the annual payent? Exaple 2: You borrow $10,000 to purchase a car and agree to repay the loan over four years of onthly payents. The APR (quoted rate) is 10.58% copounded quarterly. What is the aount of your onthly payent? Exaple 3: A coercial loan in the aount of $50,000 will be repaid with seiannual payents for four years. The quoted interest rate is 8.96% copounded quarterly. What will be the aount of the seiannual payent? F371 Page 4 of 6
PERPETUITIES Perpetuity: A series of level cash flows which continues forever. Perpetuity Present value: CF1 / r where CF1 is the aount of the cash flow per period, and r is the periodic interest rate. Present value at Tie Zero = Cash flow at Tie 1 r Exaple: Suppose that starting in one year, you will receive a perpetuity of $100 each year, forever. What is the present value of the series of cash flows at a 10% yearly interest rate? Exaple 2: Suppose that starting four years fro now, you will start receiving a perpetuity cash flow of $5,000 per year. What is the present value of this perpetuity, assuing a 12% yearly interest rate? Exaple 3: A new investent opportunity is forecast to produce a cash outflow next year of $8,000. In Year 2, a cash inflow of $2,000 is forecast. After that, starting in Year 3, it is expected to generate a cash inflow of $3,000 per year. At a discount rate of 6% per year, what should be the price of this investent today? F371 Page 5 of 6
AMORTIZATION SCHEDULES An aortization schedule is a repayent schedule for a typical consuer loan that shows: 1) the nuber of payents 2) the aount of each payent 3) the interest paid per period 4) the reduction in principal per period 5) the reaining loan balance Exaple: Suppose you borrow $4,000 and agree to repay the loan in 5 equal installents over a 5-year period. The interest rate on the loan is 10% per year. Step 1: Solve for payent using the PVIFA forula: Step 2: Coplete the aortization table as follows: Calculations Initial loan aount; then ending balance fro prior period PMT Loan balance periodic interest rate Periodic payent interest charge Loan balance reduction in principal Loan Periodic Interest Reduction in Ending Period Balance Payent Charge Principal Balance 1 $4,000.00 2 3 4 5 F371 Page 6 of 6