### CFA Practice Question

There are 410 practice questions for this study session.

### CFA Practice Question

What is the present value of 15 payments of \$100 each received every 18 months at a discount rate of 9%?

A. \$951.28
B. \$1209.10
C. \$620.43

(1+i)18/12 = 1.137993 or 13.8% CALC: N=15 FV=0 PV=? I=13.7993 PMT=100; PV = 620.43

User Comment
Laurel Don't understand the calculation above. Need more practice on the Present Value of an Annuity Calculation.
stranger How did we calculate this interest? Since 9% is annual rate cant we straight away take 9*1.5=13.5%
lna1717 I ve got A. 951.28
lna1717 you have to use the effective annual rate (EAR) instead of the quoted interest rate (9*1.5). The EAR allows to take full account of compounding within one year.
(1+periodic interest rate)^(18/12*15)
synner I used BAII plus, and set P/Y=2/3, N=15,I/Y=9,PMT = 100,

CPT PV, and i got 629. why is it different from the correct answer?
jimmymh synner, set your ba2 plus as p/y=1, N=15,
I/Y= 13.79 becuz the interest annual would be 9% plus half a year, so using the formula above
(1+i)^18/12 you'll get 13.79. Pmt= -100, CPT PV.
snowwa Well, I still don't understand how do you guys get the 13.79.
stefdunk 9%

1.09^(18/12) = 1.1379

13.79%
ipallete \$1 during 12 months at i(12) should produce the same that \$1 at i(18) during the same 12 month expressed as a part of the 18 months period
(1+i(12))=(1+i(18))^(12/18) =>
i(18) = ((1+i(12))^(18/12))-1
PeterW2006 EAR = (1 + periodic rate)^m - 1
periodic rate = 9% per year
m = number of compounding periods per year
= 18/12
= 1.5
EAR = 13.799%

Using TI BAII,
PMT = -100, N = 15, I/Y = 13.799%,
PV = CPT = \$620.44
tao001jp Why P/Y = 1? even payment is made every 18 month?
Rotigga I don't think the answer shown above is Correct. Remember that m is the number of compounding periods per year, so for example if we compound monthly, m=12, and at i = 9%, monthly compounding interest = 9/12% = 0.75%. Likewise, if we pay \$100 every 18 months, that means m = 1/(18/12) = 0.6667, so interest compounding every 18 months = 9/(0.6667)% = 13.5%. Use your calculator and plug in i=13.5, n=15, PMT=-100, hit the [PV] button, and you get \$629.89 . This is the right answer.
sunilcfa I dont agree...the answer is correct! Rottiga has simply calculated the monthly stated rate for the one and half yearly payments. He forgot to calculate the effective rate. His method is also correct but he should have calculated the ear after calculating 13.5% and would have come to the same answer.
octavianus Based on ROTIGGA and SUNILCFA comments:

EAR = [[1 + (stated rate/m)]^m]-1
EAR = [[1 + (.09/(12/18))]^(12/18)-1
EAR = 0.08808 or 8.808%

Then:
I=8.808 N=15 PMT=100;
PV=-815.28

Negative PV refers to an outflow of cash
o123 ** 2nd P/Y= .6667 (12/18), 2nd C/Y=1
PMT=100, N=15, I/Y=9 --> PV=620.429
AndressaS I didnt get it.
bentang since EAR = [(1 + Periodic Rate)^n]-1
EAR = discount rate
n = 12/18, discounted every 18 months
we get:
9% = [(1+periodic rate)^2/3]-1
periodic rate = 13.79%
ravdo anyone who made it using HP12C?
Yurik74 Guys, still dont get it. Could someone explain, please. We take 9% as periodic interest rate cause we consider period equal to one year and that's why number of peridos is less than 1 (=12/18). Correct? Kind of similar task in is given a but later with monthly payments for 30 months.
JKiro ravdo - Using HP12c
first the formular for effective annual rate:-'cause the payments are made every 18 months... EAR = (1+r)^n - 1
where r = rate; n = # of periods in a year
plagging the numbers:
(1+0.09)^(18/12) - 1 = 0.13799
going back to HP and calculating PV of an annuity:
100[CHS][PMT]; 13.799[i]; 15[n]; [PV] --> 620.44
gazelle Many thanks JKiro
Spawellian if compounding more than once a year makes the EAR higher, how does compounding LESS than once a year also make the EAR higher

Surely it's
<2nd>ICONV
NOM 9
C/Y 0.667
EFF CPT
8.80%

So in other words, getting a stated rate of 9% a year is effectively 8.80% if it's paid every 1.5 years
Ratego I think it is basically annualising the 9%.Because the 9% is paid every 18 months you are basically getting the EAR annually.Therefore you are annualizing
uformula the question changes the single unit of time from 12 months to 18 months, therefore you need to get the effective rate for 18 months and the use this as your periodic rate.

Effective rate for 18 months = ((1+.09)^(18/12)-1)b= .138

The you use .138 as your periodic rate.... with 15 being the amount of payments.

It's tricky...I hope this helps a little
SANTOSHPRABHU Using BA II Plus:

Discount rate is 9% pa, hence for 18 months is (1+0.09)yx 1.5 = 1.1137993 = 13.80%= I/Y
FV=0
N=15
PMT=100
CPT > PV = -620.43
8thlegend The greatest confusion that a lot of people are having is Discount rate.

Periodic rate and discount rate are 2 different rates.

Meaning, No = (1+r/n) because it is the discount rate.

The ^n still applies though because it is not in a year but in 18 months.

That is why its 1.08^(18/12)= 13.8%
bhaynes A calculation based on 18 months in not even needed. If you were to do the calculation based on annual payments rather than every 18 months, you get \$806.07. Being that the payment is stretched out to 18 months, you know that the value MUST be less than this annual amount \$806.07. Answer C is the only answer that qualifies.
hoyleng how to do this calculation using BAII. pls. thanks. i can't figure out about the 15 payment & the 18 mths each.
gill15 Simple. Change your discount rate to always match the timing of payments. Better not to mismatch.
Here.
(1 + .09)^1 = (1 + x)^2/3
solve for x and do your regular way with TI
jorgeman81 so for BAII you can't calculate the EAR, 13,80% with the ICONV function??
Rchan89 I am confused that this as well - why does the BA2 calculation not give us the effective rate of 13.8? What does that 8.08 it give us represent??
hopeipass 2ND P/Y .6667 ENTER
2ND QUIT
2ND TVM
15 N
100 +|- PMT
9 I/Y
CPT PV
629.9142 (so C is best answer)
hopeipass EAR = 9%
With BAII:
2ND ICONV
EFF = 9
C/Y = .6667
NOM = 9.199530
so NOM * 1.5 = 13.7993% which is the periodic
rate you use for your time value money "TVM"
calculation.
andyblake This one is very difficult. Can anybody offer a super-straight-forward explanation? Thanks.
Emily1119 Is it possible to use the medthod of iconv to get ear? If so, what data should i enter?
johntan1979 Receiving payments every 18 months is similar to compounding interest every 1.5 years. Therefore, convert the I/Y to 1.5 years:

(1+0.09)^1.5 - 1 = 13.7993%

N=15, PMT=100, CPT PV and you'll get the answer.

By the way, I noticed that Rotigga consistently produced erroneous comments. So please ignore if possible.
johntan1979 ANYWAYS, I have to say that this is a very difficult and confusing question.

If you noticed, the I/Y is converted to using the effective annual rate. You normally don't use EAR in calculations. I guess this is probably because the number of payments is fixed in this question, i.e. N=15

On normal cases, if it is semi-annual payments, you divide the I/Y and the PMT by 2 and multiply the N (in years) by 2. And if it is every 1.5 years, you multiply the I/Y and PMT by 1.5 and divide the N (in years) by 1.5.

Based on my last sentence, if you multiply by 1.5, then you get 13.5%, which by normal standards should be used in calculations. BUT you can't do this in this question, because you can't divide the N=15 by 1.5, because it is in number of payments, not number of years.

Very confusing, overall. Hope something I said made any sense.
Shaan23 Did not want to read thru this - im an actuary so if you got this wrong - read this

Using calculator you want N as the # of payments. The interest rate given is as an stated rate or quoted rate or discount rate <--- these are all the same thing different names.

so use (1 + 0.09) = (1 + j) ^ 12/18 and solve j
We want the 18 month rate(j) not the annual rate to use in calculator. To equate the two take the exponent for the right side to 12/18. j = 13.8%

rest is just calculator sub ins.
sgossett86 Thanks Shaan. I read the first few posts and they indicated what you said... Then everything went off into a tangent. Good job getting it back on track as to the solution of this problem.
bfeitosa Rule of thumb for effective rate is:

Continuous Compounding:
(2.721828)^.09 = 1.09417

Quarterly Compouding:
(1.02250)^4 = 1.09308

Semi-Annual Compouding:
(1.04500)^2= 1.09203

Annual Compouding:
(1.09)^1= 1.09

Once every 18 months (1.5 years) Compouding:
(1.09)^1.5= 1.13799

Once every 2 years compouding:

(1.09)^2= 1.18810

Forget all the rest and focus on this rule. In order to find the effective rate for a compounding period of less than a year, just divide the rate by the amount of times the money will be compounded, and take that to the power of the same number. E.g Quarterly- divide rate by 4 and raise it to the power of 4.

If the money will only be compounded in periods longer than a year, do NOT divide the stated interest rate, but raise it to the power of the compounding period (1.5 for 1.5 yrs, 2 for 2 yrs) and so on.

To solve the problem above, use the EFFECTIVE rate either on the regular annuity formula or on the BAII and you will get the correct answer.
sumaiyak @hopipass, can you explain why C/Y = .6667? and not 1.5?
dbedford EAR=(1+[r/m])^m) - 1
Discount rate = r/m =9%
m = 18/12

EAR = (1.09)^[18/12] - 1
mcbreatz Only way conceptually I could get this to stick in my brain is that the 18 month interest rate should be higher than the 1 year interest rate.
wawymtisi i dont understand this question