page-1 Time Value of Money Effective Annual Rates

page-1 Time Value of Money Effective Annual Rates Nominal Rates Periodic Rates

page-2 We will deal with 3 different rates: i. Nom = nominal, or stated, or quoted, rate per year. i. Per = periodic rate. effective annual EAR = EFF% =. rate

page-3 n i. Nom is stated in contracts. Periods per year (m) must also be given. n Examples: n 8%; Quarterly n 8%, Daily interest (365 days)

page-4 n Periodic rate = i. Per = i. Nom/m, where m is number of compounding periods per year. m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding. n Examples: 8% quarterly: i. Per = 8%/4 = 2%. 8% daily (365): i. Per = 8%/365 = 0. 021918%.

page-5 n Effective Annual Rate (EAR = EFF%): Our bank offers a Certificate of Deposit (CD) with a 10% nominal rate, compounded semiannually. What is the true rate of interest that they pay annually (the effective annual rate)? 0 $1 5% 6 mo 12 mo 5%

page-6 How do we find EFF% for a nominal rate of 10%, compounded semiannually? ( ) -1 = (1 + 0. 10) - 1. 0 2 i. Nom EFF% = 1 + m m 2 = (1. 05)2 - 1. 0 = 0. 1025 = 10. 25%. Any PV would grow to the same FV at 10. 25% annually or 10% semiannually.

page-7 Financial Calculator Using interest conversion: 2 nd IConv NOM%= 10 ENTER C/Y = 2 ENTER CPT EFF%= 10. 25

page-8 EAR = EFF% of 10% EARAnnual = 10%. EARQ = (1 + 0. 10/4)4 - 1 = 10. 38%. EARM = (1 + 0. 10/12)12 - 1 = 10. 47%. EARD(365) = (1 + 0. 10/365)365 - 1 = 10. 52%.

Calculations using Excel (in edit mode, click on the insert below for Excel) page-9

page-10 What is the FV of $100 after 3 years under 10% semiannual compounding? æ 1 + i. Nomö FVn = PVç ÷ è mø FV 3 S mn . æ 1 + 0. 10ö = $100ç ÷ è 2 ø = $100(1. 05)6 2 x 3 = $134. 01.

page-11 Using a Financial Calculator Using semi-annual periods: INPUTS N I/YR PV PMT FV OUTPUT Using annual periods: INPUTS N OUTPUT

Calculations using Excel (in edit mode, double click on the insert below for Excel) page-12

page-13 What’s the value at the end of Year 3 of 3 $100 payments made at the end of each year. The quoted interest rate is 10%, compounded semiannually? 0 1 1 yr 2 3 2 yrs 4 5 5% 100 3 yrs 6 6 -mos. periods 100

page-14 b. Use EAR = 10. 25% as the annual rate in your calculator: INPUTS N OUTPUT I/YR PV PMT FV

Calculations using Excel (in edit mode, double click on the insert below for Excel) page-15

page-16 Nominal, Periodic, or Effective Annual Rate? n 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. n If PMT is 0, either the periodic rate or the effective rate may be used. l 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. 14 E. A. R. : FV = 100, N = 5, I = _____; compute PV= $74. 14

page-17 n 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. l 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 l 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

Evaluating the typical mortgage (in edit mode, double click on the insert below for Excel) page-18

page-19 Amortization Schedule