Discussions paper series interest calculation

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Discussion Paper Series: Interest Calculation

Karnen:

By the way, I am looking into one formula that I am a bit confused.

Debt interest rate = 8% per annum

Quarters in one year = 4

Debt periodic interest rate?

I thought it should be = (1+8%)^(1/4) -1 = 1.94%, since if we (1+1.94%)^4 - 1 = 8%, yet

the answer it should be:

1 - (1/(1+8%)^(1/4)) = 1.91%, but this is if we compounded it four times (1+1.91%)^4- 1, we

won't get 8% per annum. It is said "Don't assume that an annual rate is always de-compounded

to give a periodic rate"..

What do you think?

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



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IVP:

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).

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:


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Assume you have several people with different ways to "keep" the money and you will tell me

which their opportunity cost..Are They having 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

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:

Thanks as usual for your detailed reply. However, seems to me you don't kill the beast at its

head, meaning you don't really say whether:

(1+8%)^(1/4) -1 = 1.94% or 1 - (1/(1+8%)^(1/4)) = 1.91% is the correct one for three-month

interest?


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I am reading carefully your reply, I guess, the way you are challenging that compounded rate is

a bit a surprise to me....

Why I said this?

As we know, money has no royalty, and in most cases, we could say, money is operating in a

[very] competitive market [and normal market as well in which arbitrage opportunities could be

said could only exist for a very short period. Note: there of course is a limit to arbitrage, read

paper by Shleifer and Vishny, but it is more related to behavioral finance which is another long

topic to talk about). If you believe such market does exist, then we have no choice than left to

accept the way interest is compared to each other is thru effective interest rate, in this case, on

an annual basis, to make it a bit convenient to get apple-to-apple comparison. In such market,

personal preference about risk, and or whether you want to spend your money now or later,

doesn't matter at all.

About assuming lending and borrowing rate is the same, I guess it has been discussed in many

investment textbooks, and though in practice, each rate will be different, but the conclusion will

stay the same.

What do you think?

IVP:

I thought I had answered that!

If the bank defines its lending rate as compounded, yes, you are right. Back here, they offer a

non-compounded rate stipulating the period of payments, say. Month, quarter whatever in this

way, the borrower knows how much she has to pay. Hence, the periodical rate which is the one

the bank uses to define the interest, is crystal clear: rate_not compunded/#periods. Otherwise

the layman will not easily know how much he will have to pay.

Given the non-compounded and the period you can calculate the compounded. The Kd in wacc

should be the periodical which is the one used to calculate interests to be deduced from income

and pay taxes. There is no firm at all that pays interest based on compunded rate, unless you

make all those calculations very strange to the layman.


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Karnen:

Thanks for the reply.

forget about the layman or not...bank will not tell you exactly their effective or not effective, that's

really dependent whether we are talking about deposit or lending...

if lending...bank will use lower rate (in this case, quoted rate)

if deposit ---- bank will use higher rate (in this case, effective rate)

so that's the practice that we should not argue about...they are selling money...

Back to my question:

Compared

(i) (1+8%)^(1/4) -1 = 1.94%

(ii) 1 - (1/(1+8%)^(1/4)) = 1.91%

I understand (i) but I am not too sure I am following (ii) calculation, this is the first time I saw the

way the quarterly interest rate is calculated from yearly interest rate?

IVP:

I see a very strange formula in your message.

What I know is:

Given a non-compounded, you get the compounded using This formula:

(1+i_nocomp/n)^n -1

And, yes. When the bank announces a lending rate they show the non-compounded rate. When

they promote a savings accounts.

Karnen:

Though it is not relevant to what I am asking about, but since you are giving an example of

They have two options:


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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.

Anyway, my question is more about one single and not two options...

IVP:

The two options have the same non-compounded 2k1

Karnen:

Formula (ii) below is made by a well-known consultant, yet seems strange to me, since as far as

I know, it is formula (i) below that it is correct. The other consultant just said "Don't assume that

an annual rate is always de-compounded to give a periodic rate".

(i) (1+8%)^(1/4) -1 = 1.94%

(ii) 1 - (1/(1+8%)^(1/4)) = 1.91%

What do you think, which one is correct?

IVP:

You have to define what the 8% is. If you define 8% is the effective rate, then formula i) is

correct to define the periodical rate.

What is usual, at least in Colombia and other countries, is to define 8% as a non-compounded

rate and specify the period of liquidation of interest. If that is the situation, then periodical rate is

calculated as Non-compounded rate/#periods and effective rate is

(1+ Non-compounded/#periods)^#periods-1 = (1+ periodical)^#periods-1

This is what I can tell you.

I don't know what (ii) is.


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Karnen:

Thanks for the reply.

Seems formula (ii) is pretty new...I will do research to see this formula is taken from where?

To recap what you said and from what I understood so far:

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...Once 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.

IVP:

I think I got it!

In some places, Colombia, for instance, interest is paid in advance in some deals. The formula

(ii) is just this:

Periodical interest rate in advance,

i_ad = i/(1+i) and

i= i_ad/(1-i_ad)

Where i is periodical interest at the end of period and i_ad is in advance.

If i_ad=1.91% then i=1.947%

(1+i)^(1/4) is (1+i_periodical)

That's all!

In fact I teach that and am considered in my Spanish book.

Do you use that in Indonesia?


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Karnen:

You meant the periodical interest is paid at beginning of the period, for example on early

January 2015 (instead of end of March 2015), early April 2015 (instead of end of June

2015)...and so on?

You said:

i_ad = i/(1+i) and

i= i_ad/(1-i_ad)

Is it not that i = i_ad * (1+i)?

Why we have i = i_ad/(1-i_ad)?

IVP:

Yes. Paid in advance.

My formulas (that I use since many years ago) can be derived from the strange formula you

found!

I don't explain that using that formula, but I derive those using simple numbers:

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 = i_ad/(1-i_ad)

The other way around

i_ad = i/(1+i)

That's all!!!


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Karnen:

Second case,

Why, the bold one (in the numerator) "suddenly" changes from P to (P - P(1-i_ad))?

When you said "in advance" it is like interest paid in full in the beginning of the term of the bond,

such as zero coupon bond?

Or is it normal bond, but the interest paid in the beginning of the quarter instead of end of the

quarter? of course the main difference is the payment of the first interest, which is not paid at

end of the first of quarter (three months after bond issuance), but that first quarter interest is

paid on the same date of bond issuance.

Which one that you meant?

IVP:

Simple; you pay ANY interest in advance. The same numbers of payments but start at 0 and

end at N-1. Total, N payments

Put the denominator multiplied by -1 and keep the denominator. Jus arithmetic!

Karnen:

Gosh...yes so easy, I miss that -1 in that formula.

so the "mystery" is solved.

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Economy & Finance

Discussions paper series interest calculation