University:
California State University, Northridge
Course:
MSE 604 | Engineering Economy and Financial Analysis
Academic year:

2024
Views:

365

Pages:

12
Author:

mylovaboycv2

Section 3.16. Nominal and Effective Interest Rates • Nominal interest (r) = interest compounded more than one interest period per year but quoted on an annual basis. • Example: 16%, compounded quarterly • Effective interest (i) = actual interest rate earned or charged for a specific time period. • Example: 16%/4 = 4% effective interest for each of the four quarters during the year. 61 Relationship • Relation between nominal interest and effective interest: i=(1+r/M)M -1, where • i = effective annual interest rate • r = nominal interest rate per year • M = number of compounding periods per year • r/M = interest rate per interest period 62 Nominal and Effective Interest Rates –Examples • Find the effective interest rate per year at a nominal rate of 18% compounded (1) quarterly, (2) semiannually, and (3) monthly. • (1) Quarterly compounding; i=(1+0.18/4)4 -1=0.1925 or 19.25% • (2) Semiannual compounding; i=(1+0.18/2)2 -1=0.1881 or 18.81% • (3) Monthly compounding ... 63 Nominal and Effective Interest Rates –Example • A credit card company advertises an A.P.R. of 16.9% compounded daily on unpaid balances. What is the effective interest rate per year being charged? r = 16.9% M = 365 • Solution: ieff = (1+0.169/365)365 -1=0.184 or 18.4% per year 64 Nominal and Effective Interest Rates • Two situations we’ll deal with in Chapter 3: • (1) Cash flows are annual. We’re given r per year and M. Procedure: find i/yr = (1+r/M)M-1 and discount/compound annual cash flows at i/yr. • (2) Cash flows occur M times per year. We’re given r per year and M. Find the interest rate that corresponds to M, which is r/M per time period (e.g., quarter, month). Then discount/compound the M cash flows per year at r/M for the time period given. 65 Example: 12% Nominal Annual semi-annual Quartertly Bi-monthly Monthly Weekly Daily Hourly Minutes seconds No. of Comp. Per. 1 2 4 6 12 52 365 8760 525600 31536000 EAIR (Decimal) 0.1200000 0.1236000 0.1255088 0.1261624 0.1268250 0.1273410 0.1274746 0.1274959 0.1274968 0.1274969 EAIR (per cent) 12.00000% 12.36000% 12.55088% 12.61624% 12.68250% 12.73410% 12.74746% 12.74959% 12.74968% 12.74969% 12% nominal for various compounding periods 66 Interest Problems with Compounding more often than once per Year – Example A • If you deposit $1,000 now, $3,000 four years from now followed by five quarterly deposits decreasing by $500 per quarter at an interest rate of 12% per year compounded quarterly, how much money will you have in you account 10 years from now? r/M = 3% per quarter and year 3.75 = 15th Quarter P @yr. 3.75 = P qtr. 15 = 3000(P/A, 3%, 6) - 500(P/G, 3%, 6) = $9713.60 F yr. 10 = F qtr. 40 = 9713.60(F/P, 3%, 25) + 1000(F/P, 3%, 40) = = $23,600.34 67 Interest Problems with Compounding more often than once per Year – Example B • If you deposit $1,000 now, $3,000 four years from now, and $1,500 six years from now at an interest rate of 12% per year compounded semiannually, how much money will you have in your account 10 years from now? • i per year = (1+0.12/2)12-1 = 0.1236 • F = $1,000(F/P, 12.36%, 10) + $3,000(F/P, 12.36%, 6) +$1,500(F/P, 12.36%, 4) or r/M = 6% per half-year • F = 1000(F/P, 6%, 20) + 3000(F/P, 6%, 12)+ 1500(F/P, 6%, 8) • = $11,634.50 68 Derivation of Continuous Compounding • We can state, in general terms for the EAIR: r m (1 ) −1 i =+ m Now, examine the impact of letting “m” approach infinity. 69 Derivation of Continuous Compounding • We re-define the general form as: r m (1 + ) − 1= m   1 + r    m    m r r   −1   •From the calculus of limits there is an important limit that is quite useful. h 1  lim  1 +  = 2.71828 e= h →∞ h  m r r  r – 1 = e lim  1 + , i = e eff.  m →∞ m  70 Derivation of Continuous Compounding • Example: • What is the true, effective annual interest rate if the nominal rate is given as: – r = 18%, compounded continuously Solve e0.18 – 1 = 1.1972 – 1 = 19.72%/year The 19.72% represents the MAXIMUM effective interest rate for 18% compounded anyway you choose! 71 Example • An investor requires an effective return of at least 15% per year. What is the minimum annual nominal rate that is acceptable if interest on his investment is compounded continuously? • Solution: er – 1 = 0.15 er = 1.15 ln(er) = ln(1.15) r = ln(1.15) = 0.1398 = 13.98% 72