Even I can't make this up!

Quantitative Finance Level 1

RAC is offering a Presidents Day (US holiday) special on a Samsung Galaxy.

You can choose between 2 payment plans:
1) 78 weekly worry-free payments of $29.99 each week, starting today.
2) A one-time payment of $829.00 due in 90 days.

Assuming that you want to buy this phone, what is the minimum (positive) daily interest rate that would make a person be indifferent between these payment plans?

0.01% 0.1% 1% 10% 100% None of the rest

1 solution

Chew-Seong Cheong
Mar 26, 2015

For an interest rate of $i$ , the present values of the two payments are as follows:

78 weeks of $29.99, $P_1 = \dfrac {1-\left( \frac{1}{(1+i)^7} \right)^{78}}{1-\frac{1}{(1+i)^7}}\times \$29.99$
One-time of $829.00, $P_2 = \dfrac {\$829.00}{(1+i)^{90}}$

Calculating $P_1$ and $P_2$ for various interest rates, we find that $P_1 > P_2$ for all $i$ (see table below).

Therefore the answer is $\boxed {\text{None of the rest}}$

What spreadsheet do you use and how do you insert this?

Assuming your table the daily interest should be between 0.1%-1% for the NPV of both choices to be equal right? The question would have been more precise if instead of interest rate they would have (also) mentioned inflation as a discount rate.

Patrick B - 5 years, 4 months ago

Yes, you are right to include inflation to make it more precise. But I think it would not make any different because $P_1 > P_2$ for all $i$ .

Chew-Seong Cheong - 5 years, 4 months ago

Patrick, I used Excel spreadsheet. Copy and waste it in Paint and saved it as Jpeg. The solution writing box has an insert image button for you to upload.

Chew-Seong Cheong - 5 years, 4 months ago

In your formula the i related terms in numerator and denominator needs parentheses...it is not 1 +i 7 it is (1+i) 7

Greg Grapsas - 4 years, 9 months ago

@Greg Grapsas – Thanks. I have amended them.

Chew-Seong Cheong - 4 years, 9 months ago

These figures might not have unique way to calculate but we could try rough figures as:

1
2
3

546  $4.28   $2,339.22 (Price for plan 1)
90   $9.21   $829.00  (Price for plan 2 with A = 829) 
456  $3.31   $1,510.22 (Rough difference for A (1 + p)^t = B; B = 1510.22)

$(\Large \frac{1510.22}{829})^{\frac{1}{456}} - 1 =$ 0.1316%

Other than nearest 0.1% which is not taken an answer, none of them is 0.13%. The fact is figures involved do not seem likely to be related by simple percentage.

Answer: $\boxed{None~of~the~rest}$

Lu Chee Ket - 5 years, 4 months ago

Yes, we can only get guesswork from you.

Chew-Seong Cheong - 5 years, 4 months ago

Sometime yes and sometime no, therefore the resultant is no. For this question, I just wrote with no claim of correct facts. However, I feel that finance counting like this is not easily and not obviously well definable. I actually would like to know how to conclude for a fixed answer. I am not teaching but asking.

Lu Chee Ket - 5 years, 4 months ago

@Lu Chee Ket – But of course, some problems need no guesswork, then it is pure calculator skills. There are Wikies on Quantitative Finance, mainly by Calvin Lin, for those who are willing to learn. Just do a search in Brilliant.

Chew-Seong Cheong - 5 years, 4 months ago

@Chew-Seong Cheong – Yes! I have noticed that Calvin seems to be a resemblance to Quantitative Finance in Brilliant. Before I wanted to get a new icon for this topic, I would like to survey for whether I am able to achieve until I obtain another blue. Since you didn't answer my previous question, I actually struggle for where to find required questions which I am able to solve for this topic. Wikies do not mean questions.

Lu Chee Ket - 5 years, 4 months ago

@Lu Chee Ket – There are problems attached in wikis. Not right to say it is not well-defined else there are no wikis possible. Sorry, I thought you asked for fixed answers not guessed ones so you needed wikis.

Chew-Seong Cheong - 5 years, 4 months ago

@Chew-Seong Cheong – I think all I can do is to try to read wikis mentioned. Thanks!

Lu Chee Ket - 5 years, 4 months ago

@Lu Chee Ket – You can google on a topic. There are many pages of info available. You can find worked examples and few problems to solve. I learned some of it in my MBA studies.

Chew-Seong Cheong - 5 years, 4 months ago

@Chew-Seong Cheong – Seems this is a reason why many people in Brilliant prefer Quantitative Finance rather than Chemistry.

Lu Chee Ket - 5 years, 4 months ago

@Lu Chee Ket – Calculation problems for Chemistry are few. Theory problems need the member to know Chemistry. I solved most of it through googling and relearning Chemistry which I did up to form 6.

Chew-Seong Cheong - 5 years, 4 months ago

@Chew-Seong Cheong – Situation very similar to each other. Yet, I couldn't see your icon assigned by Brilliant for Chemistry on the page about you.

Lu Chee Ket - 5 years, 4 months ago

@Lu Chee Ket – It is not in "main" topics yet. The is no stat and level computation for it yet. QF was the last to be a main. It was on beta version earlier. Chemistry is not even in beta yet.

Chew-Seong Cheong - 5 years, 4 months ago

@Chew-Seong Cheong – Could have some difference among people. You could ask Calvin perhaps.

Lu Chee Ket - 5 years, 4 months ago

@Lu Chee Ket – It is alright. I will just try some Chemistry problems to find out. Have not been doing them lately. They should also be announcements.

Chew-Seong Cheong - 5 years, 4 months ago

I solved it right using the same method (spreadsheet and plugging in the different values of interest provided in the posible solutions). The trivial solution is "None of the rest" because the probability that one of those "round" numbers for interest yield the same present value in both plans is very remote. From the spreadsheet seems obvious that the PV of weekly payments never catch up that of the one-payment. However, I chose the option 1% since both present values are in the same order of magnitude.

Félix Pérez Haoñie - 1 year, 8 months ago

My problem here is that I don’t understand the question to begin with

ViralTaco Capobianco - 2 days, 18 hours ago

also by 72 rule we may realize that none of the options are right

Mr Yovan - 5 years, 7 months ago