![www.futurumcorfinan.com
Page 4
For the credit card example, 1.5 percent per month interest has an effective annual rate of
[(1.015)^12 – 1] = 0.1956, or 19.56 percent. If the stated rate is 1.75 percent per month, as is
the case with many U.S. credit card accounts, the APY is a whopping 23.14 percent. In other
words, if you are carrying a positive credit card balance with an interest rate like this, pay it off
as soon as possible!
Discussions
Unfortunately, most of these corporate finance textbooks just stop there without exploring
further and don’t even give words of caution to all those undergraduate students, which might
be the first time being exposed to the calculation of SAIR and EAIR.
There are two things I would like to “add” to the explanation of SAIR and EAIR.
First, the book doesn’t tell you that the “interest” is paid at the end of the period (known
as “in arrears”), and not at the beginning of the period (known as “in advance”).
In certain situation, the “interest” is collected in advance.
If this is the case, then how to calculate this periodic rate?
For instance,
Debt interest rate = 8% per annum
Quarters in one year = 4
Debt periodic interest rate?
Per textbook, it should be = (1+8%)^(1/4) -1 = 1.94%, since if we (1+1.94%)^4 - 1 = 8%.
To give you a bit expanded idea about this periodical interest rate, we have:
a) 8% per annum (this is effective rate compounded 1x)...and the question is how much
the effective rate for one year if it is compounded 4 times...then effective one year =
(1+8%/4)^4 - 1...Then we have 8.24% effective per year, or 2% per quarter.
b) 8% per annum is the effective rate for 4 times compounded, then per quarter,
(1+8%)^(1/4) -1 = 1.94% per quarter.
But, all above calculation as per textbook is standing on the assumption that the interest is
collected or paid at the end of the period/quarter, and not in advance or at the beginning of
the quarter.](https://image.slidesharecdn.com/futurum-statedandeffectiveinterestrate-151108165407-lva1-app6892/85/Futurum-stated-and-effective-interest-rate-4-320.jpg)
![www.futurumcorfinan.com
Page 5
Periodical interest rate in advance (= iPad) is determined as follows:
iPad = i/(1+i) and
i= iPad/(1-i_ad)
t=0 t=1
P P(1+i) at the end.
P(1-i_ad) P in advance
First case i = P(1+i)/P -1 = i (paid at the end of period).
Second case
i = P/P(1-i_ad) -1 = (P - P(1-i_ad))/[P(1-i_ad)] - 1 = i = iPad/(1-i_ad).
Or,
iPad = i/(1+i), then
i = iPad (1+i), then
i = iPad + iPad * i, then
i – (iPad * i) = iPad, then
i * (1 – iPad) = iPad, then
i = iPad/(1-i_ad)
The other way around,
iPad = i/(1+i)
So, if we put into the above example, the periodical/quarterly interest rate paid/collected in
advance is:
1 - (1/(1+8%)^(1/4)) = 1.91%, which if we compounded it four times (1+1.91%)^4- 1, we won't
get 8% per annum, as this interest is paid/collected in advance instead of in arrears.](https://image.slidesharecdn.com/futurum-statedandeffectiveinterestrate-151108165407-lva1-app6892/85/Futurum-stated-and-effective-interest-rate-5-320.jpg)
Futurum stated and effective interest rate
- 1. www.futurumcorfinan.com Page 1 What Corporate Finance Textbooks Don’t Tell You about Stated/Nominal vs Effective Annual Interest Rates Time is money (Benjamin Franklin) “If time is money, shouldn’t we count those benjamins?” Note: IVP = Ignacio Velez-Pareja (Associate Professor of Finance at Universidad Tecnológica de Bolívar in Cartagena, Colombia) Karnen : Sukarnen (a student in corporate finance) Introduction We could find the explanations about the calculation of Stated and Effective Annual Interest Rates (SAIR vs EAIR) in most of standard corporate finance textbooks, mainly put under the Sukarnen DILARANG MENG-COPY, MENYALIN, ATAU MENDISTRIBUSIKAN SEBAGIAN ATAU SELURUH TULISAN INI TANPA PERSETUJUAN TERTULIS DARI PENULIS Untuk pertanyaan atau komentar bisa diposting melalui website www.futurumcorfinan.com
- 2. www.futurumcorfinan.com Page 2 chapter “The Time Value of Money”. The basic idea behind SAIR and EAIR is that, it is not always be possible to assume that compounding or discounting is an annual process, that is, cash flows (inflows or outflows) arise either at the start or the end of the year. We could see this in real practice, where: the contractual payment for interest charge on loan is incurred on a semi-annual basis or a quarterly basis; interest charge on credit cards is applied on a monthly basis; a fixed deposit scheme may offer daily compounding; a car dealer may quote an interest rate on a monthly basis. How should we compare interest rates that are quoted for different periods? Thus, to have the “apples with apples” comparison, it is necessary to determine the effective annual percentage rate, or effective annual interest rate. The classic example of the section in the standard corporate finance textbooks regarding the conversion of stated into the effective annual interest rate is as follows1 : 3-5b Stated Versus Effective Annual Interest Rates Both consumers and businesses need to make objective comparisons of loan costs or investment returns over different compounding periods. To put interest rates on a common basis for comparison, we must distinguish between stated and effective annual interest rates. The stated annual rate is the contractual annual rate charged by a lender or promised by a borrower. The effective annual rate (EAR), also known as the true annual return, is the annual rate of interest actually paid or earned. The effective annual rate reflects the effect of compounding frequency, whereas the stated annual rate does not. We can best illustrate the differences between stated and effective rates with numerical examples. Using the notation introduced earlier, we can calculate the effective annual rate by substituting values for the stated annual rate (r) and the compounding frequency (m) into Equation 3.14: 1 Megginson, William L., and Scott B. Smart. Introduction to Corporate Finance. Mason (USA): South-Western, a part of Cengage Learning. 2009. Chapter 3 : The Time Value of Money. Page 109-110.
- 3. www.futurumcorfinan.com Page 3 We can apply this equation using data from preceding examples. Not surprisingly, the maximum effective annual rate for a given stated annual rate occurs when interest compounds continuously. The effective annual rate for this extreme case can be found by using the following equation: For the 8 percent stated annual rate (r = 0.08), substitution into Equation 3.14a results in an effective annual rate of 8.33 percent, as follows: At the consumer level in the United States, “truth-in-lending laws” require disclosure on credit cards and loans of the annual percentage rate (APR).The APR is the stated annual rate found by multiplying the periodic rate by the number of periods in one year. For example, a bank credit card that charges 1.5 per-cent per month has an APR of 18 percent (1.5% per month x 12 months per year). However, the actual cost of this credit card account is determined by calculating the annual percentage yield (APY ), which is the same as the effective annual rate.
- 4. www.futurumcorfinan.com Page 4 For the credit card example, 1.5 percent per month interest has an effective annual rate of [(1.015)^12 – 1] = 0.1956, or 19.56 percent. If the stated rate is 1.75 percent per month, as is the case with many U.S. credit card accounts, the APY is a whopping 23.14 percent. In other words, if you are carrying a positive credit card balance with an interest rate like this, pay it off as soon as possible! Discussions Unfortunately, most of these corporate finance textbooks just stop there without exploring further and don’t even give words of caution to all those undergraduate students, which might be the first time being exposed to the calculation of SAIR and EAIR. There are two things I would like to “add” to the explanation of SAIR and EAIR. First, the book doesn’t tell you that the “interest” is paid at the end of the period (known as “in arrears”), and not at the beginning of the period (known as “in advance”). In certain situation, the “interest” is collected in advance. If this is the case, then how to calculate this periodic rate? For instance, Debt interest rate = 8% per annum Quarters in one year = 4 Debt periodic interest rate? Per textbook, it should be = (1+8%)^(1/4) -1 = 1.94%, since if we (1+1.94%)^4 - 1 = 8%. To give you a bit expanded idea about this periodical interest rate, we have: a) 8% per annum (this is effective rate compounded 1x)...and the question is how much the effective rate for one year if it is compounded 4 times...then effective one year = (1+8%/4)^4 - 1...Then we have 8.24% effective per year, or 2% per quarter. b) 8% per annum is the effective rate for 4 times compounded, then per quarter, (1+8%)^(1/4) -1 = 1.94% per quarter. But, all above calculation as per textbook is standing on the assumption that the interest is collected or paid at the end of the period/quarter, and not in advance or at the beginning of the quarter.
- 5. www.futurumcorfinan.com Page 5 Periodical interest rate in advance (= iPad) is determined as follows: iPad = i/(1+i) and i= iPad/(1-i_ad) t=0 t=1 P P(1+i) at the end. P(1-i_ad) P in advance First case i = P(1+i)/P -1 = i (paid at the end of period). Second case i = P/P(1-i_ad) -1 = (P - P(1-i_ad))/[P(1-i_ad)] - 1 = i = iPad/(1-i_ad). Or, iPad = i/(1+i), then i = iPad (1+i), then i = iPad + iPad * i, then i – (iPad * i) = iPad, then i * (1 – iPad) = iPad, then i = iPad/(1-i_ad) The other way around, iPad = i/(1+i) So, if we put into the above example, the periodical/quarterly interest rate paid/collected in advance is: 1 - (1/(1+8%)^(1/4)) = 1.91%, which if we compounded it four times (1+1.91%)^4- 1, we won't get 8% per annum, as this interest is paid/collected in advance instead of in arrears.
- 6. www.futurumcorfinan.com Page 6 As a recap, where: i is periodical interest at the end of period, and iPad is in advance. If iPad=1.91% then i=1.947% (1+i)^(1/4) is (1+i_periodical). Second, by doing the conversion between compounding methods as explained in the corporate finance textbooks, are they really “equivalent”? I give one extreme example. Suppose: r1 is the annual rate with continuous compounding. r2is the equivalent compounded m times per annum times per annum. Then we have: = (1 + r2/m)^m = e^r1 = r1 = m * ln (1 + r2/m), then = r2 = m (e^(r1/m) – 1) If we carefully look at the cash flows for this interest, then r1 and r2 above are based on different cash flows, and in what financial sense, we could say that they are equivalent? Think about it next time before just jumping to use all formulas given in the standard corporate finance textbooks. Ignacio Velez-Pareja (IVP) comments: When you contract a loan, usually they specify the non-compounded rate (in Spanish, we say nominal rate). However, if you have contracted the loan on a monthly or quarterly basis, then you find the periodic rate (8%/12, 8%/4, etc.) It is this way of paying the interest that makes the artificial compounded (we call it in Spanish, effective rate).
- 7. www.futurumcorfinan.com Page 7 Hence, you have on a quarterly basis, 8% as non-compounded rate, 2% as a periodic (quarterly) rate and the compounded rate. There is a very simple relationship between non- compounded and periodical rates, as follows 1.Compounded: periodical rate times number of periods 2.Periodical; compounded rate /number of periods. I don't give a penny for the compounded rate. That is a mathematical fiction. The most relevant rate is the periodical rate (many people think it is the compounded rate and you could tell me how many firms you know that pay interest on the basis of a compounded rate?) The rate that should be used in WACC (for instance) should be the periodical. That one is the most important rate, because that rate is the one you need to calculate the actual interest payment and the sum of all those interest payments are what you deduct from the Income tax report. Follow? Hence, If you have 8% per annum, compounded quarterly, you already know that the periodical is 2%. That rate is the one the bank uses to calculate the interest you have to pay. The compounded is (1+8%/4)^4-1= 8.2432% BUT that 8.2432% has no real meaning. In fact, do you know that the assumption behind that calculation is that you can save (or invest) exactly at the same rate you borrow money? It is as if the bank has one window where it gives you the loan at, say 2% per quarter, and another one where they pay you 2% per quarter. HOWEVER, that is true for the bank, because in equilibrium, the money it receives from you is invested (most times) at the same rate you pay. This is the considerations I give to my students: Assume you have several people with different ways to "keep" the money and you will tell me which is their opportunity cost. They have two options: a) To pay a loan of 1,000 at the end of year with interest of 8% (you will pay 1000+80 interest). b) to pay 20 per quarter and 1000 at the end of year. (20, 20, 20, 1020). For instance: 1.Keeps the money in a safe box. Opportunity cost = 0% 2.Keeps the money in a savings account Opportunity cost 0.5% per month
- 8. www.futurumcorfinan.com Page 8 3.Keeps the money in a CD Opportunity cost 1.2% per month 4.Keeps the money in a savings account Opportunity cost 2% per quarter 5.Keeps the money in a CD maturity 1 year Opportunity cost 8% per annum 6.Keeps the money in a savings account Opportunity cost 2.5% per quarter If each of them contracts a loan to be paid quarterly at (% per annum with quarterly payments of 2% interest). What each individual will prefer, a) or b)?. The capitalized cost is as said, 8.2432% per annum. Will that loan cost the same to all of them? Figure out the case of the person with his money in the safe box: will it cost more if she pays the loan in a lump sum at the end of the year (1,080) or if she pays 20, 20, 20, 1020?. The assumption in the compounded rate is that it is the same for ALL of them: 8.2432% per annum. Is that true? Will it cost more or less for case 6? For case 1? For case 5? Would you say that the extra cost of paying a) or b) is the same for all of them? I think it is not the same and yet, the compounded rate is the same for all!!! Karnen: Ignacio, interesting, as you showed above that they have two options: a) To pay a loan of 1,000 at the end of year with interest of 8% (you will pay 1000+80 interest). b) To pay 20 per quarter and 1000 at the end of year. (20, 20, 20, 1020). I don't think that two options could have the same interest rate, the risk of cash flows could be different as far as I could see, with option b) looks safer, yet the compounded rate as the textbooks taught us, that the rates for both options are the same. IVP: Dear Karnen I see you are picking the most fictional case! Perpetuities! However, that continuous interest applies to formulas such as the Black-Scholes model for financial options.
- 9. www.futurumcorfinan.com Page 9 This is a cash flow that you receive instantaneously, every Nano second. Can you even imagine that? Yet, I have seen, occasionally, a bank offering that interest rate. The difference with a practical daily rate is nil. It is a mathematical conception that exists only in the imagination. It says the effective rate of a non-compounded rate of, say, 12% per annum, compounded instantaneously. It is as if money were a liquid that flows through a Cane into your bank account. Just science fiction. Note: The first perpetuities were issued in the 12th century in Italy, France and Spain. They were initially and intentionally to circumvent the usury laws of the Catholic Church. that is because no loan principal repayment, they were not considered as loans. ~~~~~~ ####### ~~~~~~
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