The new position is different from the previous position.   On paper, my role is much the same, but in practice the mode of operation has significant differences.  Whereas the previous agency was a lightly-staffed satellite office of a purportedly trans-continental but seemingly slipshod operation, the present agency is the worldwide headquarters of a major global advertising powerhouse.  From the windows, one side has a beautiful view of mid-town perched on a hill overlooking the downtown section of the city.  The other side looks across the Hudson river to New Jersey.  For such a magnanimous building, the agency’s projects seem surprisingly small, and the budgets, based on as much as I am able to infer about them, reflect this.

Whereas every creative decision at the previous position went through a sometimes unhealthily intense debate with the Creative Director and Art Director, decisions in this firm are mostly passed like radon gas unnoticed without smell.  Perhaps a behemoth advertising firm has difficulty adapting to the Internet era.

The digital group in which I operate consists of a handful of people at best: three Project Managers of one sort or another who sit together at a series of giant desks stacked side by side as far as the eye can see in the main open area.  These people keep calm and organize periodic meetings with up to three Creative Directors, each of whose office has a more stunning view than the next, to extract the wisdom to be shuttled over to two or three Art Directors, who sit in slightly more sheltered sections of the open area.  The Art Directors produce pretty graphics representing the visionary concepts lavished upon them by the Creative Directors (in theory).  All work must pass approval from the Account Managers, who must ultimately ensure that the client is happy and that the client’s strategic goals are met by the work.  None of the personnel are used to working with someone of my obscure skills.  I am a sort of bridge between the Creative personnel, the Project Managers, and the Account people, stepping on the tips of each of their toes until they stop bothering to shine their shoes.

Only the Project Managers appear to devote their entire time to Internet projects.  The others have decidedly non-interactive backgrounds and tendencies.  Everyone seems to be a Vice-President of This-And-That-Other-Thing, although they perform roles no different from non-executive production staff I have encountered in other firms.  Each bathroom is replete with Scope mouthwash, hand lotion, and gentle facial tissues.

As it nears almost a month that I have been helping them, certain aspects of the business are becoming clear.  There are a handful of big clients who must fund the majority of the operation.  The Internet group could not be profitable, so this must mean that there are major accounts in other media, such as print and television, pulling the train along – areas of the business to which I have little-to-no exposure.  Judging by the sounds I hear while in the toilet, I am one of the few people to use the readily-available disposable splash-suppressing sanitary toilet seat covers with folded annular and bridging inner portions.

Because they are owned by a French parent organization, and, wink-wink, we all know the French have a different outlook on labor, or so I am told, the accounting system to determine the operating costs of any account (i.e. client project) is purportedly based on net revenue divided by the number of employees.  Since outside vendor companies are not considered employees, a variety of accounting tricks of using employees vs. outside contractors ensue.  At the moment, I am hired as an employee, although I may have to switch to using my corporation since I have recently begun working on an in-house project that is not bringing in any revenue.

As I was one day early on in this job instant messengering with a disgruntled employee from my previous client, I found myself recommending to them that they switch to a larger agency, where I said with new-found surprise, there is less politicking.  Having been there longer, I now know that the atmosphere in this agency is politically toxic.

The agency’s head Creative Director left the job during my first week.  People who live a life in these types of environs apparently consider such events to be monumental.  He asked his creative underlings out to a steak dinner – all except one or two.  One of the snubbed underlings, who happens to be the Art Director I am working most with, sits directly across from a Copywriter every day of the week, all year long.  They are very close and collaborate on everything.  Copywriter J got quietly up to leave for the lunch invitation without saying a word to Art Director J.

Art Director J came to confide in me (I was a safe confidant given my independent status) wide-eyed in some sort of subdued fluorescent tube Art Director rage that it was unconscionable for Copyright J to have sat across from her and not said anything about a lunch invitation to which everyone except Art Director J was invited.  I told her to let it go – it’s hard for me to fathom the snub and even harder to sympathize with the outrage.

Another week passed before L, a more junior Copywriter, but senior in personality and wit, called me in that whiny American sort of outrage via the internal line (a clunky-looking office telephone that some people are apparently accustomed to using) to complain about the direction the project we were working on together was taking.  I happen to agree with her that the guidance from our assigned Creative Director is more-or-less worthless, and the Art Director was producing design work that did not adhere to my plans and did not make the project seem exciting.  But I’m not sure I was able to satiate her thirst for outrage and vengeance.

But finally, a few days later, we had the big presentation (via clunky telephone) with the client… ten or so of us in a room. Account Manager A asked Creative Director J who should to talk the client through the ideas for the project. Creative Director J, clearly not thinking about much beside the buttons on his cell phone and the good-standing of his hair coiffure, indicated with a hand gesture of distraught that I was probably the one to do so.  This might have made sense in a less politically charged environment, since the flow of ideas is, after all, my specialty.  Account Manager A immediately caught his faux-pas and hurriedly asked Art Director J to do the talking to the client, as if giving a cupcake to a sobbing child.

For several months now, I have been keeping an Excel document especially to track my hours at these types of positions.  Each client has its own designated sheet, where I keep a log of hours and fees for each week of work.  Immediately upon arrival in the morning, I update the starting time for the day.  Immediately after lunch (and sometimes before), I enter in the length of my break.  Before departure, I update the timesheet to reflect total hours worked for the day.  In this manner, I am aware of exactly how much money I am earning at every leg of the day, every day of the week, every week of the year, and ultimately for the year as a whole.  When I send an invoice at the beginning of each week, the date, invoice number, and amount is tallied in the Excel document.  When the client pays the invoice, I enter bold text “Paid” next to the invoice number.  A final sheet, mislabeled “Net Worth”, automatically updates with every new entry to reflect my total earnings this year.

The office has an atrium with a skylight, a jukebox, some circular tables, and a small coffee shop run alternate days by ghetto tall Showannah and a bubbly tall Latina woman.  Showannah has decided that I like green tea in the morning.  And something about being someone who has an assigned drink appeals to me, so despite my preference for non-caffeinated beverages, I often have green tea in the morning.  The afternoon is peppermint tea time, sometimes with an oatmeal raisin cookie.  The atrium is almost always empty.

The building has a cafeteria four floors below mine.  I have not yet ventured there.

Today, one Project Manager asked me for a critique of the designs produced by one of the Art Directors.  All in all, the design left a lot to be desired.  The site is obviously going to be boring and static, although Project Manager K explained that this was not likely to change given budget and timing constraints..  When I was finished with my feedback, Project Manager K said, “Thanks, I’m so glad we have someone like you around.”  Project Manager J, sitting on the other side of the desk looked up and exclaimed, “Fuckin’-a, Yes!,” followed by “I only wish the Creatives understood that.”

I said, “The Creatives will not understand.”

In an earlier post, I exposed the math of mortages in layman’s terms.  Mortgages calculations are done using the formula for calculating Net Present Value, meaning the value today of money dished out tomorrow in recurring payments to the bank.  However, there is more than just the pure interest rate of a mortgage involved in buying a home.

The economy does not stand still for the few decades it takes you to pay off your mortgage.  Inflation means that your a dollar today will be worth less tomorrow, as the price of goods and services generally increases over time.  Plus, property takes effort (and money) to keep up.  There are property taxes, condo fees, maintenance costs, and utilities to pay.  Things that you, as a renter, have never before heard or worried about.

The assumptions

Here is a link to download the Excel file for these calculations.

To do any sort of estimate, certain assumptions have to be made.  For the following calculations, here are our assumptions:

Obviously in a volatile market, these assumptions will not be reliable.  For that reason and others, we need to set limits on how far into the future we predict given these shaky assumptions.  Shorter-term predictions are more likely to hold water.

The lost decade

Here is the 10-year cash flow resulting from these assumptions:

Given the declining real estate market for the first two years (the “property appreciation rate” row), coupled with inflation (assumed to be a steady 3% in the initial variables), we’re predicting that selling the property within the first 10 years will be a loss.  Although the nominal value of the property (the “property value” row) exceeds the buying price in year 5, selling it would still lead to a loss.  This is because the nominal property value does not take into account the time value of money.

The cost of goods and services increases every year, meaning that a dollar today is more valuable than a dollar tomorrow.  This is the definition of inflation – things cost more in the future.  So even though the property value exceeds its buying price in 5 years doesn’t mean it’s necessarily worth more in terms of today’s dollar value.  In fact, if we assume 3% inflation, even with the real estate market on the rebound after 2 years of negative gains, the value of the property in terms of today’s dollars (the “present value of property” row) at the end of year 5 is still only $443,000.

So if we were to sell it, even after 10 years, we’d be taking a loss (an 11.24% loss to be exact, as shown in the “return on investment” row). In fact, with this model, it’s not until year 13 that we would be able to sell the property with anything resembling a profit.

Another thing to notice is that this model shows that our yearly expenses (the “cash outlay” row) for a property costing $500,000 are significantly more than our current rental situations.  The first year requires $88,000 due to all the costs of buying (equivalent to $7,333 per month).  Subsequent years hover around the mid-$50’s (equivalent to about $4,600 per month) for the first ten years.   To keep costs near what they are today would require purchasing a property closer to $300,000.

The second decade

The second decade sees the property start to gain positive value.

You can see that by year 13, the property’s present value, meaning its value in terms of today’s dollars, is worth $500,567, about what we paid for it.  So selling it at this point would break even.  However, this break-even point does not factor in all the fees and operating costs we would have put into the property during the first 12 years of ownership (all the stuff in red).  Those expenses, if counted, would still lead to a loss for several more years.

So after the second decade, we can optimistically be said to have barely broken even.

The third decade

In the third decade of ownership, when we’re in our mid-60’s, we see some real gains.  Assuming steady inflation of 3% and steady real-estate appreciation of 5% (very shaky assumptions), we are no longer working for our home, but our home is working for us.

At the end of 30 years, we are full owners of the property, having paid off the private loan in year 10, and the mortgage in year 30.  Things are starting to look up.  If we were to sell the property at the end of this decade, we would have made a modest profit of (38%, or $194,000 in present dollars), and have had a place to live for 30 years.

Of course, the years after the 30th year, assuming the world is not overrun by vegan environmentalists with the sane people living in yurts by that time, would be the most valuable in which to hold onto the property….

Conclusions

If my calculations are at all in the right order, then we cannot afford to buy anything more than $300,000 if we want to keep our monthly outlays similar to what they are today in our rental situations.  If you plug in “$300,000″ in the “property cost” assumption, you’ll see that our cash outlays would average around $35,000 for the first bunch of years, which is not a major change from today.   If we assume a “property cost” of $500,000, then we get into cash outlays higher than $55,000, which require significantly more commitment.

Most questionable of my assumptions are the inflation and appreciation rates.

Another important factor is the “private loan” repayment.  I currently have the assumption that we borrow $200,000 from private investors and repay it within 10 years at 3% (to keep up with inflation).  If we get rid of the private loan and take out a bigger mortgage instead, then our cash outlay actually becomes more manageable, and might allow us to cover a more expensive property, albeit at a higher overall interest rate.  If you change the “private loan amount’ field, you’ll see the changes appear in the “cash outlay” row.

So we need to determine whether the assumptions and calculations are believable.

Di Fara is considered one of the best, if not the very best, pizza in the country.  People travel from all the states just to taste its drizzled oil greasiness.

The price for a plain slice is $4.  The price for a regular pie is $20, and a sicilian pie is $25.  Each pie has 8 slices.  Di Fara is open Tuesday thru Sunday, noon to 4:30pm and 6-9pm, a total of 7.5 hours per day, 6 days per week.

A best-case scenario

I’m going to come up with a best-case scenario for Di Fara’s gross sales.  To do so, let’s assume, on average, a pie takes a half hour to produce: 15 minutes of prep time, and 15 minutes of cooking.  While the pie is cooking, another pie can be prepared.  So there are effectively 4 pies produced every hour.  Those who have visited Di Fara will realize that this is a very optimistic average, considering that waits of 2 hours or more are a regular occurrence.

Let’s also assume that all pizza is sold by the slice, since individual slices are more expensive than slices by the pie, so this will boost our projected sales.  We’ll say the average slice is $5, even though this is far above a realistic average.

Furthermore, let’s assume Di Fara is open the full 7.5 hours per day, 6 days per week (Tues thru Sunday).  We all know that it is often closed intermittently, but assuming longer hours will again give us a best-case appraisal of the business.

To complete our analysis, let’s assume that for every two slices, a customer also purchases a $3 beverage.

So, here are our assumptions:

• Cost per slice: $5 / slice
• Cost per Beverage: $3 / beverage
• Beverage sales: 0.5 beverages / slice
• Hours of operation per day: 7.5 hours / day
• Days of operation per week: 6 days / week
• Weeks of operation per year: 52 weeks / year
• Average time to prepare pie: 0.25 hours / pie
• Number slices per pie: 8 slices / pie

From this, we can calculate the number of slices sold per week:

Total number of slices produced each week = Hours of operation per day * Days of operation per week / Average time to prepare pie * Number of slices per pie
= 7.5 hours/day * 6 days/week / 0.25 hours/pie * 8 slices/pie
= 1440 slices/week

Gross pizza sales are:

Gross pizza sales = Slices per week * Cost per slice
= 1440 slices/week * $5/slice
= $7200/week

So Di Fara sells 1440 slices per week.  And for every 2 slices, they sell a $3 beverage, so total beverage sales each week are:

Gross beverage sales = Slices per week * Beverages per slice * Cost per beverage
= 1440 slices/week * 0.5 beverages/slice * $3/beverage
= $2160/week

Total gross sales per week are just pizza sales plus beverage sales.   So…

Total gross sales per week = Gross pizza sales per week + Gross beverage sales per week
= $7200/week + $2160/week
= $9360/week

Di Fara makes gross profits of, at best, $9,360 per week.  For the entire year, Di Fara would make 52 times this amount, for the 52 weeks in the year.

Gross yearly sales = Gross weekly sales * Number of weeks per year
= $9360/week * 52 weeks/year
= $486,720/year

So in a best-case scenario, Di Fara makes $486,720 in gross sales per year.

A worst case scenario

As we’ve already discussed, the total above is what we’re assuming to be a best-case scenario.  A very unrealistically great business.  The real number is probably significantly less.

Whereas in the best-case, everyone orders individual slices, our worst-case scenario will have everyone ordering plain pies.  Maybe Di Fara only produces two pies per hour…  And what if people only bought one beverage for every 4 slices, and Di Fara was forced shut for 4 weeks out of the year due to health code violations and knee surgery?

If we were to change our assumptions to reflect a worser case scenario, we might have something like this:

• Cost per slice: $2.50 / slice
• Cost per Beverage: $3 / beverage
• Beverage sales: 1 per every 0.25 beverages/slice
• Hours of operation per day: 7.5 hours / day
• Days of operation per week: 6 days / week
• Weeks of operation per year: 48 weeks / year
• Average time to prepare pie: 0.5 hour / pie
• Number slices per pie: 8 slices / pie

That would give us:

Total number of slices produced each week = Hours of operation per day * Days of operation per week / Average time to prepare pie * Number of slices per pie
= 7.5 hours/day * 6 days/week / 0.5 hours/pie * 8 slices/pie
= 720 slices/week

Gross weekly pizza sales = Slices per week * Cost per slice
= 720 slices/week * $2.50/slice
= $1800/week

Gross weekly beverage sales = Slices per week * Beverages per slice * Cost per beverage
= 720 slices/week * 0.25 beverages/slice * $3/beverage
= $540/week

Total weekly gross sales = Gross weekly pizza sales + Gross weekly beverage sales
= $1800/week + $540/week
= $2340/week

Gross yearly sales = Gross weekly sales * Number of weeks per year
= $2340/week * 48 weeks/year
= $112,320/year

So the worst-case scenario has Di Fara making gross profits of $112,320 per year, about four times less than our best-case scenario.

Conclusion

The reality probably lies somewhere in the middle.  If we take the average of best- and worst-case scenarios, we’d end up with an estimated gross yearly sales of around $299,520.

Gross sales do not take into account labor costs, supply costs, rent, taxes, waste, insurance, etc.  At any given time there are 2 or 3 people working in the pizza shop: Dom Demarco, his son, and/or his daughter.

Rates for net profit after all costs in the food industry are very low, usually in the low single digits.  But let’s say Di Fara, being a cheapo family business, has net profit somewhere between 10% and 30%.  Thirty percent would be considered amazing in the restaurant business, but maybe Dom is a genius after all and pays his children nothing while shorting his suppliers.

This conservative estimate would put Di Fara’s net profits between $11,232 in the worst-case, and $146,016 in the best-case.  If we are to take an average of these in order to get a best reasonable guesstimate of net profits, we’d have Di Fara’s earning a net profit per year of $78,624.

Conclusion: Dom Demarco is a fool

Dom Demarco of Di Fara

Risk in bonds & loans

A bond holder takes on multiple levels of risk.  In the first place. there is the possibility that the amount you loaned the bond issuer, and the interest payments you expect, will never be paid back (i.e. credit risk).  Second is the possibility, explained in great detail in a previous post, that the interest rates will move in a direction that reduces the value of your bond (i.e. interest rate risk).  Furthermore, given that some bonds may be bought in a foreign currency, there is the risk that the foreign currency will lose value, bringing the value of your bond down with it (i.e. curency risk.)  Lastly, a bond or loan always leaves open the risk of having an unexpected interruption in how much money you are earning compared to how much you owe (i.e. funding risk).

Investors in bonds and loans therefore look for ways to mitigate each or all of these risks.  Two mechanisms of hedging this exposure to risk are interest rate swaps and credit default swaps.

Interest rate swaps

At its purest, an interest rate swap is exactly as it sounds: a trade of interest rates on two separate assets between two separate parties.

For example, let’s say Nina has a bank account with Homecrest Savings Bank that accrues a fixed rate of 2% interest each year on her unemployment benefits.  Meanwhile, Homecrest Savings has loaned money to Amos to fund the purchase of his Master’s degree at Touro College at some variable rate of interest that follows a standard bank rate, such as LIBOR, or the federal funds rate.

Since the interest Amos is paying to Homecrest Savings is variable, while the interest the bank pays Nina is fixed, there remains a risk of some kind of asset/liability mismatch, or other complications for the bank as a result of having different interest rates on the various activities.  It would be better for Homecrest Savings to have better matched interest rates on its assets and liabilities, so it can be sure that payments from Amos will sufficiently cover the obligations it has to Nina.

So the bank may decide to trade the variable interest it receives from Amos for a roughly equivalent stream of fixed rate payments coming from some other financial instituttion, let’s call it Midwood Pension Partners.  The two banks will swap interest rates, thereby creating a fixed-for-floating interest rate swap.  Thus Homecrest Savings is assured of being able to always cover payments to Nina, while Midwood Pension Partners takes on the risk, and thereby the opportunity, inherent in a variable rate interest loan.

In other words,Homecrest Savings has traded away its interest rate risk.  Now if Amos defaults on his loan, it is Midwood Pension Partners that suffers the loss.  Homecrest Savings is still assured its fixed stream of payments from Midwood Pension Partners.

Credit default swaps

The interest rate swap example above showed how a bank can isolate and remove the interest rate risk from the other risks of an asset, thereby leaving only the credit and funding risks.  A credit default swap goes one step further and allows a financial institution to also remove the funding aspect of an investment, thereby leaving only the credit risk.

Let’s say Daniel A has a 20 year $10,000 bond issued by the Iraqi Treasury back when Bush was President.  Daniel isn’t so confident anymore that the Iraqi Treasury will actually be able to pay back the interest and principal on that bond, given that Obama keeps talking about withdrawing troops.  In other words, Daniel sees a lot of credit risk there.

Amos is an opportunist, and thinks that either Obama will never truly commit on his promise to pull the troops, or that by the time he actually does, the Iraqi Treasury will be secure and the oil will flow.  So he’s willing to take on a little bet with Daniel: if Daniel pays him a fixed fee every quarter, he’ll insure Daniel against the potential default of the Iraqi Treasury on its bonds.  This is a credit default swap:  a fixed stream of payments in return for an insurance policy that covers the losses incurred if (and only if) the debt issuer defaults.

Ramon hears about this deal, and thinks that both his brothers are fools.  But he smells free money.  So, although he doesn’t actually hold any Iraqi Treasury bonds, he too agrees to pay Amos a fixed fee in return for insurance against the default of the Iraqi Treasury.  Ramon pays a small fee every quarter to Amos and if the Iraqi Treasury defaults on its bonds, he has the potential to win up to $10,000.  Ramon has no funding risk because he hasn’t ever bought an Iraqi bond, so he has almost nothing to lose.

In other words, the protection buyer of a credit default swap does not actually have to own or trade the underlying reference asset.  A totally  unrelated party will still get a full insurance payout if the bond issuer defaults. The protection seller must be pretty sure that the bond issuer will not default, or else they are on the hook for a lot of money in payouts.

Bond issuers, be they corporations or countries, have credit ratings, issued by independent organizations like Moodys and Standard & Poors.  These purportedly indicate the general creditworthiness, and therefore the credit risk, involved in any issuer’s bonds.  The price of a credit default swap depends highly on the rating of the issuer of the underlying bond.

There is an active secondary market for credit default swaps.  If the bond issuer’s credit rating becomes worse, the price of credit default swaps referenced to that issuer’s bonds will increase.  This is because the chance of that issuer defaulting, and therefore the chance of the holder of a credit default swap being the recipient of a huge insurance policy payout, goes up if the reference entity looks like it’s on the way down.

The more the issuer’s credit rating drops, the higher the price of a credit default swap on the secondary market.  And conversely, the more an issuer’s credit ratings increase, the lower the price of the credit default swaps referencing that issuer.

So with every successful suicide attack of Al Qaeda in Mesopotamia against the Iraqi government, the price of credit default swaps referencing Iraqi Treasury bonds go up in value.

Recall that purchasing a bond is like giving a fixed-interest loan to a company or government. The price of bonds is determined by a variety of factors, including how likely the borrower is to pay you back (i.e. the credit rating), how long you will earn interest on the bond before it is paid back (i.e. time to maturity), how much interest the bond will earn (i.e. the coupon), the amount that will be paid back at maturity (the face value), and any other market forces at play at any given moment. But, all else being equal, interest rates, play the key role in shaping bond prices.

If you had to choose between two similar bonds with different interest rates, you’d take the bond with the higher interest rate, of course.   Take for example, a 10 year bond with a face value of $1,000.

At 5% interest, it will earn you $50 per year.

$1,000 x 0.05 = $50

At 10% interest, it will earn you $100 per year.

$1,000 x 0.10 = $100

Why would anyone buy the bond with the lower interest rate?

People selling bonds try to make them attractive to buyers.  On the secondary market where bonds are re-sold, a seller who is not the original issuer of the bond cannot change the interest rate, maturity date, or the face value of the bond – these things are set only by the issuing institution.  But a re-seller can change the price they are asking for the bond, which may make it worthwhile to a buyer.

To compare the “worthwhileness” of two bond, it’s useful compare their current yields, which is a way of evaluating the return on investment.

current yield = coupon / spot price

In the example above, the 10 year $1,000 bond at 10% has a $100 coupon (the amount paid in interest each year).  This means the current yield (return on investment) is $100 / $1,000, or 10%.

The 5% bond has a current yield of $50 / $1,000, or 5%, which makes it obviously a worse investment.

But if the seller of the 5% bond offers it for only $500 (half its original price), that puts the yield of the 5% bond at $50/$500, or 10%.  This makes it just as good an investment as the 10% bond.

So by dropping the price of the bond, the seller is able to compensate for a less attractive interest rate.

Conversely, a seller who has a bond with a very high interest rate compared to the competition will sell the bond at a higher price than its face value.  Take the example where a seller has a 10 year $1,000 bond at 10%.  Imagine that all the other bonds on the market are only offering 5%.  The yield that the competition are offering is $50 / $1,000 = 5%.  The seller of the 10% bond can offer to sell her bond for $2,000, and she will still be offering an attractive 5% yield, which is in tune with the competition.

yield = $100 / $2,000 = 0.05

She is set to make a $1,000 profit.

And thus interest rates and bond prices are inversely correlated.

QED

For a home buyer, a mortgage loan is a loan that you pay back over time (in theory).  For a bank, a mortgage loan is an investment that earns a stream of interest income.

Let’s say there’s a beautiful house in historic Stony Point, Rockland County going for the bargain price of $120,000, that you can’t wait to buy while the market is down and interest rates are low.  You borrow $100,000 at 5% interest compounded monthly, repayable over 20 years.   From the bank’s perspective, the present value of their investment is $100,000, and it will produce a stream of revenue for 240 months.

To calculate the monthly payments you should expect, consider that you know the present value, the annual interest rate, and the total number of conversion periods.  The formula linking these all these variables together with a regular stream of payments is:

PV = R x [{1 - (1 + r)-n} / r]

where PV is the present value, R is the regular payment amount, r is the interest rate per conversion period, and n is the total number of conversion periods.

First we have to convert the annual interest rate into the interest rate per conversion period.  There are 12 conversion periods per year (one each month), since the interest is compounded monthly, so the monthly interest rate is r = 0.05/12 = 0.0042, or 0.4%.

Rearranging the formula to solve for R, the monthly payment amount, we get:

R = PV / [{1 - (1 + r)-n} / r]

Plugging in the numbers, we have:

R = $100,000 / [{1 - (1 + 0.0042)-240} / 0.0042]

and…

R = $662

Paying $662 per month for 240 months will eventually add up to $660 x 240 = $158,880.  The borrower is able to purchase the house, but is ultimately paying $58,880 more for the house as a result of borrowing money.  But this added cost is spread out over 20 years, which makes it more manageable.  From the bank’s perspective, the loan brings in $158,880 – $100,000 = $58,880 in revenue over 20 years, a 58.88% return.

Maybe you should go for a bigger house.  If you think you want to borrow $500,000, the only part of the formula that changes is the present value, PV.  The other part, [{1 - (1 + r)^-n} / r], known as the accumulation factor, won’t change unless the interest rate and number of conversion periods in your loan change. With r=0.0042 and n=240, the accumulation factor is equal to 151.

A $500,000 loan, also at 5% annual interest over 20 years, compounded monthly, would have regular payments of R = PV / accumulation_factor.  Plugging in he numbers, we have R = $500,000 / 151 = $3,311.  A $200,000 loan, with the same conditions, would have regular payments of $200,000 / 151 = $1,324.

Say you calculate that you can afford to make mortgage payments of $1,500 per month.  What price house should you buy?  At 5% annual interest over 20 years, compounded monthly, the accumulation factor is still 151.  The formula is still R = PV / accumulation_factor.  In this case, we want to solve for PV, so we rearrange the formula to be: PV = R x accumulation_factor.  Plugging in the numbers, PV = $1,500 x 151 = $226,500.

…putting money in the cookie jar

Putting money away is exactly what Obama doesn’t want you to be doing, although he’s far too tactful to say so directly.  But if you were the one to disobey El Obama’s non-orders, you may just be hiding rubles in the cookie jar.  As you can probably tell by his oversized pants, Obama doesn’t eat cookies… ever.

But if you, unlike purportedly-organic-produce-eating Barack, were to put away $1 every day in the cookie jar, at the end of the year, you’d have… $365 of course.  But with the possibility of inflation and cost of living increases each year, there are better things to do with your money.  Like…

…putting your money in a savings account

Now we’re doing a bit better. You’ve saved up $365 and realized that your cookie jar is better filled with Newman’s Own Organic Chocolate Chip Cookies.  So you’ve put the $365 into a 3% interest savings account, and you’ve even eaten some cookies, you devil…  And you earned interest! The formula for simple interest is:

SI = PV x r

Where SI is the simple interest earned.  PV is the present value (how much you deposit), and r is the interest rate.  So at the end of a year, the interest you earned is $365 x 0.03 = $11.  Congratulations!  You put your money to good use and earned $11 this year… not to mention the cookies enjoyed.

Inflation in the USA reduces the value of your money by something near 3% per year, on average.  So actually, all things considered, you just-about stayed even this year, and earned nary a cent.  Time to start thinking bigger by…

…earning compound interest in a savings account

Sure, you say.  I may have earned $11 last year, but that interest is going to be compounded, dollface!  What you mean is that the interest you made this year is going to be added to the principal amount you have in your account.  So next year, you’ll earn interest on the amount you deposited plus this year’s interest ($365 + 11 = $376), and the following year, you’ll earn interest on the interest you earned from that interest ($376 x 1.03 = $387), and after 10 years, you’ll be rich.  Poor sucker.

So, exactly how much will you have after 10 years?

FV = PV x [1 + (i / k)]n

Where FV is the “future value” of your money (e.g., the value after 10 years, in this example), PV is the present value (i.e., how much you deposit now), i is the annual interest rate, k is the number of conversion periods per year (how many times your money earns interest per year), and n is the total number of conversion periods.

In a typical savings account, your interest is compounded only once per year, so k = 1.  We want to know what the future value of your money will be after ten years, which in this case is 10 conversion cycles (ten interest cycles), so n = 10.  Our formula is thus FV = $365 x [1 + (0.03 / 1)]¹⁰

And that, my friends and fellow prisoners, is equal to $491. But you earned some compound interest!  To be exact, you earned:

CI = FV - PV

Where CI is the compound interest earned on top of principal amount you deposited.   FV is the future value, which in this case after 10 years we have just calculated to be $491.   PV is the present value, which is how much we originally deposited, i.e. $365.   So… after 10 years, you’ve earned interest of CI = $491 – $365 = $126.

As we established, with 3% interest, even if you had deposited $10,000, you would have just about kept up with inflation and earned almost nothing extra.. No matter how much you save, your 3% interest is doing nothing for you if inflation is wiping out your growth.  So what about…

…having an anxiety attack about inflation

Inflation is basically negative interest.  You can calculate the loss of value of your money over time due to inflation the same way you calculate its interest.

After one year of 3% inflation, your $365 has therefore lost $365 x 0.03 = $11 of value.  After 10 years, it has lost $126, using the same formula you used for calculating 10 years’ worth of compound interest, but using the rate of inflation (~3%, but varies each year)  instead of the rate of interest (also ~3% in our savings account examples).

If the interest you earn on your money is greater than the inflation rate for any given year, then you’re increasing the value of your money that year.  Otherwise, you’re just treading water at best.

With what we’ve already outlined, your children will be working at the pizza place in high school to pay off the hundreds of thousands of dollars in debt they will go through to get a decent education.  You should start thinking about…

…saving for your child’s education

So you’ve decided that, although your child, Wendy, will have to suffer the public education system, due to your negligence, you will at least offer with some sincerity to pay for her undergraduate degree at a state college.

New York state colleges go for about $20k per year, including room and board.  And, assuming your kid is one of the smart ones who get through undergrad in just 4 years, you’ll be shelling out $80k minimum.

Luckily for you, you don’t have kids yet.  It will be 20 years before your future kid goes to college. So you have plenty of time to start saving now.  How much should you put away in the bank right now in order to have enough to pay for her college?

FV = PV x (1 + r)n

Where FV is the future value, PV is the present value, r is the interest rate per conversion period, and n is the total number of conversion periods.

So the future value you want to have on hand in 20 years is $80k.  The annual interest rate, let’s say, is 10%, since you’ve figured out how to be very creative with your money, using bonds and stocks that earn more interest than than the standard bank savings account interest you had before.

So the future value, FV = $80,000, which is how much you want to have in the end; r = 0.10, which is the interest rate you can earn per year; n = 20, which is how many conversion periods (i.e., interest cycles) between now and when you want to have earned the money.

So, the formula above, in this case, looks like $80,000 = PV x (1 + 0.10)²⁰.  Solving for PV, we get PV = $80,000 / (1 + 0.10)²⁰.  In other words, PV = $11,891.  So if you invest $11,891 today at 10% interest per year, you’ll have enough to pay for your daughter’s undergrad education in 20 years.

But let’s be more realistic, and let’s say you can only find an investment of 5% per year… slightly better than a typical bank savings account.  That means PV = $80,000 / (1 + 0.0.05)²⁰.  With this interest rate scenario, using the same formula with an interest rate, r, of 0.05, the present value, the amount you have to invest today, is PV = $39,189.

Cancel that Tulum vacation this year.   But wait, you realize…. you won’t need the full $39,189 up-front.  Who says you need it all at once?  You can start making smaller payments every month that will contribute towards that final $80k goal in 20 years.  We’re talking a finite number of regular payments, not a lump sum.  You can still take that vacation.  You just need to start….

…contributing regular payments toward your kid’s education

So you’re poor and smart.  You’ve realized that you don’t need the whole shebang up-front.  You can put away savings every month until you save enough for the college expenses.   This is equivalent to a sinking fund.

You need to guarantee that 20 years from now, Wendy, your beautiful and chaste college-age daughter will be the lucky recipient of $80k to pay for college.  No doubt, she will fully appreciate the effort and sacrifices you’ve made in diligently saving all your profits of all your work every month in order to fund the development of her educated mind.

Let’s say you’ve again figured out how to earn 5% interest on your money, compounded yearly.  You’re getting better with your investments.  How much should you put away every month such that in 20 years you’ll have $80k saved up?  The formula linking future value, recurring payments, interest rates, and conversion periods is:

FV = R x [{(1 + r)n - 1} / r]

Where FV is the future value we are trying to achieve;  R is the payment you need to make each conversion period; r is the interest rate per conversion period.  and n is the total number of conversion periods.

In our case, FV = $80,000, the amount we want in the future; R, the recurring payment amount, is not known and is what we want to find out; r is 0.05, the annual interest rate, since we only have one conversion period per year; and n = 20, since we have 20 conversion periods: one for each year.   So the formula above looks like: $80,000 = R x [{(1 + 0.05)²⁰ - 1} / 0.05]

Rearranging this equation for R, we get R = $80,000 / [{(1 + 0.05)²⁰ - 1} / 0.05].  Solving this equation for R, we get R = $2,419  So you need only put away $2,419 every year at 5% interest for the next 20 years in order to have enough to pay for your daughter’s cheapo state college alcohol-hazed undergraduate education.

Since the interest on our money accrues yearly, not monthly, we just calculated how much we need to contribute to the fund each year.  We can always divide that number by 12 to get the necessary monthly contribution. $2,419 / 12 = $202 per month.

If we were able to find an investment that had conversion periods once every month, rather than once per year, then our interest would accrue 12 times per year.  We would have to modify our numbers a bit.  The more often the money accrues interest, the better.  With monthly conversion periods, the monthly interest rate, r, would be the annual rate divided by 12 months: 0.05 / 12 = 0.0042;  and n, the number of conversion periods would be 12 per year (one for each month), so n = 12 x 20 = 240.  The formula becomes: R = $80,000 / [{(1 + 0.0042)²⁴⁰ - 1} / 0.0042].  In this case, solving for R, the recurring payment every month would only have to be $194, which saves us $8 per month compared to the yearly interest payments.

Maybe you can afford to put away this amount each year.   Maybe you can’t.   Maybe you should start…

…putting away whatever pathetic amount you can muster to save each month

Let’s say you’re poor.  You can only afford to put away $100 per month towards your kid’s college education.   $100 per month is equal to $1200 per year.  How much will that measly amount be worth in 20 years?  Using the same formula as above, we have FV = R x [{(1 + r)ⁿ - 1} / r].

In our current example, we know the value of R, what we can contribute each year.  It’s $1,200.  And we want to figure out what FV, the future value, will be after 20 years of contributing this amount.  If we get an interest rate, r,  of just 5% per year, compounded yearly, we’ll again have one conversion period per year, so n, the total number of conversion periods, is 20.

Our formula is now: FV = $1200 x [{(1 + 0.0.5)²⁰ - 1} / 0.0.5].  Solving for FV, we get FV = $39,679.  So your deposits of $1200 every year will be worth $39,679 in 20 years at 5% interest.  This is about half of what we need to pay for our baby’s undergrad.  Pay double this amount, and Wendy will be ok.

If we can put away $500 per month, that equals $6,000 per year.  Using the same formula, the future value of $6,000 per year over 20 years, at 5% interest, is FV = $6,000 x [{(1 + 0.0.5)²⁰ - 1} / 0.0.5].  Solving this equation, FV = $198,396.  So putting away $6,000 per year for 20 years at 5% interest leaves your baby with $198,396 when it comes time for college.

Don’t worry, Wendy will be just fine.