Could someone please check my math?

[Lliam]

**Funny side note: While typing this up, as a question, half way through I think I figured it out, which is why this is now a "check my math" post**

Hello,

I'm currently getting my grade 11 college math (Functions and Applications, MCF3M) through a learn from home program. Unfortunately I ran into a little trouble with this problem, which had me stumped . I would be very thankful to anyone who could check my math

Nader plans to buy a used car. He can afford to pay $280 at the end of each month for three years. The best interest rate he can find is 9.8%/a, compounded monthly. For this interest rate, the most he could spend on a vehicle is $8702.85.

Determine the amount he could spend on the purchase of a car if the interest rate is 9.8%/compounded annually.

First of all, I understand the first part about him being able to spend $8702.85. I know this can be figured out using the present value formula:

PV=R[1-(1+i)^-n]/i
=$280[1-(1+[.098/12])^-36]/(.098/12)
=8702.849629
Which, rounded, gives you $8702.85

And I think this is how you figure out the question (though like I said I'm unsure):

1) Find the (monthly?) interest rate (of the compounded annually plan?)
(1+i)^12=(1+[.098/12])
(1+i)^12=(1.008166667)
1+i=12√1.008166667
1+i=1.000678021
i=1.000678021-1
i=.000678021

2)Use this new interest rate in the present value formula used earlier
PV=R[1-(1+i)^-n]/i
=280[1-(1+.000678021)^-36]/.000678021
=9954.641382
Which, rounded, gives you $9954.64

Therefore Nate could spend $9954.64 on the purchase of a car.

*NOTE: I am also unsure whether I should be using the future value formula in step 2 instead of the present value formula.*

Like I said, if someone could look this over and tell me if I did it right or what I did wrong I would be EXTREMELY thankful!

- Lliam

 

[Yogi]

Is your answer of $8702.85 something you came up with or was this given info in the problem? The conversion from annual to monthly interest rates by dividing by 12 is called "nominal rate of interest convertible monthly" and yet I do not see the words "nominal" or "convertible monthly" anywhere in the problem. Without this wording I would think you would use [(1+i)^(1/p)]-1 for the effective monthly rate given p time periods per year so...
[1.098^(1/12)]-1=.00782129 effective monthly.

For the second part, the interest is only compounded annually and you are given an annual interest rate so why convert to monthly? You have 3 payments of 12*280 at the end of each year and apply PV formula using the annual rate.

As for whether to use PV or FV, it is pretty unclear the way they word it. "Determine the present/accumulated value of the payment stream" would be much more clear to me than "Determine the amount he could spend". The amount he spends is worth different values at different moments in time and in the question there is no mention of time. I would just go with PV.

 

[Lliam]

Thank you very much for the help guys! I think I'm beginning to understand Sorry if I forgot to use the proper terms for what i was doing in my explanation. The book I have to learn from is not very descriptive :neutral:

@Yogi The answer $8702.85 is part of the given info in the problem. I was just explaining that I understood how to get that number (which is why I also assumed that later on you use the PV formula not the FV formula).

So based off what you guys are saying it should be done like this(?):

1) Calculate monthly rate i when nominal rate r compounds annually
i = [(1 + r)^(1/12)] - 1
=[(1 + .098)^(1/12)] - 1
=.00782129

2) Put this through the present value formula
PV=R[1-(1+i)^-n]/i
=280[1-(1+.00782129)^(-36)]/.00782129
=8755.615821

Therefore: Nader will be able to spend $8755.61 on the purchase of a car.

Btw does anyone know of an online calculator for this sort of thing so that I can check my answer?

Thanks!

Lliam

P.S. Thanks for the job advice Denis. I'll let you know how that goes

 

[Yogi]

To calculate monthly rate i when nominal rate r compounds annually:
i = (1 + r)^(1/12) - 1
So:
i = 1.098^(1/12) - 1 = .00782128... (as per Yogi).

And present value formula:
PV = P(1 - v) / i where v = 1 / (1 + i)^n
(with your problem, P = 280 and n = 36)

Assuming an annual effective interest rate i and p time periods per year, the corresponding pthly effective interest rate for a pthly period is: (1+i)^(1/p)-1

Denis, you associate "nominal rates" with a formula that does not contain nominal rates, the formula for nominal rates is different and now the OP is copying you -_-. Your "r" (usually you use i for annual rates) is called annual effective interest rate. In the given problem we use this formula to convert from annual effective interest rate to monthly effective interest rate. Had they wanted nominal they would have either used the word "nominal" somewhere, or they would have said 9.8% convertible monthly, in which the calculations for effective monthly interest is just: .098/12=.00816666 This is what the OP tried to use originally but without the keywords "nominal" or "convertible" there is no indication that they are talking about nominal rates.

In your PV formula, v is typically just 1/(1+i) and you just write PV = P(1 - v^n) / i . For example, if I have a payment stream of $100 at the end of each year for 3 years I can just write 100(v+v^2+v^3) which is nice and clean looking.

 

[Deleted member 4993]

Btw does anyone know of an online calculator for this sort of thing so that I can check my answer?

Your regular calculator may have buttons like FV, PV and PMT

 

[Lliam]

Denis, you associate "nominal rates" with a formula that does not contain nominal rates, the formula for nominal rates is different and now the OP is copying you -_-.

Thank you for clarifying this. Other than that, was the rest of my answer correct or were there still further errors?

Your regular calculator may have buttons like FV, PV and PMT

Unfortunately it doesn't appear to have those. :?


- Lliam

 

[Yogi]

The book I have to learn from is not very descriptive :neutral: THAT SOUNDS ABOUT RIGHT
@Yogi The answer $8702.85 is part of the given info in the problem.

Based on that..."The best interest rate he can find is 9.8%/a, compounded monthly" would be much more clear if written as "The best nominal interest rate he can find is 9.8% convertible monthly." So yes they are using nominal rates in the first part and you just use .098/12 and plug into PV like you did originally. I was just unclear if that was part of the problem because it seemed odd that they even include this information.

Part 2: "the interest rate is 9.8%/compounded annually."
In your book this means "convertible annually" so it is still using nominal rates and instead of dividing by 12 we divide by one.
.098/1=.098 effective annually. There are 3 yearly payments of 280*12, so your PV formula becomes:
3360[1-v^3]/.098=$8,385.33 where v=1/1.098

Now that we know your text was actually talking about nominal rates we didn't need the effective annual->effective monthly formula which I originally included. Does this answer make sense? We are now compounding interest far less than before so the value should be less than it was in part 1 which it is. Maybe thats why they included the convertible monthly answer so you can compare them.

From an actuarial science perspective the wording of that problem is pretty horrendous. I would have wrote the WHOLE problem as: "Nader plans to buy a used car. He can afford to pay $280 at the end of each month for three years. Determine the present value of these payments if the nominal interest rate is 9.8% convertible annually."

Oh, Texas Instruments BA-35 Solar is great for these and not that expensive.

 

[Lliam]

THANK YOU YOGI!! :mrgreen:

You have made the question so much simpler then it seemed! I totally understand it now!

One of the things that had me confused was that, by the way the question was worded, I though the value would be more than part 1.

The way you explained it makes so much more sense. You're a life saver!

Oh, Texas Instruments BA-35 Solar is great for these and not that expensive.

I will keep that in mind the next time I'm at Staples

THANKS!

Lliam

 

[Yogi]

Maybe this example can clear some confusion:

Deposit $100 into an account that pays an effective interest rate of i=5% a year. Suppose that after each quarter of a year the interest earned is withdrawn.

At time 0 we have $100.
At time 1/4 we accumulate 100*1.05^(1/4)=$101.23, withdraw the interest $1.23 and we have $100 again.
At time 1/2 we accumulate 100*1.05^(1/4)=$101.23, withdraw the interest $1.23 and we have $100 again.
At time 3/4 we accumulate 100*1.05^(1/4)=$101.23, withdraw the interest $1.23 and we have $100 again.
At time 1 we accumulate 100*1.05^(1/4)=$101.23, withdraw the interest $1.23 and we have $100 again.

The total interest earned in this way is 1.23*4=4.92 so the interest earned over the year is 4.92%. This is the nominal rate of interest convertible quarterly and can be written as i superscript(4) and pronounced "i-upper-4". Divide it by 4 to get the quarterly effective rate 1.23%
============================
So basically quarterly effective-->annual effective 1.0123^4 - 1 =.05
------------annual effective-->quarterly effective 1.05^.25 - 1=.0123 (opposite of above)
quarterly effective-->nominal rate convertible quarterly .0123*4=.0492
nominal rate convertible quarterly-->quarterly effective .0492/4=.0123