Presentation on theme: "Chapter 3 Understanding Money Management"— Presentation transcript:
1 Chapter 3 Understanding Money Management Nominal and Effective Interest RatesEquivalence Calculations using Effective Interest RatesDebt Management
2 Focus1. If payments occur more frequently than annual, how do you calculate economic equivalence?If interest period is other than annual, how do you calculate economic equivalence?How are commercial loans structured?How should you manage your debt?
3 Nominal Versus Effective Interest Rates Nominal Interest Rate:Interest rate quoted based on an annual periodEffective Interest Rate:Actual interest earned or paid in a year or some other time period
5 What It Really Means? In words, 18% Compounded MonthlyWhat It Really Means?Interest rate per month (i) = 18% / 12 = 1.5%Number of interest periods per year (N) = 12In words,Bank will charge 1.5% interest each month on your unpaid balance, if you borrowed moneyYou will earn 1.5% interest each month on your remaining balance, if you deposited money
6 18% compounded monthly = 1.5% = $1.1956 0.1956 or 19.56% Question: Suppose that you invest $1 for 1 year at 18% compounded monthly. How much interest would you earn?Solution:= $1.1956or 19.56%18%= 1.5%
7 Effective Annual Interest Rate (Yield) r = nominal interest rate per yearia = effective annual interest rateM = number of interest periods per year
8 = : 1.5% 18% compounded monthly or 1.5% per month for 12 months 19.56 % compounded annually
9 Practice ProblemIf your credit card calculates the interest based on 12.5% APR, what is your monthly interest rate and annual effective interest rate, respectively?Your current outstanding balance is $2,000 and skips payments for 2 months. What would be the total balance 2 months from now?
11 Practice ProblemSuppose your savings account pays 9% interest compounded quarterly. If you deposit $10,000 for one year, how much would you have?
12 Effective Annual Interest Rates (9% compounded quarterly) First quarterBase amount+ Interest (2.25%)$10,000+ $225Second quarter= New base amount= $10,225+$230.06Third quarter= $10,455.06+$235.24Fourth quarter+ Interest (2.25 %)= Value after one year= $10,690.30+ $240.53= $10,930.83
13 Compounding Semi-annually Compounding Quarterly Nominal and Effective Interest Rates with Different Compounding PeriodsEffective RatesNominal RateCompounding AnnuallyCompounding Semi-annuallyCompounding QuarterlyCompounding MonthlyCompounding Daily4%4.00%4.04%4.06%4.07%4.08%55.005.065.095.125.1366.006.096.146.176.1877.007.127.197.237.2588.008.168.248.308.3399.009.209.319.389.421010.0010.2510.3810.4710.521111.0011.3011.4611.5711.621212.0012.3612.5512.6812.74
14 Effective Interest Rate per Payment Period (i) C = number of interest periods per payment periodK = number of payment periods per yearCK = total number of interest periods per year, or Mr /K = nominal interest rate per payment period
15 12% compounded monthly Payment Period = Quarter Compounding Period = Month One-year1st Qtr2nd Qtr3rd Qtr4th Qtr1%1%1%3.030 %Effective interest rate per quarterEffective annual interest rate
16 Effective Interest Rate per Payment Period with Continuous Compounding where CK = number of compounding periods per yearcontinuous compounding =>
17 Payment Period = Quarter Case 0: 8% compounded quarterlyPayment Period = QuarterInterest Period = Quarterly1st Q2nd Q3rd Q4th Q1 interest periodGiven r = 8%,K = 4 payments per yearC = 1 interest period per quarterM = 4 interest periods per year
18 Payment Period = Quarter Case 1: 8% compounded monthlyPayment Period = QuarterInterest Period = Monthly1st Q2nd Q3rd Q4th Q3 interest periodsGiven r = 8%,K = 4 payments per yearC = 3 interest periods per quarterM = 12 interest periods per year
19 Payment Period = Quarter Case 2: 8% compounded weeklyPayment Period = QuarterInterest Period = Weekly1st Q2nd Q3rd Q4th Q13 interest periodsGiven r = 8%,K = 4 payments per yearC = 13 interest periods per quarterM = 52 interest periods per year
20 Payment Period = Quarter Case 3: 8% compounded continuouslyPayment Period = QuarterInterest Period = Continuously1st Q2nd Q3rd Q4th Q interest periodsGiven r = 8%,K = 4 payments per year
21 Summary: Effective interest rate per quarter Case 0Case 1Case 2Case 38% compounded quarterly8% compounded monthly8% compounded weekly8% compounded continuouslyPayments occur quarterly2.000% per quarter2.013% per quarter2.0186% per quarter2.0201% per quarter
22 Equivalence Analysis using Effective Interest Rates Step 1: Identify the payment period (e.g., annual, quarter, month, week, etc)Step 2: Identify the interest period (e.g., annually, quarterly, monthly, etc)Step 3: Find the effective interest rate that covers the payment period.
23 Case I: When Payment Periods and Compounding periods coincide Step 1: Identify the number of compounding periods (M) per yearStep 2: Compute the effective interest rate per payment period (i)i = r / MStep 3: Determine the total number of payment periods (N)N = M (number of years)Step 4: Use the appropriate interest formula using i and N above
24 Example 3.4: Calculating Auto Loan Payments Given:Invoice Price = $21,599Sales tax at 4% = $21,599 (0.04) = $863.96Dealer’s freight = $21,599 (0.01) = $215.99Total purchase price = $22,678.95Down payment = $2,678.95Dealer’s interest rate = 8.5% APRLength of financing = 48 monthsFind: the monthly payment
25 Solution: Payment Period = Interest Period $20,00048AGiven: P = $20,000, r = 8.5% per yearK = 12 payments per yearN = 48 payment periodsFind AStep 1: M = 12Step 2: i = r / M = 8.5% / 12 = % per monthStep 3: N = (12)(4) = 48 monthsStep 4: A = $20,000(A/P, %,48) = $492.97
26 Suppose you want to pay off the remaining loan in lump sum right after making the 25th payment. How much would this lump be?$20,00048122524$492.97$492.9725 payments that were already made23 payments that are still outstandingP = $ (P/A, %, 23)= $10,428.96
27 Practice ProblemYou have a habit of drinking a cup of Starbuck coffee ($2.00 a cup) on the way to work every morning for 30 years. If you put the money in the bank for the same period, how much would you have, assuming your accounts earns 5% interest compounded daily.NOTE: Assume you drink a cup of coffee every day including weekends.
29 Case II: When Payment Periods Differ from Compounding Periods Step 1: Identify the following parametersM = No. of compounding periodsK = No. of payment periodsC = No. of interest periods per payment periodStep 2: Compute the effective interest rate per payment periodFor discrete compoundingFor continuous compoundingStep 3: Find the total no. of payment periodsN = K (no. of years)Step 4: Use i and N in the appropriate equivalence formula
30 Example 3.5 Discrete Case: Quarterly deposits with Monthly compounding F = ?Year 1Year 2Year 312QuartersA = $1,000Step 1: M = 12 compounding periods/yearK = 4 payment periods/yearC = 3 interest periods per quarterStep 2:Step 3: N = 4(3) = 12Step 4: F = $1,000 (F/A, 3.030%, 12)= $14,216.24
31 Continuous Case: Quarterly deposits with Continuous compounding F = ?Year 1Year 2Year 312QuartersA = $1,000Step 1: K = 4 payment periods/yearC = interest periods per quarterStep 2:Step 3: N = 4(3) = 12Step 4: F = $1,000 (F/A, 3.045%, 12)= $14,228.37
32 Practice ProblemA series of equal quarterly payments of $5,000 for 10 years is equivalent to what present amount at an interest rate of 9% compounded(a) quarterly(b) monthly(c) continuously
42 Example 3.9 Buying versus Lease Decision Option 1Debt FinancingOption 2Lease FinancingPrice$14,695Down payment$2,000APR (%)3.6%Monthly payment$372.55$236.45Length36 monthsFees$495Cash due at lease end$300Purchase option at lease end$Cash due at signing$731.45
43 Which Interest Rate to Use to Compare These Options?
45 SummaryFinancial institutions often quote interest rate based on an APR.In all financial analysis, we need to convert the APR into an appropriate effective interest rate based on a payment period.When payment period and interest period differ, calculate an effective interest rate that covers the payment period. Then use the appropriate interest formulas to determine the equivalent values