effective periodic rates?

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BMiller12's picture

Are effective rates like the effective periodic rate (EPR) or effective annual rate (EAR) actually used for, say,  discounting cash flows or are these merely rates that have been transformed so that one can compare rates with different compounding periods (monthly, quarterly, etc) and to know actually what you are paying/receiving as a result of compounding ??

I realize the effective periodic rate of 15% annually with semiannual compounding is approx 7.238%, not 7.5%.  But if you were running a DCF with semiannual cash flows, which rate would you use to discount the semiannual cash flows ??  

I’ve always thought when using a non-annual compounding rate the present value factor would be, in this case:

1 / (1+r/2)^(t*2)    =   1 / (1.075)^(t*2)   =    .930232    if at the end of year 1

but if you use the 7.238% you get a different PV factor (.9325)…so which is it ?

I’ve always thought to capture different compounding, you just divide the annual rate by the compounding period and multiply the time (t) by the compounding periods.  Especially for bonds, don’t we just divide the rate by the compounding periods??  

Thanks in advance!

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aaronhotchner's picture

Use the relevant effective rate for the time period between payments. A 15% coupon is a nominal annual rate corresponding to an effective 7.5% six-month rate. The effective annual rate is 1.075^2-1, which would obviously be slightly above 15%. If you were discounting on an annual basis, you’d use the effective annual rate. If you were discounting payments when they come (every six months), you’d use 7.5% since it is the effective rate on a six-month basis.

Your 7.238% figure is the effective six-month rate of an effective 15% annual rate, but your original 15% is a nominal rate. Be careful to avoid that kind of mistake.

BMiller12's picture

I’m sorry, i don’t follow. so in my example, would i discount by the semiannual amount by 7.5% ??

How/when do you know when to use an effective annual or periodic rate when discounting or compounding?

BMiller12's picture

I know this is not right, but why would you not use the 7.238% to discount the semiannual coupon for a bond with a 15% annual coupon? Why is it (1+ 15/2) ?

aaronhotchner's picture

BMiller12 wrote:
I’m sorry, i don’t follow. so in my example, would i discount by the semiannual amount by 7.5% ??

How/when do you know when to use an effective annual or periodic rate when discounting or compounding?

If you’re talking about discounting a bond that has a coupon rate of 15% and pays coupons semiannually, then yes, you would discount by 7.5%. But you also need to be aware that 7.5% refers to one period, which is six months. Annual rates don’t enter into these situations at all.

The rate you should use will depend on how frequently cash flows come in. If cash flows are paid annually but you are given a semiannual rate, then you would compound it to get the annual rate. If cash flows are paid every quarter but you are given an effective annual rate, you would need to find (1 + r)^(1/4) - 1 for the effective quarterly rate. If instead you are given a “quoted,” or nominal, annual rate (like a bond’s coupon rate) you would simply divide by the number of periods (here, 4) to get the effective quarterly rate.

aaronhotchner's picture

BMiller12 wrote:
I know this is not right, but why would you not use the 7.238% to discount the semiannual coupon for a bond with a 15% annual coupon? Why is it (1+ 15/2) ?

Bond coupon rates are *always* stated in nominal terms.

You refer to the bond as having a “15% annual coupon.” This literally means that the bond pays out at a rate of 15% once every year, and you would therefore use an interest rate of 15% for discounting.

If, instead, you want a 15% semiannual coupon, you would just divide by 2 to get 7.5% effective rate every 6 months. So the first payment would be valued at 1/1.075 and the second payment, occurring at t = 1 year, would be valued at 1/(1.075^2).

You will pretty much never see a normal situation where a semiannual bond is quoted with an effective annual coupon rate, because that’s simply not how bonds are quoted. It’s an industry standard.

BMiller12's picture

But if 7.238% compounded gets you back to 15% annually, why use 7.5% which actually gives you more than 15% annually, shouldn’t the two match and get you back to one another ?

So it depends if the rate is a stated / nominal rate or if it’s an effective rate? If given an effective rate then you need to match the periodic cash flows with the effective periodic rate….and if given a nominal rate then you match the CF with the nominal periodic rate (simply divide)? Just seems odd bc if you divide to get the nominal periodic rate, compounding it forward gives you a rate bigger than the nominal rate you started with, so how do they line up? Im clearly either over thinking this or outright confused.

Another example: If you have a DCF for a company and are using quarterly cash flows, and you want to discount those cash flows to find the PV, what rate would you use if your discount rate (WACC) is say 15% ?
Would it be 15%/4 = 3.75 with a first PV factor of: 1 / (1.0375)^1
or, (1.15)^(1/4)-1 = 3.55% with a first PV factor of 1/ (1.0355)^1

Thanks, Aaron!

aaronhotchner's picture

BMiller12 wrote:
But if 7.238% compounded gets you back to 15% annually, why use 7.5% which actually gives you more than 15% annually, shouldn’t the two match and get you back to one another ?

The bond’s actual return is 1.075^2 - 1 = 15.5625% annually. This assumes you have a bond that pays 7.5% six months in, then you reinvest your coupon at 7.5%, so you are compounding your returns. If a bond pays out more frequently than once per year, then its effective annual return is always higher than the quoted interest rate due to compounding. It may help you to look at some material on future and forward interest rates. A 15% annual coupon bond must return less than a 15% semiannual coupon bond because you get to take your coupon (thus you have more money than you had before) and make a new investment with it. You have a 6-month interest advantage with semiannual bonds, so your annual return is effectively higher.

BMiller12 wrote:
So it depends if the rate is a stated / nominal rate or if it’s an effective rate? If given an effective rate then you need to match the periodic cash flows with the effective periodic rate….and if given a nominal rate then you match the CF with the nominal periodic rate (simply divide)? Just seems odd bc if you divide to get the nominal periodic rate, compounding it forward gives you a rate bigger than the nominal rate you started with, so how do they line up? Im clearly either over thinking this or outright confused.

As above, nominal rates and effective rates shouldn’t be equal unless your payment period is the same as your discounting period. If you are given a nominal annual rate on an investment that pays out once per year, then the effective rate is the same as the nominal rate. If you have a nominal annual rate on a payment that pays out more than once per year, then the effective rate will be higher than the nominal rate. And if the nominal annual rate is for an investment that pays less frequently than once per year, the effective rate will be lower than the nominal rate.

You’re right that you simply divide to get the effective period rate, and then you compound to get the effective annual rate. Think of mortgages. A typical mortgage right now might be 4% APR, but you pay your mortgage monthly. 4% is a nominal rate, so the monthly interest rate is 4% / 12 = 0.333%. Unlike a bond, you’re also paying off principal, so your effective annual rate is actually likely to be lower than (1.00333)^12 - 1. Nonetheless, the mortgage is quoted at 4% annually, just because. You don’t care about that 4% beyond converting it to an effective rate. In short: when doing calculations, never use nominal rates; when reporting interest rates (especially for bonds), you will generally use a nominal rate unless asked for an effective rate. When you start calculating a bond’s yield to maturity, for a semiannual bond you will report 2 * the effective yield. It seems counterintuitive, but everyone follows the same rules so you just have to get used to it.

BMiller12 wrote:
Another example: If you have a DCF for a company and are using quarterly cash flows, and you want to discount those cash flows to find the PV, what rate would you use if your discount rate (WACC) is say 15% ?
Would it be 15%/4 = 3.75 with a first PV factor of: 1 / (1.0375)^1
or, (1.15)^(1/4)-1 = 3.55% with a first PV factor of 1/ (1.0355)^1

Thanks, Aaron!

The WACC is likely going to be quoted as a nominal annual rate, so you would discount by 15% / 4 = 3.75%, and compound in quarterly increments. That is, the PV factor of the payment in the 3rd quarter of the first year is 1.0375^(-3) = 0.8954.

BMiller12's picture

Had a quick second to see if any responses, but haven’t fully read your post. But thanks so much for your help, Aaron. It’s much appreciated!

BMiller12's picture

First sentence, 2nd paragraph:
“You’re right that you simply divide to get the effective period rate, and then you compound to get the effective annual rate.”

I guess i tend to think that dividing the nominal annual rate will give you the nominal periodic rate, not effective periodic rate; perhaps this is just semantics. The sentence above says effective. When i think of effective periodic rate i think of: (1+ r)^(1/m) -1, or the 7.238% from earlier for example, and not 7.5%.

” In short: when doing calculations, never use nominal rates” - Maybe the answer to the above will address this, but i thought nominal rates is what we used and nominal rates is what we plug into the formula ( 1+ (r/m))^(n*m)) as ‘r’ or the PV factor equation. I guess i never even tend to think of effective rates when making calculations (ie PV or FV factors). I tend to think of them as helpful in the sense that you can readily compare two rates with different compounding frequency to determine the “effective” return. And nominal rates are usually how things are stated and the rates that you use mostly. Is this wrong ??

Just curious, why did you think that the WACC would be a nominal rate?

And, for the quarterly dcf example. Just want to ask again, why wouldn’t you use the effective periodic rate of 1.03728 to discount the 3rd quarter?

You’ve been much help, i think i’m almost there. B

aaronhotchner's picture

Quote:
I guess i tend to think that dividing the nominal annual rate will give you the nominal periodic rate, not effective periodic rate; perhaps this is just semantics.

It’s not a matter of semantics. Effective and nominal rates are very different, and must be treated as such. A nominal quarterly rate would be 1/4 of the annual effective rate. Nominal rates are just multiples of another time horizon’s effective rate. In the case of the semiannual bond, the interest rate you see is nominal, and it was found by taking the effective semiannual rate (which is predetermined and more important) and doubling it. The effective periodic rate of 7.238% applies only when the effective annual rate is 15%. The formula you’re using is for taking effective rates and changing the time period. When you start with a nominal rate, you just divide or multiply by the number of periods.

Quote:
I guess i never even tend to think of effective rates when making calculations (ie PV or FV factors). I tend to think of them as helpful in the sense that you can readily compare two rates with different compounding frequency to determine the “effective” return. And nominal rates are usually how things are stated and the rates that you use mostly. Is this wrong ??

Yes, that’s wrong. Nominal rates must be converted to effective rates to be useful. ALL present value calculations should be done using effective rates. If you use a financial calculator, when you price a bond the interest rate you input is the effective rate. A 5-year car loan at 6%, for example, would be priced with 60 periods (5 years * 12 months/year) and an interest rate of 0.5% (6%/year nominally / 12 months/year). The relevant interest rate for TVM is always, always, always the effective rate – this does not mean converting the 0.5% per month to an effective annual rate, if that’s where you’re getting confused. 0.5% per month is itself an effective rate in my example.

Quote:
Just curious, why did you think that the WACC would be a nominal rate?

Like any other financial rate, we must assume that it is reported as a nominal annual rate unless told otherwise. A credit card with a 24% APR really means that you are charged 2% interest on your balance every month. The 24% is nominal; the 2% is effective. Likewise, the effective annual rate of the credit card is 1.02^12 - 1 = 26.82%, as opposed to the 24% APR.

Quote:
And, for the quarterly dcf example. Just want to ask again, why wouldn’t you use the effective periodic rate of 1.03728 to discount the 3rd quarter?

Again, you’re assuming the 15% is effective and then converting it to a quarterly rate. But if the 15% is nominal, like I claim (and which would generally be the case if the interest rate is reported annually but the cash flows come in quarterly), you would divide by 4 and then compound on a quarterly basis. You can’t convert a nominal rate to an effective rate with compounding, so it comes down to being able to recognize what’s nominal and what’s effective.

BMiller12's picture

Then why do you call the 7.5% semi rate the “effective” rate and the 7.238% the “effective periodic rate”. Aren’t we labeling them the same thing?

aaronhotchner's picture

I don’t use the term “effective periodic rate,” if only because it adds confusion. I always just call it the “effective rate.” Any rate is either effective or nominal. A 7.5% effective six-month rate is larger than a 15% effective annual rate and exactly equal to a 15% nominal annual rate. A 15% effective annual rate is exactly equal to a 7.238% effective six-month rate. In the case of semiannual bonds, the 15% is nominal, and so it corresponds to a 7.5% effective six-month rate.

As far as why 7.5% is effective but 7.238% is effective periodic, if you must use those terms, the difference comes from where you start. An effective rate is the nominal annual rate divided by the number of periods. The effective periodic rate is the effective annual rate converted to an effective rate for a different period. So you absolutely cannot start with the same annual rate (15% in our case) and convert it to both an effective rate and an effective periodic rate without taking an intermediate step for one of them. If the 15% is a nominal annual rate with semiannual compounding, you cannot directly convert it to an effective periodic rate, because you are not starting with an effective annual rate. If the 15% is already an effective rate, then you would use (1 + r)^(1/m) - 1 to find the effective periodic rate.

In short: For a 7.238% semiannual rate, 15% represents the effective annual rate. For a 7.5% semiannual rate, 15% represents the nominal annual rate. They are two very different 15% figures, which is why you always need to be aware of whether the rate you’re told is nominal or effective.

BMiller12's picture

So you can go from nominal to effective by dividing/multiplying. But you can’t get from 15% nominal to 7.238% effective unless the annual rate is an effective rate? This is because 7.238% is derived from an effective annual rate and not a nominal rate, right? So to your point, what’s the intermediate step..just assuming that the 15% is effective and not nominal? Or how do you convert 15% nominal to effective?

I know some of this is silly, but do you at least see/understand some of my confusion? I think the texts casually throw these terms around without fully stipulating the difference, or maybe im just missing it. Just seems weird that 7.5% can be the effective rate for 6 months and 7.238% can also be the effective rate for the same period. I now realize that they’re based on different starting points. So if you’re were given a 15% effective annual rate (say a WACC) and needed to discount the 6 month cash flow in a dcf, would you use 7.238% ? Or would you first need to convert to a nominal annual rate and then use the effective period rate - how would you do that?

You can’t just divide/multiply effective rates like you can convert effective rates to/from nominal rates?

I think you mentioned before, but is it just standard convention that rates, especially those for bonds but also discount rates, are nominal rates ? Otherwise it would be specified.

I guess from the beginning, and the reason for some of the confusion, is when i said use nominal rates in a calculation i meant that’s what you plug into (1 / (1+r/m))^m*n) for r. The r/m part converts it to an effective rate which is why you said to always use effective rates. Seems like that’s where you and I broke down, right? Seems like the formulas (1+r/m)^n*m and (1+r)^(1/m)-1 are all the same formulas, albeit slightly different variations.

Aaron, thanks so much for your patience!

aaronhotchner's picture

Quote:
So you can go from nominal to effective by dividing/multiplying. But you can’t get from 15% nominal to 7.238% effective unless the annual rate is an effective rate? This is because 7.238% is derived from an effective annual rate and not a nominal rate, right? So to your point, what’s the intermediate step..just assuming that the 15% is effective and not nominal? Or how do you convert 15% nominal to effective?

Your statements are correct. If you know the 15% is effective, you can go straight to 7.238% using compounding. If you know the 15% is nominal, you must divide by the number of periods, which will give you the effective rate for each period. You would then compound that new rate by the number of periods if you needed to find the effective annual rate. If you don’t know whether it’s effective or nominal, and payments are made more frequently than once per year, assume the annual rate you’re given is nominal.

Quote:
I know some of this is silly, but do you at least see/understand some of my confusion? I think the texts casually throw these terms around without fully stipulating the difference, or maybe im just missing it. Just seems weird that 7.5% can be the effective rate for 6 months and 7.238% can also be the effective rate for the same period.

One thing you’ll come to realize is that discount factors are guesswork in reality more than anything else. There is no “correct” discount rate, because it is unpredictable. So don’t get too hung up on the idea that different discount rates exist for the same time period, because that happens all the time. It’s a matter of how good your predictions are. I do understand your confusion, though.

Quote:
So if you’re were given a 15% effective annual rate (say a WACC) and needed to discount the 6 month cash flow in a dcf, would you use 7.238% ? Or would you first need to convert to a nominal annual rate and then use the effective period rate - how would you do that?

Yes, you would use 7.238%, because you have an effective rate already and you want an effective rate for a different period.

Quote:
You can’t just divide/multiply effective rates like you can convert effective rates to/from nominal rates?

Correct, you must always use compounding to convert between effective rates.

Quote:
I think you mentioned before, but is it just standard convention that rates, especially those for bonds but also discount rates, are nominal rates ? Otherwise it would be specified.

Yes, if you are given an annual rate it will almost certainly be nominal if cash flows occur in between years, unless specified otherwise. If you are given, say, a monthly rate, and payments are made monthly, that rate is almost certainly the effective monthly rate, again unless specified otherwise. Sometimes you will also see statements like, “The interest rate is 10% compounded quarterly.” If you ever see a “… compounded” however frequently, it is definitely a nominal rate.

Quote:
I guess from the beginning, and the reason for some of the confusion, is when i said use nominal rates in a calculation i meant that’s what you plug into (1 / (1+r/m))^m*n) for r. The r/m part converts it to an effective rate which is why you said to always use effective rates. Seems like that’s where you and I broke down, right? Seems like the formulas (1+r/m)^n*m and (1+r)^(1/m)-1 are all the same formulas, albeit slightly different variations.

That’s all correct. (1 + r/m)^(n * m) - 1 will take a nominal rate and provide an effective rate for the correct period. (1 + r)^(1/m) - 1 will take an effective rate and provide the corresponding effective rate for the correct period.

BMiller12's picture

This is all great. Thanks again for your help!!! A lot of back and forth, so i appreciate you checking/responding so frequently. Means a lot.

One last question: why does “Compounding” definitively tell you that the rate is a stated nominal rate?

aaronhotchner's picture

You’re welcome.

The compounded language indicates a nominal rate simply by convention – you just need to know that it means the rate is nominal. It shows that the interest rate is annual, but the relevant period isn’t, which should immediately signal to you that the quoted rate is nominal. Almost no annualized interest rates you’ll find in finance are reported as an effective rate, with the exception of forward rates.

http://en.wikipedia.org/wiki/Compound_interest
http://en.wikipedia.org/wiki/Forward_rate

BMiller12's picture

Thanks bro!

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