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We will deal with 3 different rates: i Nom = nominal, or stated, or quoted, rate per year.

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## We will deal with 3 different rates: i Nom = nominal, or stated, or quoted, rate per year.

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**We will deal with 3 different rates:**iNom = nominal, or stated, or quoted, rate per year. iPer = periodic rate. EAR = EFF% = . effective annual rate**iNom is stated in contracts. Periods per year (m) must also**be given. This is also frequently referred to as an Annual Percentage Rate (APR). • Examples: • 8%; Quarterly • 8%, Daily interest (365 days)**Periodic rate = iPer = iNom/m, where m is number of**compounding periods per year. m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding. • Examples: 8% quarterly: iPer = 8%/4 = 2%. 8% daily (365): iPer = 8%/365 = 0.021918%.**Effective Annual Rate (EAR = EFF%):**Our bank offers a Certificate of Deposit (CD) with a 10% nominal rate, compounded semi-annually. What is the true rate of interest that they pay annually (the effective annual rate)? 0 6mo 12mo 5% 5% $1**(1 + )**iNom m m EFF% = - 1 How do we find EFF% for a nominal rate of 10%, compounded semiannually? (1 + ) 2 0.10 2 = - 1.0 = (1.05)2 - 1.0 = 0.1025 = 10.25%. Any PV would grow to the same FV at 10.25% annually or 10% semiannually. In both instances, our money earns 10.25 percent each year.**For a given rate, more frequent compounding increases**returns 12 percent compounded annually EARA = 12%. EARs = (1 + 0.12/2)2 - 1 = 12.36%. EARM = (1 + 0.12/12)12 - 1 = 12.68%. EARD(365) = (1 + 0.12/365)365 - 1 = 12.75%. 12 percent compounded semi-annually 12 percent compounded monthly 12 percent compounded daily**What is the FV of $100, if you earn 10 percent compounded**semi-annually? When using semi-annual periods we use the periodic rate: 6 5 100 0 INPUTS N I/YR PV PMT FV OUTPUT Cpt = 134.01 When using annual periods we use the EAR: 10.25 100 0 INPUTS 3 N I/YR PV PMT FV OUTPUT Cpt = 134.01 The nominal rate is never used in calculations!**Can the effective rate ever be equal to the nominal rate?**• Yes, but only if annual compounding is used, i.e., if m = 1. • If m > 1, EFF% will always be greater than the nominal rate.**When is each rate used?**The use of the periodic rate and the effective annual rate is determined by the choice of N. If N is the number of years, the EAR is used. If N is the number of periods (shorterthan 1 year), the periodic rate is used. The nominal rate is not used in calculations. As shown later, the choice of N will be determined by the number of payments (PMT). Written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines. iNom:**Nominal, Periodic, or Effective Annual Rate?**• We never use the nominal rate calculating future or present values. We always use either the effective annual rate or the periodic rate of return. • If PMT is 0, either the periodic rate or the effective rate may be used. • Ex: Find the present value of $100 to be received in 5 years. The nominal rate is 6 percent, compounded monthly.Periodic: FV = 100, N = 5*12, I = _______; compute PV= $74.14E.A.R.: FV = 100, N = 5, I = __________; compute PV= $74.14 6/12 6.1678**If PMT has a value, N must equal the number of payments.**Since N is determined by the number of payments, we must make I (the interest rate) consistent with how we defined N. • Ex: Find the present value of $100 per year for 5 years. The nominal rate is 6 percent, compounded monthly.PMT = 100, N = 5, I = _________; compute PV = $419.32 • Ex: Find the present value of $100 per month for 5 years. The nominal rate is 6 percent, compounded monthly.PMT = 100, N = 5*12, I = ________; compute PV = $5,172.56 6.1678 6/12**A few other points to note.**• If the rate is 12% compounded monthly, your EAR is 12.6825%, this is used if the N is in number of years. • If the rate is quoted at 12% compounded monthly, the periodic rate would be 1% (12%/12). The 1% is used if the N is in number of months. • If the rate is 12% compounded monthly, you cannot divide the 12% by anything besides its number of compounding periods (in this case 12 monthly periods), or you change the interest rate.**Example**• What is the future value of 4 quarterly payments of $100 each, given a nominal rate of 12 percent, compounded monthly. • Note that the quarterly payments and monthly compounding are different, so we have to transform the monthly compounded rate to a quarterly compounded rate. This takes some work.First, find the EAR of the monthly rate:EARM = (1 + 0.12/12)12 - 1 = 12.68%.Second, find the nominal rate with quarterly compounding that has an effective annual rate of 12.68%(1 + Inom/4)4 - 1 = 12.68%.Inom = 12.12%Third, compute the future value of the paymentsN = 4, I = 12.12/4, PV = 0, PMT = 100, cpt FV = $418.55 • See the next slide for the Excel Example