Question about calculating Effective Annual Yield

[allalongthewatchtowe]

Hello All,

Let’s say I am being given a semi-annual yield of 4.5130%. Therefore, if I have to find yield per quarter, I would first find the yield for six months, i.e. 4.5130/2 => r = 0.022565. Now, to calculate yield per quarter (rq), I would do {(1+rq)^.5 -1} => rq=1.12195% Hence, EAY = 4.4880. I am good with this.

However, why can’t we calculate like this:

We are given semi-annual yield for six months, r = 0.022565. Therefore, we can calculate r/2 = 0.0112825. This is yield for 3 months. Hence, EAY using this newly calculated,r, is equal to ((1+0.0112825)^4-1) = 4.5899%. What’s wrong with this method? Essentially, what I am asking is that if we can halve the annual yield to get a semi-annual yield (i.e. we divided 4.5130 by 2), why can’t we divide the given annual yield by 4 and then annualize it using ()^4? I am curious. Moreover, I see that EAY using this method is a little higher than the previous method because we are compounding over 4 periods, as opposed to compounding over 2 periods (i.e. while computing the square root) in the earlier method. It’s not that we never divide the given annual yield to get a lower multiple yield. We do so. For instance, given a 90-day T-bill with annualized discount of 1.2%, I would calculate 90-day discount rate by dividing 1.2% by 4. So, I think we do divide the annualized rate to get a rate for a lower interval, but I am not sure why we don’t use this method for calculating EAY as shown above.

I would appreciate any help.

[S2000magician]

@allalongthewatchtowe

Hello All,

Let’s say I am being given a semi-annual yield of 4.5130%. Therefore, if I have to find yield per quarter, I would first find the yield for six months, i.e. 4.5130/2 => r = 0.022565.

Given your calculation, 4.5130% is not the semiannual yield; it’s the bond equivalent yield (BEY), which is twice the semiannual effective yield. It’s a nominal (annual) rate, compounded semiannually.

@allalongthewatchtowe

Now, to calculate yield per quarter (rq), I would do {(1+rq)^.5 -1} => rq=1.12195%

Yes, if 2.2565% is the effective semiannual yield, then 1.12195% is the effective quarterly yield. (Actually, it’s 1.12196%, but you’re close enough.)

@allalongthewatchtowe

Hence, EAY = 4.4880. I am good with this.

This is very sad news. Sad because you’ve admitted being “good” with something that’s completely wrong. This is not the effective annual yield. This is the nominal annual yield, compounded quarterly.

If you invest some money in an investment, come back one year later, and compare the amount of money in your account to the original amount you invested, the percentage by which it has grown is the effective annual yield: it’s the amount of interest you actually earn, as a percentage of your original investment.

To go from an effective yield for one length of time to an effective yield for another length of time, you compound. If you merely multiply an effective yield by a number to get another yield, the latter is a nominal yield, not an effective yield.

@allalongthewatchtowe

However, why can’t we calculate like this:

We are given semi-annual yield for six months, r = 0.022565. Therefore, we can calculate r/2 = 0.0112825. This is yield for 3 months. Hence, EAY using this newly calculated,r, is equal to ((1+0.0112825)^4-1) = 4.5899%. What’s wrong with this method? Essentially, what I am asking is that if we can halve the annual yield to get a semi-annual yield (i.e. we divided 4.5130 by 2), why can’t we divide the given annual yield by 4 and then annualize it using ()^4? I am curious. Moreover, I see that EAY using this method is a little higher than the previous method because we are compounding over 4 periods, as opposed to compounding over 2 periods (i.e. while computing the square root) in the earlier method.

This is a nominal 3-month yield, not an effective 3-month yield. You cannot compound a nominal yield to get an effective yield; you must compound an effective yield to get an effective yield.

@allalongthewatchtowe

It’s not that we never divide the given annual yield to get a lower multiple yield. We do so. For instance, given a 90-day T-bill with annualized discount of 1.2%, I would calculate 90-day discount rate by dividing 1.2% by 4. So, I think we do divide the annualized rate to get a rate for a lower interval, but I am not sure why we don’t use this method for calculating EAY as shown above.

We do this because the 1.2% is a nominal rate, not an effective rate. (Furthermore, it’s a discount rate, not an add-on rate; that complicates things even more.)

@allalongthewatchtowe

I would appreciate any help.

I wrote an article that may be of some help here: http://financialexamhelp123.com/nominal-vs-effective-interest-rates/.

[allalongthewatchtowe]

Hello S2000magician, Thank you so much for your response. After reading your post, I realized that there was so big hole in my understanding that one could have passed a truck. I did go through the excellent blog you have posted. I have a follow-up on one of the statements in your blog, which is also posted here.

@S2000magician

Given your calculation, 4.5130% is not the semiannual yield; it’s the bond equivalent yield (BEY), which is twice the semiannual effective yield. It’s a nominal (annual) rate, compounded semiannually.

Let’s say A bank deposit is quoted add-on yield of 1.5% based on a 360-day year. Now I am being required to calculate BEY and the yield on a semi-annual bond basis. Here’s what I did: To calculate BEY, I would annualize add-on yield to a 365-day year. Therefore, BEY = 365/360*1.5 = 1.5208%. Now, this is as per the official answer. However, if I consider the statement above and the one in your blog “BEY is twice the semiannual effective interest rate” then semi-annual yield would be 2*BEY = 3.0416%. However, this is not the correct answer. Hence, I am definitely missing something. Moreover, I noticed that if I approach the calculation of semi-annual yield a little differently, then I can arrive at the official answer. So, I know that I am doing something wrong, as I was doing above. HEnce, please correct me. Here’s what I did : (I mainly used the formulas in Quant method chapter) From BEY, I can calcuate holding period return as: BEY = HPY * 365/t => HPY = 0.41665% (I used the BEY calculated above; t =100) Now, once I have HPY, I can calculate annualized, compounded yield, aka EAY, (I hope I am not messing up “nominal” yield and “compounded” or “Effective” yield) as : [(1 + HPY) ^ 365/100)]-1 = 1.5292%. Now, because I have an effective yield, I can calculate effective yield for six-months by finding square root: as {(1+EAY)^.5} - 1 = 0.761706%. Now, I can find nominal rate by multiplying this by 2: 1.523413%, which matches the official answer. Can you please explain what I am missing here? Secondly, I have another question on your excellent blog. I think your blog is the only one on the Internet that explains how different securities treat nominal vs. effective annual rates. Do you know any other securities, other than US Treasury rates, LIBOR and Money market instruments, that use nominal rates? I think Fixed income, such as bond, calculate effective annual yield but then annualize it to find APR, or nominal rate, YTM. Thanks in advance. I look forward to hearing from you.

[S2000magician]

@allalongthewatchtowe

Hello S2000magician,

Thank you so much for your response. After reading your post, I realized that there was so big hole in my understanding that one could have passed a truck. I did go through the excellent blog you have posted. I have a follow-up on one of the statements in your blog, which is also posted here.

@S2000magician

Given your calculation, 4.5130% is not the semiannual yield; it’s the bond equivalent yield (BEY), which is twice the semiannual effective yield. It’s a nominal (annual) rate, compounded semiannually.

Let’s say A bank deposit is quoted add-on yield of 1.5% based on a 360-day year.

I’m not certain (exactly) what you mean by, “quoted add-on yield of 1.5% based on a 360-day year”, but I’ll assume (probably reasonably) that you mean that 1.5% is the annual yield, based on a 360-day year (i.e., if you invest $100 and wait 360 days, your account will be worth $101.50), and my discussion will be predicated on that assumption. If you meant something else, let me know and I’ll revise the discussion appropriately.

@allalongthewatchtowe

Now I am being required to calculate BEY and the yield on a semi-annual bond basis. Here’s what I did:

To calculate BEY, I would annualize add-on yield to a 365-day year. Therefore, BEY = 365/360*1.5 = 1.5208%.

This is not correct. To calculate BEY you have to calculate the effective semi-annual yield, then double it.

You would start by computing the EAY: the effective yield for 365 days. Because I’m assuming that 1.5% is the effective 360-day yield, we would get the EAY by compounding, not by multiplying by 365/360:

1 + EAY = 1.015^(365/360)

1 + EAY = 1.015210

EAY = 1.5210%

To compute the effective semiannual yield, we _un_compound the EAY:

1 + ESAY = (1 + EAY)^½

1 + ESAY = 1.015210^½

1 + ESAY = 1.007576

ESAY = 0.7576%

To compute BEY, we double the effective semiannual yield:

BEY = 2 × ESAY = 2 × 0.7576% = 1.5153%

@allalongthewatchtowe

Now, this is as per the official answer.

If that’s per the official answer, then the official answer’s wonky. Even if the 1.5% were a nominal 360-day rate, multiplying it by 365/360 would give you the EAY, not the BEY.

I’ll stop here for now and let you digest this. Let me know when you’re ready for a continuation.

[allalongthewatchtowe]

Hello S2000magician,

Thank you so much for your help.

@S2000magician

My discussion will be predicated on that assumption. If you meant something else, let me know and I’ll revise the discussion appropriately.

My apologies for posting a confusing question. I am such an id1ot. I must slap myself for this. The question states that A bank deposit for 100 days is quoted with an add-on yield of 1.5% based on a 360-day year. Calculate BEY and the yield on a semi-annual bond basis.

Now, that you have explained me the process, here’s how I would do it:

Add-on yield = (HPY) * (360/t) ; with yield = 1.5%; t=100;

HPY = 0.41667%

Now, I would get EAY from HPY.

1 + EAY = (1 + HPY)^(365/100)

EAY = 1.52925%

Now, I will get ESAY (uncompounding EAY):

ESAY = (1 + EAY)^½ - 1

Therefore, ESAY = 0.7616%

Therefore, BEY is 2* ESAY = 1.52344%.

My answers are not equal to the official answers, which are BEY = 1.5208% and the annual yield on a semi-annual bond basis = 1.5236%.

Do you think I am correct?

@S2000magician

I’ll stop here for now and let you digest this.

I was able to fully understand your explanation. However, I noticed one difference between how BEY is calculated in 2013 curriculum and in 2014 curriculum. For instance, 2013 curriculum, on page 494 of Book 5 calculates BEY exactly how you have done. However, 2014 curriculum, on page 427 Book 5, calculates BEY very differently. Curriculum 2014 states that _the 90-day commercial paper discount rate of 5.76% quoted for a 360-day year converts to an add-on rate for a 365- day year of 5.925%. This converted rate is called a BEY, or sometimes just an “investment yield.” (_I believe this merely does a linear conversion from a 360-day year to 365-day year. ) Moreover, there are tonnes of question in the curriculum that follow this concept. This is very different from what was in 2013 curriculum, which uses the method that you have taught me.

Hence, very respectfully, coud you please explain the difference between calculating BEY in 2013 and calculation BEY in 2014. I did some homework before asking this question to you. I foundthis article where the writer has calculated BEY as per what’s in 2014 curriculum. I also found your excellent blogon this topic, which calculates BEY as per what’s in 2013 curriculum. Now, I am really lost.

@S2000magician

Let me know when you’re ready for a continuation.

I am ready for continuation. I look forward to hearing from you.

With utmost regards,

Allalongthewatchtower

[allalongthewatchtowe]

Hello S2000magician,

Can you please help me with your above response. I am not sure whether you saw my analysis. I am eagerly looking forward to hearing from you.

Thanks in advance.

[allalongthewatchtowe]

@S2000magician

I’ll stop here for now and let you digest this. Let me know when you’re ready for a continuation.

Hello S2000magician,

First of all, I am sorry for re-bouncing this thread. I would really appreciate if you could help me with my analysis above. And I am ready for additional stuff.

Thanks in advance.