- CFA Exams
- 2021 CFA Level I Exam
- Study Session 2. Quantitative Methods (1)
- Reading 6. The Time Value of Money
- Subject 4. The Future Value and Present Value of a Series of Equal Cash Flows (Ordinary Annuities, Annuity Dues, and Perpetuities)

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**CFA Practice Question**

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

B. $1209.10

C. $620.43

A. $951.28

B. $1209.10

C. $620.43

Correct Answer: C

(1+i)

^{18/12}= 1.137993 or 13.8% CALC: N=15 FV=0 PV=? I=13.7993 PMT=100; PV = 620.43###
**User Contributed Comments**
44

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 |