Loan math problem
The price inflation isn't a percentage, it's a fixed amount. If the dealer adds $R to the price of both the trade-in and the purchased car, then everyone ends up with the right amount of money in their pockets. So your formula should be: D + T + R = 0.1 * (P + R) "Lachlan has $600 cash and a car worth $500. That's $1,100. The new car is priced at $21,800. Lachlan needs a loan for $20,700. However, the finance company insists that the buyer must pay a 10% deposit, which is $2,180. Lachlan only has $1,100, so no loan.  The car dealer wants to make a sale, so suggests some tricks. The car dealer could buy Lachlan's old banger for $1,500 instead of $500, and sell the new car for $22,800 instead of $21,800. Doesn't make a difference to the dealer, he gets the same amount of cash. Now Lachlan has $600 cash and $1,500 for his car or $2,100 in total. He needs 10% of $22,800 as deposit which is $2,280. That's not quite there but you see how the principle works. Lachlan is about $200 short. So the dealer adds $1,200 to both car prices. Lachlan has $600 cash and a car ""worth"" $1,700, total $2,300. The new car is sold for $23,000 requiring a $2,300 deposit which works out exactly.  How could we have found the right amount without guessing?  Lachlan had $1,100. The new car costs $21,800. The dealer increases both prices by x dollars. Lachlan has now $1,100 + x deposit. The car now costs $21,800 + x. The deposit should be 10%, so $1,100 + x = 10% of ($21,800 + x) = $2,180 + 0.1 x.  $1,100 + x = $2,180 + 0.1 x : Subtract $1,100 x = $1,080 + 0.1 x : Subtract 0.1 x 0.9 x = $1,080 : Divide by 0.9 x = $1,080 / 0.9 = $1,200 The dealer inflates the cost of the new car and the value of the old car by $1,200. Now that's the theory. In practice I don't know how the finance company feels about this, and if they would be happy if they found out."
"Despite having a math degree, I basically only use basic algebra/probability/calculus on a day to day basis as my career has gone a different direction away from the modelling/quanty stuff.  Some fun reading:  * The SABR Model - [SABR/LIBOR Model](http://www.amazon.com/SABR-LIBOR-Market-Model-Interest-Rate/dp/0470740051) * Shevre's Stochastic Calculus for Finance - [Book 1](http://www.amazon.com/Stochastic-Calculus-Finance-Binomial-Textbooks/dp/0387249680/) &amp; [Book 2](http://www.amazon.com/Stochastic-Calculus-Finance-Continuous-Time-Textbooks/dp/144192311X/)  One of the big 'hard' problems is calibrating a swap curve w/ what's known as the 3s6s Basis. As a number of true quants have said to me it is a ""non-trivial problem"". Its basically trying to match two curves with different compounding over a number of different knot points.  SABR Model, listed above, is all about calibrating and figuring out how the current rate enviroment is behaving, is it normal or lognormal? What is the blend between the two, how do you know when you are in a different environment etc. Can rates go negative?" You have to wonder, don't you? Is he leaving because he finally realises the business is entirely a scam? That it's nothing more than a payday loan scheme, charging 25% interest, but the loan comes from the driver!  They effectively borrow large sums of money (which are car maintenance costs deferred for the future) from themselves, and have to work like a dog to do it.  A payday loan is a much more ethical thing, and not hidden behind math that Uber drivers are incapable of performing. Even if you ahve a very efficient car, the most you can ever make driving fro Uber is 6 cents a mile, net profit.  If you think 6 cents a mile is an equitable payment, then by all means, become an Uber driver! Driving 100 miles will net you $6.  On the odd occasion I have had to use my car for work, the accepted rate to charge it out at (just to cover the running costs!) is 90 cents per kilometre, or $1.45 per mile.  That figure is what it actually costs to run a car. It does not include any payment for actually driving the car. Even post meltdown, there are banks that will lend money based on a low loan to value, so 50% might not be a problem. But such loans come at a price. The current 30 year fixed rated is 4.5% or so, but you might see quite a bit higher than this on the loan. "There are two solutions. One is financially better, the other is psychologically better.  Financially better: You should pay off loans in the order of interest rate, so the 9% first, then the six percent, then the 3-4%. If you pay back $1000, the first one saves you $90 a year in interest, the second saves you $60, the third $30-$40 a year. Every year until everything is paid back.  Psychologically better: Some people have psychological problems paying back debt, and it is better for them to pay back small loans first. If you feel better having two loans instead of three, and feeling better makes you able to save money and pay back the other loans, while having three loans would make you depressed and unable to make savings - pay back the smaller loan.  If you don't have that kind of problem, use the ""financially better"" method." "The formula you are looking for is pretty complicated. It's given here: http://itl.nist.gov/div898/handbook/eda/section3/eda3661.htm You might prefer to let somebody else do the grunt work for you. This page will calculate the probability for you: http://stattrek.com/online-calculator/normal.aspx. In your case, you'd enter mean=.114, standard deviation=.132, and ""standard score""= ... oh, you didn't say what you're paying on your debt. Let's say it's 6%, i.e. .06. Note that this page will give you the probability that the actual number will be less than or equal to the ""standard score"". Enter all that and click the magic button and the probability that the investment will produce less than 6% is ... .34124, or 34%. The handy rule of thumb is that the probability is about 68% that the actual number will be within 1 standard deviation of the mean, 95% that it will be within 2 standard deviations, and 99.7% that it will be within 3. Which isn't exactly what you want because you don't want ""within"" but ""less than"". But you could get that by just adding half the difference from 100% for each of the above, i.e. instead of 68-95-99.7 it would be 84-98-99.9. Oh, I missed that in a follow-up comment you say you are paying 4% on a mortgage which you are adjusting to 3% because of tax implications. Probability based on mean and SD you gave of getting less than 3% is 26%. I didn't read the article you cite. I assume the standard deviation given is for the rate of return for one year. If you stretch that over many years, the SD goes down, as many factors tend to even out. So while the probability that money in a given, say, mutual fund will grow by less than 3% in one year is fairly high -- the 25 - 35% we're talking here sounds plausible to me -- the probability that it will grow by an average of less than 3% over a period of 10 or 15 or 20 years is much less. Further Thought There is, of course, no provably-true formula for what makes a reasonable risk. Suppose I offered you an investment that had a 99% chance of showing a $5,000 profit and a 1% chance of a $495,000 loss. Would you take it? I wouldn't. Even though the chance of a loss is small, if it happened, I'd lose everything I have. Is it worth that risk for the modest potential profit? I'd say no. Of course to someone who has a billion dollars, this might be a very reasonable risk. If it fails, oh well, that could really cut in to what he can spend on lunch tomorrow." But how do I bring the initial deposit into the equation? Basically, you can't.  Unless you combine two different formulas from Math of Finance into a single expression. The single initial deposit of $1000 will compound for 20 years at 5% compounded annually. The final amount for this part of the deposit will be: V1 = 1000 x (1.05)^20 In addition the series of 20 payments will be an ordinary annuity with a regular payment of $100, with the value on the occasion of the 20th payment given by:  So the final total amount in the account at the end of 20 years will be the sum of these two values... What you are looking here is the cost of capital, because that is what you are effectively giving up in order to invest in those loans. In a discounted cash-flow, it would be the *i* in the denominator (1+*i*). For instance, instead of purchasing those loans you could have lent your money at the risk-free rate (not 100% true, but typical assumption), and therefore you are taking a slight risk in those loans for a higher return.  There are several ways to compute that number, the one most often used would be the rate if the bank were to lend that money. In this way, it would be the Fed Funds rate plus some additional risk premium. Not that I doubted everyone's assumption but I wanted to see the math so I did some spreadsheet hacking. I assumed a monthly payments for 30 years which left us with total payments of 483.89.  I then assumed we'd pay an extra $200/month in one of two scenarios.  Scenario 1 we just paid that $200 directly to the lender.  In scenario 2 we set the extra $200 aside every month until we were able to pay off the $10k at 7%.  I assumed that the minimum payments were allocated proportionately and the overpayments were allocated evenly.  That meant we paid off loan 5 at about month 77, loan 4 in month 88, loan 3 in month 120, loan 2 in month 165, and loan 1 in month 170. Getting over to scenario 2 where we pay $483.89 to lender and save $200 separately.  In month 48 we've saved $9600 relative to the principle remaining in loan 3 of $9547.  We pay that off and we're left with loan 1,2,4,5 with a combined principle of about $60930.  At this point we are now going to make payments of 683.89 instead of saving towards principle.  Now our weighted average interest rate is 6.800% instead of 6.824%.  We can calculate the number of payments left given a principle of 60930, interest of 6.8%, and payment of 683.89 to be 124.4 months left for a total of 172.4 months Conclusion: Scenario 1 pays off the debt 3 months sooner with the same monthly expenditure as scenario 2.