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Engineering Economy Chapter 4 – 1

Nominal and Effective Interest Rates School of Mechanical Engineering Henan Polytechnic University Winter 2014 LEARNING OUTCOMES 1. Understand interest rate statements 2. Use formula for effective interest rates 3. Determine interest rate for any time period 4. Determine payment period (PP) and compounding period (CP) for equivalence calculations 5. Make calculations for single cash flows 6. Make calculations for series and gradient cash flows with PP ≥ CP 7. Perform equivalence calculations when PP < CP 8. Use interest rate formula for continuous compounding 9. Make calculations for varying interest rates 4-2 Interest Rate Statements The terms ?nominal? and ?effective? enter into consideration when the interest period is less than one year. New time-based definitions to understand and remember Interest period (t) – period of time over which interest is expressed. For example, 1% per month. Compounding period (CP) – Shortest time unit over which interest is charged or earned. For example,10% per year compounded monthly. Compounding frequency (m) – Number of times compounding occurs within the interest period t. For example, at i = 10% per year, compounded monthly, interest would be compounded 12 times during the one year interest period. 4-3 Understanding Interest Rate Terminology A nominal interest rate(r), which is the same as APR(Annual Percentage Rate), is obtained by multiplying an interest rate that is expressed over a short time period by the number of compounding periods in a longer time period

That is

r = interest rate per period x number of compounding periods Ex

If i = 1% per month, nominal rate per year is r = (1) * (12) = 12%/yr Effective interest rate( ia ), which is the same as APY(Annual Percentage Yield), takes compounding into account (effective rates can be obtained from nominal rates via a formula). IMPORTANT

? Nominal interest rates are essentially simple interest rates. Therefore, they can never be used in interest formulas. ? Effective interest rates must always be used hereafter in all interest formulas. 4-4 1-5 Effective Annual Interest Rate (EAIR) Formula ? Effective interest rate is the actual rate that applies for a stated period of time. ? These are by far the most commonly quoted rates in everyday business and industry. ? The EAIR is easily generalized to any nominal rate by substituting “r/m” for the period interest rate i Effective i = (1 + i)n – 1 = [(1 + r/m)m – 1] where: r = nominal interest rate per year m = number of compounding periods per year i = effective interest rate per compounding period = r/m 6 year ia = effective interest rate per Statement

9% Compounded Monthly ? r = 9% (the nominal rate). ? Compounded monthly means m = 12. ? The effective monthly rate is: 0.09/12 = 0.0075 = 0.75% per month 0.75% 0.75% 0.75% 0.75% 0.75% 0.75% 0.75% 0.75% 0.75% 0.75% 0.75% 0.75% 1 2 3 4 5 6 7 8 9 10 11 12 One Year Duration (12 months) 7 Effective Annual Interest Rates Nominal rates are converted into effective annual rates via the equation: Effective interest per year ( ia ) = [(1 + r/m)m – 1], ia = effective annual interest rate i = effective rate for one compounding period (r/m) m = number times interest is compounded per year Example

For a nominal interest rate(r) of 12% per year, determine the nominal and effective rates per year for (a) quarterly, and (b) monthly compounding Solution

(a) Nominal( r )/ year = 12% per year Nominal( r )/ quarter = 12/4 = 3.0% per quarter Effective( ia) / year = (1 + 0.03)4 – 1 = 0.1255 ==? 12.55% per year Nominal( r )/month = 12/12 = 1.0% per year Effective( ia )/ year = (1 + 0.01)12 – 1 = 0.1268 ==? 12.68% per year 4-8 (b) Example 1 I put $2,000 in a bank at 6% per year compounded monthly. I will leave this in for 12 months. How much money will I have in 12 months ? Basic Formula: Solution by modified basic formula 0 1 2 3 4 5 6 7 8 9 10 11 12 F = P(1+i)n ia = [(1 + r/m)m – 1], Effective interest per year ( ia ) = [(1 + r/m)m – 1] F = $2,000(1 + 0.06/12)12 = $2,000(1.0617) = $2,123.40 9 Effective Interest Rates for Any Period of Time Example 2

For an interest rate of 1.2% per month, determine the nominal and effective rates (a) per quarter, and (b) per year Solution: ia = [(1 + r/m)m – 1] a) Nominal i / quarter = (1.2)(3) = 3.6% per quarter Effective 作文i / quarter = (1 + 0.036/3)3 – 1 = 3.64% per quarter b) Nominal i /year = (1.2)(12) = 14.4% per year Effective i / year = (1 + 0.144 / 12)12 – 1 = 15.39% per year 10 Example 3

For a Federal Credit Union that offers an interest rate of 8% per year, compounded quarterly, determine the nominal rate per 6 months. Solution

i = 8/4 = 2% per quarter ; r = 2(2%) = 4% per six months Example 4

The Second National Bank of Fullertum advertises an APR of 14% compounded monthly for student loans. Determine the APY. Show hand and spreadsheet solutions. Solution: Hand solution

i = (1 + 0.14/12)12 - 1 = 14.93% per year Spreadsheet solution

= EFFECT(14%,12) displays 14.93% 1-11 EAIR (Effective Aunnal Interest Rate) Given a Nominal Rate Example 5

Given

Interest is 8% per year “compounded quarterly”. What is the true annual interest rate ? Solution

EAIR = [(1 + r/m)m – 1] EAIR = [(1 + 0.08/4)4 – 1] EAIR = [(1 + 0.02)4 – 1] = 0.0824 = 8.24%/year Example 5

An interest rate of 13% per year compounded monthly is equivalent to what effective interest rate per year. Solution

[( 1 + 0.13/12)12 – 1] = 0.138 or 13.8% 12 Example 6 I borrow $15,000 and agree to pay the money back in one year at 25% interest compounded weekly. How much do I have to pay back in a year ? Solution

EAIR = [(1 + r/m)m – 1] EAIR (ia ) = [(1 + 0.25/52)52 – 1)] = 0.28326 EAIR (ia ) = 28.326% F1 = P(1 + ia ) = $15,000 * (1 + 0.28326)1 = $19,248.90 13 Example 7 An interest rate of 8% per 6 months, compounded monthly, is equivalent to what effective rate per quarter ? Solution

8% per 6 months = 0.08/6 = 0.0133 per month i = (1 + 0.0133)3 – 1 = 0.0405 or 4.05% per quarter 1-14 Example 8 A company plans to place money in a new venture capital fund that currently returns 18% per year, compounded daily. What effective rate is (a) yearly and (b) semiannually? Solution

(a) yearly

r = 0.18 and m = 365 EAIR = [(1 + r/m)m – 1] EAIR = ia = [(1+0.18/365)365 – 1] = 0.19716 19.716% (b) semiyearly

r = 0.18/2 = 0.09 and m = 182 EAIR = i6 months = [(1+0.09/182)182 – 1] = 0.09415 9.415% 15 Example 9 Voltex Corp, an auto parts supplier, is buying a heavy duty machine. As the site engineer, you are called to evaluate the bids from three vendors. Voltex will make payments on a semiannual basis only. What are the effective annual rates? Which bid has the lowest effective annual rate? Three different interest charging plans. Bid # 1

9.0% per year, compounded quarterly Bid # 2

3.0% per quarter, compounded quarterly Bid # 3

8.8% per year, compounded monthly Which Plan has the lowest annual interest rate? 16 Example 9

The Matching Rule ? The interest must be consistent with the payment period ! ? To compare the 3 plans: ? Compute the true, effective 6-month rate ? Compute the true effective 1 year rate. ? Then, one can compare the 3 plans ! ? We need a 6-month effective rate and then calculate the 1 year true effective rate! 17 Example 9

Bid # 1 9% per year, Compounded Quarterly 0 CP-1 CP-2 CP-3 CP-4 Payment Period 1 Payment Period 2 1 Payment Payment ? PP = 6 months, r = 9% per year; r = 4.5% per 6 months ? m = 2 quarters per 6 months or m = 4 quarters per year ? Effective i = (1 + i)n – 1 = [(1 + r/m)m – 1] Effective i % per 6 months = [(1+0.045/2)2 – 1] = 0.0455 = 4.55% Effective i % per year = [(1+0.09/4)4 – 1] = 0.0931 = 9.31% Rule

The interest rate must match the payment period ! 18 Example 9

Bid # 2 3% per quarter, compounded quarterly ? PP = 6 months ? r = 3% per quarter, then r per year = 12% ? m = 2 quarters per 6 months ? m = 4 quarters per year Solution

Effective i = (1 + i)n – 1 = [(1 + r/m)m – 1] Effective i6 months = [(1 + 0.03)2 – 1] = 0.0609 ? 6.09% Effective i12 months = [(1 + 0.03)4 – 1] = 0.1255 ? 12.55% 19 Example 9

Bid # 3 8.8% per year, Compounded Monthly ? PP = 6 months ? r = 8.8% per year = 4.4% per 6 months ? m = 6 for semiannual rates ? m = 12 for annual rates Solution: Effective i = (1 + i)n – 1 = [(1 + r/m)m – 1] Effective i6-months = [(1 + 0.044/6)6 – 1] = 0.048 =?4.48% Effective i12 months = [(1 + 0.088/12)12 – 1] =0.0916 =?9.16% 20 Summarizing Example 9

The 3 Bids Bid No. 1 6-month 4.55% 1-year 9.31% 2 3 6.09% 4.48% 12.55% 9.16% The Plan 3 presents the lowest interest rate. 21 Class Work # 1 For a Federal Credit Union that offers an interest rate of 8% per year, compounded quarterly, determine the nominal rate per 6 months. Solution: 1-22 Class Work # 2 In ?N Out Payday Loans advertises that for a fee of only $10 per month, you can immediately borrow $200 for one month. If a person accepts the offer, what are ( a ) the nominal interest rate per year and ( b ) the effective rate per year? Solution: 1-23 Class Work #3 I borrow $15,000 from a loan shark. I agree to pay the money back in one year at 25% interest compounded weekly. How much do I have to pay back in a year. Solution: Effective i = (1 + i)n – 1 = [(1 + r/m)m – 1] 24 Class Work #4 I am in desperate need of money. I get a cash advance on my credit card. The interest rate is 18% compounded monthly. What is the effective annual interest rate? The effective annual rate is

Effective i = (1 + i)n – 1 = [(1 + r/m)m – 1] ? Note: r = Nominal Rate is 18%; i = r/m = The effective monthly rate is 1.5% / month; ia = The effective annual rate is 19.56% /year. 25 Class Work # 5 An interest rate of 8% per 6 months, compounded monthly, is equivalent to what effective rate per quarter? Solution: 1-26 Homework for Chapter 4 – 1 ? Handout Problems for Chapter 4-1 ? 4.5, 4.6, 4.12, and 4.18 1-27

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