Engineering Economic Analysis Chapter 4: Part 3 Other Interest Formulas Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Nominal and Effective Interest Nominal interest rate/year: the annual interest rate w/o considering the effect of any compounding. 12%/year Interest rate/period: the nominal interest rate/year divided by the number of interest compounding periods. 12%/year/12 months/year = 1%/period Effective interest rate/year: the annual interest rate taking into account the effect of the compounding periods in the year. 12%/year compounded monthly is equivalent to 12.68%/year compounded yearly Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Nominal and Effective Interest Assume i = 5%, compounded semi-annually. How much interest accrues on $100 over 1 year at this rate? Interest accrued = $100 · (F/P, i, n) - $100 Here n = 2 We get $100 · (1 + 0.025)2 – $100 = $100 · (1.051) - $100 = $105.1 - $100 = $5.10 Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Nominal and Effective Interest What interest rate, compounded annually results in F = $105.10, when P = 100 over 1 year Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Nominal and Effective Interest How does 5.1% per year relate to 5% compounded semiannually? 5% is the nominal interest rate 5.1% is the effective interest rate Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Nominal and Effective Interest r is the nominal interest rate per interest period. It is the rate not considering compounding i is the effective interest rate per interest period. It is the rate considering compounding ia is the effective interest rate per year m number of compounding periods per time period Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Nominal and Effective Interest The effective interest rate ia is defined to be: This is why 5% compounded semiannually means that ia is 5.1% per year m = 2, r = 5% and i = 2.5% In Excel: EFFECT(nom, nper) nom = the nominal rate for an entire year, r nper = the number of compoundings in a year, m Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Example 1 You go to a check cashing store to get a $100 advance on your pay check. They are willing to do so provided you write a check for $120 dated for 1 week from now. What nominal interest rate per year are you paying? What effective interest rate per year are you paying? Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Example 1 What nominal interest rate per year are you paying? First determine the weekly interest. Remember nominal means no interest Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Example 1 What effective interest rate per year are you paying? Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Example 1 Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Example 2 A bank pays 8% nominal interest per year, compounded quarterly. A person deposits $5,000 at time 0. After the end of each of the next 5 years she wants to withdraw an equal amount. What is this amount? If compounding interval doesn’t equal cash flow period, we need to be sure to notice Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Example 2 This is not a uniform series problem because we don’t have a cash flow each time, but interest is compounded each time. There are at least two ways to solve this. Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Example 2 Find Then Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Example 2 Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Example 2 Translate 8% nominal interest compounded quarterly into an effective annual rate so we have 5 one year periods. r = 8% m = 4 Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Example 2 We now know that we have an effective annual interest of 8.24% and now we can use the following cash flow diagram. Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Example 2 Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Example 2 Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Continuous Compounding As m, the number of compounding subperiods per year grows to ∞, the interest compounds continuously. m↑ and r/m ↓ Under continuous compounding, the effective interest rate per year is: In Excel, just use the exponential directly Many credit cards use continuous compounding Where e is the exponential function Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Single Payment – Continuous Compounding When we have single payments with continuous compounding we replace the (1+i) with e and n with r · n So we see that the formulas are very similar. The bracket instead of the parenthesis for the functional notation imply continuous compounding. For these we have to use a different table. Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Example 3 What is the amount of interest earned on $2000 for 2 years earning 5% nominal interest compounded continuously? Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Example 3 Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Example 4 How long will it take $2,000 to double to $4,000 at 10% nominal interest compounded continuously? Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Example 4 This is a perfect application of Goal Seek Use a value of 1 to make sure the formula in B9 works Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Example 5 What is the effective annual interest rate of 6% compounded continuously? With Excel, you can see how the non-continuous compounding formula performs with more and more compounding periods: Do the differences in these values matter? Probably, if you are Bank of America getting mortgage payments, yes. Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Uniform Series – Continuous Compounding at Nominal Rate r per Period Just use i = er – 1 in the uniform series equations: Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Example 6 How much money will accrue in an account earning 5% compounded continuously for 5 years if $500 is deposited each year? Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Money flows continuously over one period, like: Continuous Uniform Cash Flow (1 Period) with Continuous Compounding at rate r Money flows continuously over one period, like: Money out of an ATM Money into a credit card company in the form of payments Let the continuous, uniform cash flow totaling be distributed over m subperiods within one period (n = 1). Thus is the cash flow at the end of each subperiod Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Continuous Uniform Cash Flow (1 Period) with Continuous Compounding at rate r We use the following formula to find the future value n periods in the future. Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Continuous Uniform Cash Flow (1 Period) with Continuous Compounding at rate r We use the following formula to find the present of a continuous payment n periods in the future. Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Example 7 A gas station ATM takes payments. The gas station receives $40,000 per month, credited continuously to their account during each month. If they earn 9% nominal interest compounded continuously, how much money will the bank account have, if they start at 0 at time 0, after one month? Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Example 7 Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

Example 7 No special tricks for Excel Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.