Download Understanding Interest Rates: Nominal vs Effective Rates and Compounding - Prof. Jeremy Re and more Study notes Civil Engineering in PDF only on Docsity!
Nominal and Effective Interest Rates
Thus far, all interest rates have been compounded annually. However, many interest rates are compounded more frequently. Many projects raise capital through loans, mortgages, bonds and stocks, which have interest rates compounded more frequently than annually:
- semiannually
- quarterly
- monthly
- weekly
- daily
- continuous We introduce two new terms Nominal and Effective interest rates to account for those interest periods shorter than 1 year. Nominal interest rate, r , is an interest rate that does not include any consideration of compounding r = interest rate per period * number of periods if nominal r = 1.5% per month then for a year, r = 1.5% / month * 12 months / year = 18% per year for a quarter, r = 1.5%/month * 3 months/quarter = 4.5% per quarter for a week, r = 1.5%/month * 0.231 months/week = 0.346% per week These are all equivalent nominal interest rates. Also known as the APR: annual percentage rate = nominal interest rate Effective interest rate is the ACTUAL rate that applies for the stated time period. The compounding of interest during the time period of the corresponding nominal rate is accounted for by the effective interest rate. It is commonly expressed as an effective annual rate, ia. If compounding frequency not stated then assume it to be same as the time period of r. Also known as they APY: annual percentage yield = effective interest rate For example: 4% per year, compounded monthly 4% per year is the nominal interest rate, the fact that it is compounded monthly gives us the
necessary information to determine the effective annual rate. All effective interest rates all have the form of โr% per period t, compounded m-lyโ, where m is the time frequency. Most importantly: ALL formulas, factors, tabulated values, use the EFFECTIVE INTEREST RATE to properly account for the time value of money. We must ALWAYS determine the effective interest rate before any time value of money calculations. Particularly if the cash flows occur at intervals other than annual.
Time units
We need to keep in mind 3 time units associated with interest rate statements: time period (t)โ period over which the interest is expressed (e.g., 5% per year) compounding period (CP) โ shortest unit over which interest is charged or earned (5% per year compounded monthly) compounding frequency (m) โ number of times that m compounding occurs within time period t. 8% per year, compounded monthly time period, t = 1 year compounding period, CP of 1 month compounding frequency, m = 12 times per year Previously, interest rates were expressed โper yearโ and the compounding period was 1 year, thus the compounding frequency was 1. We can write an expression: Effective rate per CP = r% per time period t / m compounding periods per t = r / m Consider: 9% per year, compounded quarterly r% per time period t = 9% per year CP = quarter m = 4 effective rate per CP = 9% / 4 = 2.25% per quarter 9% per year, compounded monthly r% per time period t = 9% per year CP = month m = 12 effective rate per CP = 9% / 12 = 0.75% per month 4.5% per 6-months, compounded weekly r% per time period t = 4.5% per 6 months CP = weekly m = 26
F = P(1 + ia)n^ = 1000(1 + 0.19562)^1 = $1195. Let's compare compound periods: Assume 18% per year compounded:
Period Times
compounded per
year, m
Rate per
compound
period, i
Effective annual rate, ia
Yearly 1 18% (1.18)^1-1=18%
6 months 2 9% (1.09)^2-1=18.81%
quarterly 4 4.5% (1.045)^4-1=19.
Monthly 12 1.5% (1.015)^12-1=19.562%
Weekly 52 0.346% (1.00346)^52-1=19.684%
Daily 365 0.0493 (1.000493)^365-1=19.716%
Moral of story, always pay attention to the compounding period. The effective interest rate of 19.7% is a bit higher than the nominal rate of 18%. The downside to effective interest rates?
- They are not integers
- there are no factor tables for non-integer values
- can interpolate between values
- can use factor formula with ia instead of i Interpolation: What is the present worth factor, (P/F,14.81%,5)? Look up P/F,14 and P/F, 15 0.5194 0. 0.4972โ0. 15 โ 14
y โ0. x โ 14 0.4972โ0. 15 โ 14 ๎14.81โ 14 ๎๎0.5194= y y =0. formula:
๎ 1 ๎ ia ๎ n 1 ๎ 1 ๎0.1481๎ 5
Thus, effective annual interest rates make the use of the tables a bit trickier. So, let's just redefine our cash flow diagrams and use convenient periods.
Effective Interest Rates for Any period
CP is the period over which interest compounds. New term: PP โ payment period, frequency of the payments or receipts. Interest may compound monthly, but payments made yearly, thus CP and PP are not equal. For example, a company deposits money on a monthly basis into an account with a nominal interest rate of 14% per year, compounded semiannually. To evaluate cash flows that occur more frequently than annually (PP < 1 year), the effective rate over the PP must be used. We can generalize the previous formula: recall that r = i * m ia =๎ 1 ๎ i ๎ m โ 1 effective i =๎ 1 ๎ r / m ๎ m โ 1 r = nominal interest rate per payment period (PP) m = number of compounding periods per payment period (CP per PP) Prodecure: โ Convert given interest rate into the nominal rate for the payment period โ then determine the effective interest rate.
Determine effective interest rate on semiannual and annual basis: effective i =๎ 1 ๎ r / m ๎ m โ 1 where r = nominal interest rate per payment period m = # of compounding periods per payment period Semiannually PP= 6 months r = 3% /quarter * (2 quarters / 1 6-month period) = 6% per 6-months m = 2 quarters per 6 months Effective i% per 6 months = (1+0.06/2)^2 -1 = 6.09% per 6-months Annually PP = 6 months r = 3% / quarter * 4 quarters / year = 12% per year m = 4 quarters per year eff i per year = (1 + 0.12/4)^4 -1 = 12.55% per year
Equivalence Relations
Quite often PP<>CP. That is, frequency of cash flows is not equal to the period over which interest is compounded. Cash flows may be monthly but interest compounds quarterly. PP โ payment period CP โ compounding period 8% per 6-months, compounded quarterly PP in this case is 6 months. You get your interest every 6 months. However, the CP is 3 months, as they compound interest quarterly. As we saw in the examples last week, we can calculate effective interest rates for any period. This is necessary because payment periods and compounding period frequencies differ. So, let's apply what we've learned. Consider 2 general cases: PP > CP and PP < CP
PP > CP, Single amounts.
how to determine correct i, n values? 2 options
- CP basis Determine effective interest rate over the compounding period and set n equal to the number of compounding periods between P and F. (effective rate per cp = r/m where both r and m have the same period t)
P = F (P/F, effective i% per CP, # of periods n) e.g., 15% per year, compounded monthly CP = 1 month effective i% per month = r/m = 15% per year / 12 months per year = 1.25% per month thus, for 2 years, n = 2*12 = 24
- Yearly basis Determine effective interest rate for the time period t of the nominal rate, and set n equal to the total number periods using this same time period. P = F (P/F, effective i% per t, # of periods n) e.g., 15% per year, compounded monthly t = 1 year effective i% per year = (1 + 0.15 per year /12 months per year)^12-1 = 16.076% thus for 2 years, n = 2 In the first case, we are calculating the effective rate per month and looking at 24 months. In the 2nd, we are calculating the same effective rate per month, but then looking at how that compounds over a year So for the 2 examples, the present worth factor: P=F(P/F,1.25%, 24) = 0. P = F(P/F,16.076%,2) = 0. while 1.25% IS tabulated, 16.076% is not. Thus, for the lazy there still may be shortcuts. Example: Which bank gives better rate of return on a 3 year investment? 3% per quarter, compounded semiannually or 6% semiannually, compounded yearly? F = P(F/P,i,n) situation 1 CP = 6 months effective i per 6 months = 3% /quarter * 2 quarters / 6-mo = 6% per 6-mo for 3 years, n = 2 6-month/yr * 3 yr = 6 F = P(F/P,6%,6) = 1. versus situation 2 CP = 1 year effective i per year = 6% per 6-mo * 2 6-mo/year = 12% per year for 3 years, n = 3 F = P(F/P,12%, 3) = 1. Thus, the first option is a better investment. Note, the Nominal interest rate is the same in both problems.
the beginning of the period, so it earns no interest for the partial period it was there.. In this case, PP = CP Converse case, when PP < CP and compound interest is allowed. Then equivalent P, F, or A values are determined using effective interest rate per compounding period. For example: Weekly cash flows and quarterly compounding at 12% per year In this case PP = 1 week CP = 1 quarter Thus 12 % per year, compounded quarterly. Interest rate per CP = 12% per year / 4 quarters/year = 3% per quarter. m = 1 week / (1 quarter * 13 weeks/quarter) = 1/13. Effective weekly interest rate = (1 + 0.03)^(1/13) โ 1 = 0.228% So a P/A problem for a year: P = A (P/A,0.288%, 52)
Continuous Compounding
What happens if we allow interest to be compounded on shorter and shorter compounding frequencies. This occurs on businesses that have many cash flows per day and interest is compounded continuously. Thus if CP -> 0, m -> infinity lim m->8 i = lim (1 + r/m)^m - m = hr, then = lim h->8 (1+1/h)^hr = lim h->8 [(1+1/h)h]r - recall from calculus that lim h->8 (1+1/h)^h = e thus i = e^r- this equation can be used to compute the effective continuous interest rate when the time periods on i and r are the same. So, our 18% per year, compounded continuously, what is effective annual interest rate: i = e^0.18 โ 1 = 19.72% effective monthly rate: nominal monthly rate r = 18% / 12 = 1.5% per month i = e^0.015 โ 1 = 1.511%
