Help with an equation

[Slaine]

Have a look at the attached little graphic.

Can you tell me if this can be solved for 'r' (ie r= ....)

and if so, what is the answer.

Many thanks

[Guv]

It seems too messy to solve for r analytically, although maybe it could be done. The expression is suggestive of a financial formula, which might be solved via a good hand calculator.

If I had to know the value, I would use the Newton-Raphson successive approximation approach, which works as follows.

• Function(r) = Messy expression I forgot.
  
• Determine Derivative(r) = Some messy expression.
  
• Guess at an a solution.
  
• NextGuess = LastGuess - Function(LastGuess)/Derivative(LastGuess)
  
• Use NextGuess as LastGuess, and compute a better NextGuess.
  
• Keep it up until some NextGuess is almost equal to the LastGuess

There are some examples of this method posted at this forum. It is a common method of finding Square Root (or higher roots) with a computer. The examples show it being used for square root.

Do a search using one or more of the following as search keys: Newton, squareroot, Raphson, approximation, successive.

[Guv]

BTW: If I were to try solving it analytically, I would do some algebraic manipulation to get all the terms involving r on one side of an equation. Then take ln (natural log of both sides) and see if it helps.

The above will not help much if you cannot get all multiply and divide operations on one side of the equals sigen.

[NotLKH]

Does that equation relate to anything in particular?

I'd suggest trying some trig substitution, such as given a right
triangle whose two sides are 1 and sqr(r), then its Hypotenuese
is sqr(1+r). In referance to the angle between the side whose
length = 1 and the hypotenuse , then r = Tan^2, (1+r)^a is
sec^2a, and so on, then manipulate from there.

-Lou

[NotLKH]

BTW, What does the Period in your equation signify?

[mathmaster]

Guv said it looks too messy to solve analytically,
so being a very messy math major i tried...

i messed with the equation and got to this. mleh.

i could not solve it, but some equations came up: there are three ways this can be solved for r, as far as i found

r(r+1)^a * RV - F(r+1)^b * RT + r(aFR^2 * T - ES) + FRT = 0

OR

(r+1)^b - arR-1 = [infinity]

OR

(r+1)^b - arR-1 =- [infinity]

last two look like they are not the solutions you are looking for, because if the other vars are held constant, implies that r -> [infinity]

looking at the first one again, analytically solving that one gives us nothing as it cannot be resolved. Even MathCad bailed out.

good luck with approximations

Cheers,

Tom Kartmazov

[Slaine]

Thanks for your replies.

Guv:

I am currently using successive approximation to calculate r but just wondered if there was an easier method.

It is in fact a financial calculation used to work out the Nominal Interest on some car finance.

I am calculating a periodic rental based on an set rate, but the NPV Input has some bonuses applied to it. The above equation is to calculate what interest rate would need to be applied, to get the same periodic rental if the bonuses were not applied. So in effect

NPV - Bonus @ rate y1% over x years gives a rental of z

So what does rate have to be so that:

NPV @ rate y2& over x years gives same rental z

[NotLKH]

Originally posted by Slaine
..................
NPV - Bonus @ rate y1% over x years gives a rental of z

So what does rate have to be so that:

NPV @ rate y2& over x years gives same rental z

Your equation doesn't seem correct.

1) it seems that a and b are the years over which your rates will apply. BUT you state that they will both be the same time period. so that means a = b = x. yes?

2) r appears to be your interest rate. BUT, you claim there are 2 interest rates, y1 and y2. where are y1 and y2 in your equation?

3) What does TFR Mean? ES? RV.? AR?

-Lou

[Slaine]

Your equation doesn't seem correct

This is quite probable!

I am putting these together from a number of sources including -digging them out of a nasty spreadsheet, interpreting some source code, and working out (guessing!) what a program is doing based only on the results it shows. It's a right pain in the butt.

Anyway here is more information.

TFR = True finance Rental (The rental calculated for the lease when the bonus fingures are included)

ES = Sales price of the vehicle, excluding VAT.

RV = Residual Value of the vehicle (The value of the vehicle at the end of the contract)

AR = Number of advance rentals made (a contract might be over 36 months, but in the first month the customer pays 3 months of rentals, leaving 33 outstanding).

r is the new interest rate I want to calculate.

I used a and b as replacements for some other equations.

I didn't expect so much interest, only a single reply saying it's not possible but if anyone is really interested in the challenge I am happy to post all the equations/calcs I have.

[da_silvy]

Slaine, do u play TFC?

[Slaine]

No. In fact I have no idea what TFC is!

[sql_lall]

I'll try, buty probably fail where others have failed before me
Also, for those who don't know:
THE . (Full stop) STANDS FOR MULTIPLY!!!!
I.e. RV.(1+r)^a = RV times (1+r)^a

And yes, i think that a = b, otherwise it seems odd to have (1+r) to the power of a AND b in different places.