One of the problems asks to obtain the equivalent quarterly rate from 6% semiannual rate and I dont understand the logic given in the answer: 1.03^1/2-1=14.9% and then it uses this 14.9% to discount quarterly coupon payments!! Shouldn’t it be divided by 4 firstly, and then be used as a discount factor?! Or may it be just a mistake in the problem? And one off-topic question, probability of using ChiSquare test statistic formula is too low right? 
thanks in advance
You need to take compounding and the conventions of bond yield quotes into account:
Recall first of all that the semiannual rate is typically obtained by multiplying the 6-month effective rate by 2. If you want to obtain the quarterly effective rate, you need to calculate the 3 month effective rate and multiply that by 4. You proceed as follows:
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If the seminannual rate is 6% then the 6 month effective rate is 3%.
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The 6-month effective rate includes compounding in each period, thus to get the month rate, you cannot just divide by 2, but rather need to take the square root, => 1.03 ^(1/2)-1=1.49%=0.0149
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This quarterly rate then needs to be multiplied by 4 to get the annual rate.
The Chi Square qestions I encountered so far were pretty simple and typically computing the test statistic was half if not the entire battle. So even without putting much effort into it, just memorize it quick and you might have some easy points in the exam. Here it is by the way, write it down a couple times and you’ll remember :
Chi Square Test Statistic = (n-1)s^2/var
Greetings to the Caucasus by the way, my ancestors are from the area!!
Watch yer decimals…
1.03 ^(1/2) - 1 = 0.0149 = 1.49%
whoops, thanks!! I just copied from the original post without double checking the result (kind of sounds like a violation of Standard V(A) Diligence and Reasonable Basis)
I corrected my post.
Thanks again.
Many thanks from Georgia guys! You are always welcome here
You are very welcome.
Thanks so much, hopefully some day…
so from my understanding in situations like this, you would always take the annual yield and divide it by 2 to get the effective yield for six months and from there you would raise it by the power (or fraction of a power) in order to get it to quarterly periodicity correct?
Lets say the question wanted to find the annual rate for a monthly annual rate
you would take the 6 -> (6/2=3%) 1.03^(1/6) -1 = .00494 .00494*12= .05926 (5.926%)?
So my question is would everything first need to be computed to a 6-month effective rate and from there just convert it to either quaretly, monthly, etc?
Correct. That is, typically you are provided with the bond equivalent yield, which is constructed by multiplying the 6 month rate by 2 (most bonds pay coupons semi-annually). So if not stated otherwise, I would proceed in that manner.
Annual percentage rate, or APR, is simple interest after the fees you pay to originate the loan, However, APR doesn’t include the effects of compound interest.
APR is Annualizing with SIMPLE multiplication: a 6mo rate doubles; Quarterly rate is times 4; a monthly rate times 12….
On the other hand, effective annual percentage rate, also known as EAR, EAPR, or annual percentage yield (APY), ALSO takes the effects of compound interest into account.
Annualizing with COMPOUNDING: a 6 month rate squares; Quarterly rate is raised to the fourth; a monthly rate is raised to the 12th…
PROBLEM: A hypothetical bond has an annual percentage rate of 8% with a periodicity of 12. This bond’s semi-annual rate is closest to:
Note Start with Definitions:
8% APR with periodicity of 12 means a monthly rate was annualized by SIMPLE multiplication:
Step 1: Find the Monthly Rate
Note – if APR uses simple multiplication to annualize – to find the period we have to find the monthly rate with the same math (multiplying/dividing) rather than (compounding with exponents / nth roots)
8% = 12x; or the monthly rate is 8/12 or 0.67%
What do they want: lets go slow
First – what is another way to annualize a monthly rate of 0.75%? COMPOUNDING
So the question is what 6 month rate COMPOUNDED = my monthly rate COMPOUNDED
Step 2: Annualize the monthly rate with COMPOUNDING - Compound the 12 month rate that we are given:
x = (1.0067)^12; or the COMPOUNDED annual rate = 8.3%
NOTE: an APR of 8% compares to an APY of 8.3 % of the same monthly rate its two different ways to annualize.
Simple underlying idea – there are TWO ways to annualize:
0.67% * 12 = APR = Annualize with Simple Multiplication
(1.0067)^12 = APY = Annualize with COMPOUNDING
Step 3: Find the 6month rate that compounds to the same 8.3% rate as our monthly 0.67% did
X = (1.083)^1/2 = 1.0407
NOTE: This just means that compounding .67% monthly and compounding .0407% semi-annually will both have the same return: (1.0067)^12 = (1.0407)^2
Step 4: Find the APR for the 6 month rate
This means we annualize by SIMPLE multiplication rather than COMPOUNDING to annualize:
X = .0407 * 2; 8.13%
Question
A hypothetical bond has an annual percentage rate of 8% with a periodicity of 12. This bond’s semi-annual rate is closest to:
- 8.01%
- 8.10%
- 8.13%
Solution
The correct answer is C.
(1+0.08/12)12=(1+APR2/2)2
(1+0.08/12)^12=(1+APR2/2)^2
(1+0.0812)122=1+APR22(1+0.0812)122=1+APR22
2×(1+0.08/12)^12/2)–1=APR2
2×(1+0.0812)122)–1=APR2
APR2=0.0813
CONCLUSION:
These problems are messy – but they are actually just tedious and the equations look confusing. The underlying idea is fairly simple:
CORE IDEA: there are 2 ways to annualize, and there are different periods we can annualize and we want to compare them.
The CFA loves mixing this up – how do you compare a quarterly rate that is annualized with simple math to a monthly rate that is annualized with compounding – you can imagine all the different types of questions mixing and matching the different periods and going from one to the other.
ALSO – they love using lots of different words to say the same thing – APY and EAY and Geometric Average and Compounding are all exactly the same with different names: annualize with compounding.
APR, BEY (Bond Equivalent Yield) and YTM are all the same: annualize with simple multiplication
REMEMBER – lots of different names and words but in the end there’s just 2 ways to do it, multiply or use an exponent; simple or compound.
STRATEGY:
First Make the two numbers in the question comparable by compounding
Then depending on the question you simply find the period they are asking about and annualize that period in the way they are asking
ALWAYS start by making the two numbers comparable by annualizing them – and always annualize by compounding (its easier)
The Formula in the CFA Text looks horrendous because it smushes the two steps together but if you break it down, All it is doing is exactly this – its making two different period rates equal when they are annualized by compounding.
Converting Between Periodicities
The general formula for converting annual percentage rate from m periods per year (APRm) to an annual percentage rate for n periods per year (APRn) from the text is:
(1+APRm/m)^m=(1+APRn/n)^n
Take another look at this formula
To plug in our example above the formula is simply doing this:
NOTE: This just means that compounding .67% monthly and compounding .0407% semi-annually will both have the same return: (1.0067)^12 = (1.0407)^2
Just always remember – once you have found the two numbers that are comparable to each other, then the question will ask you to annualize it – be careful to annualize it in the way the question asks, they will disguise it by mixing up the terminology, but in the end no matter how many different names there are there can only be two different ways to annualize
Question
A hypothetical bond has an annual percentage rate of 8% with a periodicity of 12. This bond’s semi-annual rate is closest to:
- 8.01%
- 8.10%
- 8.13%
Same Solution explained in steps:
Step 1: find the period rate from the APR; APR12/12; .08/12 = 0.667
Step 2: find the COMPOUNDED annualized rate (APY) for that period rate; (1+.067)^12; = 1.083
Step 3: find the rate for a 6 month/semi-annual period that will compound to the same rate; (1.083)^1/2; 1.0407
Step 4: annualize this period rate to an APR; .0407 * 2; .0813
Conclusions:
Two things can change, how you annualize or the number of periods
So – change how you annualize:
If you have a 6 month rate of .0407 you can annualize it two way:
Simple math: .0407 * 2 = .0813
Compounding: (1.0407)^2 = 1.083
OR – change the period
If you have a 12 month period and that has been annualized by compounding to 8.3%
You earn 0.67% each month
You could also earn 4.07% semi-annually and either way you’d get 8.3%
The CFA will mix and match to try and make it complicated – the strategy is to break it down into steps: first make comparable by using same annualizing method; then find the periods/period rates.
The CFA also used different terminology to refer to each annualizing method. APR is just an instruction set for how annualize, you can also say Bond Equivalent Yield or simple interest;
APY is also just an instruction set for how to annualize; you can also say compound, exponential, geometric, EAY.
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