Lender fees - at arms length or a transfer? Plus fee-or-no-fee?

I just want to make two points about the lender fee I've introduced recently in my discussion on understanding rates of interest.  First, the world is a diverse and unpredictable place.  There's nothing to stop a lender giving a loan at seemingly crazy rates (either very high seeming, or very now seeming).  The other way of coming at this is to say that, for any given rate of return, there's some possible market environment which make this rate of return understandable.  And even if it isn't understandable, human beings are free to agree whatever rates they want between them (absent any usury laws which may be in place, of course).  If a lender lends money because they don't need to charge for it, or because they're favourably disposed towards the borrower, then the transaction is more like a transfer and less like an arms length business contract.  

The second point I want to make is that you know nothing about how much of a fee, if any, is being charged if all you know about the loan is the rate of interest.  This seems like such a strange thing to say that I'll pause a while before explaining more.  I'm saying that if I told you that a man walked into a bar and asked for a one day lend of £100 and agreed a rate $r$ with a lender, then you the listener have no clear idea of how much of a fee the lender charged, if indeed he did charge anything at all.  I will explain this more in a later post.  Looking at a rate is often a great place to work out the fee charged by the lender, but as I will show, this can quite easily be thwarted by adverse economic conditions.

How to express a loan's costs

Version 17:

A man walks into a bar and asks the people in the bar for a lend of £100 for a year, at which point he'll return and give the £100 back plus some extra cash for their trouble.  A healthy show of hands among the customers shows strong interest.  He picks one, takes the £100 and walks out again. 

And here, finally, we have it.  The borrower makes it more worth the while of the potential lender by giving them back not only their original lend amount, but something extra.    Depending on the size of this payment for lenders, you would expect to see sometimes more and sometimes fewer hands go up to volunteer themselves as lenders.

The two ways in general of expressing this something extra is in cash terms (as if it were a fixed fee for a service rendered) and in terms of a rate of return (as if it were the promised return of some investment the lender made).   There are two dimensions to this.  First, do you express the payment as a cash amount or as a rate.  The argument for expressing it as a rate seems familiar to us.  We get to compare this fee with other fees, perhaps with different amounts on loan.  But this display / formatting issue is of course entirely superficial.  You still have the same promise of the same cash at the end.  The second dimension comes when you realise that for any given final cash payment, there must be a single upfront payment which means the same thing to you now.  If you present value the terminal payment, then the present value that results also, in  a sense, represents a slightly different contract which you could have entered into and perhaps felt the same about (or quite similar about - it isn't quite the same, of course, since you have the comfort of the fee in your pocket right away).  This same line of reasoning could result in two payments, one now, one at the end.  Or three - now, in the middle, and at the end.  Or indeed a cash payment every month.  Or every day.  Or every minute.  Or every instant.

Well, if that's the case, and you could chose which kind of contract to draw up between you the lender and that borrower, which one would you chose?  Not only is it pleasant to have the certainty of some fraction of your fee in your pocket as early as possible, but there's also a benefit in seeing that your borrower has maintained a decent payment schedule.  Ignoring all of the practical costs associated with daily days of reckoning, there's clearly an advantage in seeing the money return early and regularly.  What you're doing is receiving information of some sort about that borrower.  His monthly payments are all crying out to you 'this is a man you can trust'; 'he continues to not let you down'.  So you may prefer receiving your fee in instalments during the life of the loan for at least those two reasons (cash in pocket,  confidence building).  Also, which would you prefer - to find out that your borrower has no money to pay you back only at the end of the term of the loan, or at some point earlier?  Clearly, the earlier the better.  If the man is foolish with money, then you'd want to know this sooner rather than later because he still may possess some fraction of your precious £100 in the early days than at expiry, by which point he may have spent it all.  Yet another reason why it might be nice to receive the fee in regular instalments is because the lender himself may quite like to receive regular payments and this would be one way to achieve it.  And the reverse might be the case too - you don't want any income tax based payments in the intervening period, and are happy picking up a capital gain once at the end, for tax purposes.

To cover all cases, I should mention that the payments don't need to be regular.  They can follow any kind of schedule you like, as long as they are effectively the same as the single terminal payment fee when you present value them all.  While we're at it, you might as well do the same thing for the return of capital.  This doesn't need to be paid back on the last date.  Perhaps you might find a loan structure where, rather like a repayment mortgage as opposed to an interest only one, your ongoing payments are partly fee, partly return of capital.  At the extreme, you might reach the last date and have no capital left to pay at all.  This tends to suit borrowers less well, however, since they want to hold on to the capital for the whole borrow period, to give them  the best chance of making good use of that capital.  Still, there's nothing to stop you writing a loan contract which does this.

In short, whether you think of the payment as a cash fee, or a series of instalments, or as a single coupon paid at the end (like a discount bond) or as a series of regular payments/coupons is largely irrelevant.  And whether you quote the cash payments in absolute terms or in terms of the loan size is again your decision.  It will make some of the maths easier to consider the amount as a rate but it won't make you any richer or poorer than this deal was going to make you.

There's an advantage in deciding how you're going to quote your extra amount in a way which facilitates comparison with other loans.  The quoting of choice is often to describe the fee as a rate of the initial loan, with an assumed regular set of annual (or  semi-annual) coupon  payments, with the final payment of the final coupon happening on the same day as the principal repayment.  This is indeed mostly how the world of fixed income investing quotes the fee.  So your fee is your return, often quoted on a semi-annual coupon basis or maybe on an annualised basis.  Corporate projects often run their course on a multiple-year time frame, so this frequency of compounding would probably have struck a decent balance with the excessive costs  in triggering your calculation agent (who in the olden days operated in the costly and unproductive pre-computer days. monthly, weekly or daily).  The theorists of finance, on the other hand, see advantages in calculating continuous compounding since it allows them to develop calculus based models and lines of reasoning.

At the start of this post, I introduced many permutations in describing how you might like to receive your lenders fee, but then I developed the idea that this was merely syntactic sugar; additionally, that the particular flavour of sugar used was a function of how the market in question developed at origin; that there's a finance theory use which prefers the advantages of the mathematics behind continuous compounding.  There are always the usual tax, religious and other non-finance reasons why you might consider your fee a services rendered fee versus an investment income fee.

Bearing all of that in mind, we can say that the borrower will receive a rate of return $r$ for making the loan of £100, where $r$ is paid annually, at the end of each year, and together with the principal, at expiry.  This is the classic shape of the contract for a loan which many people have in mind when they try to describe a standard loan contract.

Over the next couple of posts, I will try to start developing an equation for expressing the component parts of  $r$, the extra you get for making a lend.

A man walks into a bar...

In a recent posting I asked the question, what's in a rate as a precursor to looking at some of the major factors which make up a rate.  But before I dive fully into that subject I'd like to point out that all of the theory based constituents of a rate are just that - theory based.  There's nothing to stop any two human beings or institutions in the world offering each other any rate they want to on financial contracts such as loans, swaps, forward rate agreements, etc.

It comes back to this question of the rational man, and the maximisation of individual utility which underlies the various kinds of reasoning associated with thinking like a rational utility maximiser.  The model of a utility maximiser gives you the possibility to arrive at a framework for arriving at a fair rate, and when you look at individual rate markets, you see live rates happening out there.  The degree to which you can 'read off' or calibrate parameters of your rational man model from current market rates is also the degree to which you think markets are rational.  This belief (rational expectations) is what allows some people to imply certain parameters of their fair value rates models.

But if you believe there are times when the markets for rates are behaving irrationally, then your system for implying model parameters from market facts breaks down.

In order to explain what I meant by the four characteristics of a fair rate (inflation, the market, participant profiles, this deal) I'd like to cast them all as permutations of the traditional joke opening line: "A man walks into a bar...".  The reason why I like this opening phrase is because you get a sense of some kind of connection between the man and the other people in the bar, but they're clearly not family.  In fact, perhaps a bar environment is in the same category as a market here.  Bars also famously double up as places were stolen goods get sold on, so they can literally act as a market.  Con schemes, new business ventures can all be implemented here.  Indeed weren't the first permanent markets in England originally coffee houses, which are slightly more sober versions of bars.  The relationship with the man and his bar is thus nicely situated for me.   It appeals to my inner anarchist anthropologist.  Also there's a plethora of versions of jokes which start that way.  I hope the familiarity allows you to capture what's really different in each case.

Version 1: 

A man walks into a bar and asks the people in the bar for a lend of £100 for a year, at which point he'll return and give the £100 back.  A healthy show of hands among the customers shows strong interest.  He picks one, takes the £100 and walks out again.  Sound plausible?  Maybe.  

Version 2: 

A man walks into a bar and asks the people in the bar for a lend of £100 for a year, at which point he'll return and give the £100 back.  Despite some interest, no-one in this particular bar has that sum of money ready to hand.  They're all out with meagre drinking budgets.

Version 3:

A man walks into a bar and asks the people in the bar for a lend of £100 for a year, at which point he'll return and give the £100 back.  There's some interest, but again no-one has £100 free cash.  But though they may be poor, they are kind and communal spirited.  So they organise among themselves a collection of cash, each giving what they can, until they reach the £100 requested, and hand it to the man.

Version 4: 

A man walks into a bar and asks the people in the bar for a lend of £100 for a year.  One person sitting in a cubicle recognises the man as a distant cousin and happily hands over the cash.

Version 5: 

A man walks into a bar and asks the people in the bar for a lend of 1 penny for a single day, at which point he'll return and give the penny back.  It gets more plausible as you reduce the amount to 1 penny and 1 day. The lender probably thinks nothing much of it, and doesn't much care if the man never comes back with the penny.  Perhaps the penny becomes the price of some entertaining pub conversation which follows.

Version 6: A man walks into a bar and asks the people in the bar for a lend of £1,000 for ten years, at which point he'll return and give the £1,000 back.  It gets less plausible if you increase the amount to £1,000 and ten years.  

Version 7: A man walks into a bar and asks the people in the bar for a lend of £100 for a day, at which point he'll return and give the £100 back.  But he takes from his pocket a Rolex watch which he's happy to leave with the barman and says the lender can keep it if the borrower doesn't show up with the money.  There's a rush of people to the bar, all willing to lend him £100.

So far, none of these variants have involved a lend with interest.  You are probably thinking that most are unlikely scenarios as a result of this.  Though version 4 (a relative) and version 5 (a penny for a day) are the most likely, followed by version 7 (Rolex).  Though I'm sure you're wondering of the Rolex is a fake, aren't you?

Let's see how far I can press this without introducing a rate of interest for the lender.

Version 8: 

A man walks into a bar and asks the people in the bar for a lend of £100 for a day, at which point he'll return and give the £100 back.  One customer agrees and hands over the money.  But then that customer, several hours later, gets a call from home and needs to return in a hurry.  He asks the rest of the bar if anyone else would like to take on his obligation?  One man agrees, so, the departing customer gets £100 from the new lender, who in turn sits waiting for the original man who walked into the bar.

Version 9:

As with version 8, but this time the obligation gets passed around a whose series of customers of the bar, all coming and going during the day.  By the time the original man who walked into the bar returns, the entire bar population (including bar-staff, who've changed shift) is different.  But just by shouting 'where is the guy I owe this money to?', the original borrower is able to return the £100 to the current lender.

Version 10:

As with version 9, but the original man is confused.  He doesn't want to hand over £100 to some stranger.  No-one there is recognisable to him.  He begins to wonder if this was even the right bar.  He leaves the bar still holding the £100, vaguely promising to himself to return tomorrow to see if he could see the lender.

Version 11:

As with version 10, except this time the lender who has now inherited the loan gets quite angry, and frustrated at his lack of any ability to convince the borrower to return the cash.  

In these newer (but still rate-less) versions, the bar is falteringly beginning to operate somewhat like an exchange venue (it doesn't deserve to be called a market quite yet, but markets are many things, including being exchange venues).  Whereas version 9 looks to be a healthy exchange venue, versions 10 and 11 show it breaking down through lack of trust and operational clarity.  In the original version 1 of the story, you were immediately clear in your own heads that there's an issue of trust at play - between the potential borrower and the potential lender.  Now you can see that there's a second dimension to this issue of trust.  The re-distribution of this loan and the borrower's willingness to settle with someone else are clearly dependent on a wider sense of trust - the trust placed in the perceived fairness of the bar as honest institution.


Version 12: 
 A man walks into a bar and asks the people of the bar for £100, which he says he'll keep.



Version 13:

 A man walks into a bar and asks the people of the bar for £100.  He says he "needs a whore bad" but doesn't have the cash to pay for one.  "That's where you come in", he smiles.







Version 14:

 A man walks into a bar and asks the people of the bar for £100, which he says he'll use to buy a warm coat for that homeless man out on the bridge since it is coming in to winter and the snow's arrived.





With these variants, we get to see the reductio of any 'lending' operation, one where the money is not returned.  The act itself seems like a form of charity, though the versions where the borrower states his purpose clearly can have an effect on the lend.


Version 15: 

 A man walks into a bar and asks the people in the bar for a lend of £100 for a day after which he'll repay.  He gets his money and departs.  Fifteen minutes later, another man comes in, also asking for £100 for a day, promising to repay.  He also gets his money.  This carries on every fifteen minutes until no-one in the bar had any money left to lend.

Version 16:
 A man walks into a Zimbabwean bar and asks the people in the bar for a lend of 100 Zimbabwean dollars which he'll pay back this time next year.

From just these bar stories, it ought to be possible to develop a theory of the rate of interest, a theory which tries to explain what a fair lending rate might be.

In the next posting, I'll introduce variants of the story where forms of payment are made to the lenders.  As you will have noticed, so far, no policemen, no financial services authority, no formal contracts, no lawsuits, no government guidelines  have been brought into the story.  

What's in a rate?

Have you ever looked inside a rate?  In this post I'll try to do so, giving some ideas about how you might slice up a rate in terms of economic forces or financial forces in markets.    Remember, first and foremost a rate is just a fraction of some reference amount.  In other words it describes a unit or quantity in terms of its size relative to some reference quantity.

Let's talk money.  A money rate describes some amount of money with respect to some other reference amount of money.  In the vast majority of cases in fixed income finance, the reference amount of money represents either a starting amount or a closing amount.  Usually the rate summarises some kind of financial promise you're involved in or it represents a post hoc analysis of some investment you made in a security or portfolio of securities.

These rates are also known as returns, yields, and interest.  Return is a nice expression, conjuring up an image of the return of invested capital, with some extra capital too.  Yield is quite an agricultural sounding variant - think of it as expressing the size of a crop with respect to the size of the field.  Interest (interesse) was originally a late payment penalty built into the contracts for loans, which then morphed into contract structures where failure was built in.  This allowed the contracts to side-step Christian usury laws.  Muslims perform a similar piece of arithmetical/contractual engineering in their dealings with returns.

The clearest security with a return is probably a loan by a lender to a borrower for a fixed term, with no intervening days of reckoning, where accumulated interest is rolled into the current capital embedded in the security (compounding).  That is, in cases with just two days of reckoning - at the start day and on the last day - the day of termination of the loan.  The slightly more general case is where the rate as expressed fits into a compound growth formula, as described in another posting.  This implies all such rates must have associated with them implicitly recipes for how they are used.  These recipes are called the rate's time basis, compounding frequency, day count convention.  They flesh out how to operate with the rate.  As noted, again in a previous post, they're all fully interchangeable, so we shouldn't look here to find out what's inside a rate.

 In the next posting I'll explain why I think it is good to categorise of the constituents of a rate as follows: inflation, the market, participant profiles, this deal.  I think of this as a slow zoom camera, first of all, picking up macroeconomic effects, then market (and close-market) effects, then evaluating the states and preferences of the contract participants, before finally looking at the terms of the contract.

Anatomy of a convert - interested?

Interest has its own interest - cultural, historical, mathematical.  In this post I'd like to point out the vague irrelevancy we like to attribute to it in our everyday lives. Interest rates for loans to the safest bet have most often been somewhere around the 5% level, give or take. I think for many people this is psychologically around the 'fee' scale - we're all used to banks and other money institutions (pension providers, insurance companies, etc.) charging fees with are in the same ball park.  We are also mostly dimly aware of the role of inflation on money - we all dimly know that the meaning of 5% is itself in some general way clouded or constrained by the inflation level, so that gives us an even further excuse to consider the difference between, say 5% and 5.5% as not significant.    

For banks and those dealing with fixed income products, like bond traders, their discrimination levels need to be a lot finer.  The reason is because they're usually applying it to much larger sums.  The only time in most of our lives where we get to play with large sums is when buying a house.  Here, we often come to appreciate the difference in meaning from a 5% payback rate and a 5.5% payback rate.  What we're doing, in our heads, is spelling out the meaning of that 0.5% with respect to a large sum.  Say our house costs £100,000.  Then that 0.5% amounts to £500.  If that was an annual additional payment, then the corresponding monthly payment would be about £42, or a meal for two in a restaurant.  Whereas if you're buying £100,000,000 worth of a convertible issue, then that 0.5% amounts to half a million pounds.  And if you earn less than a pound per day, that 0.5% is the least of your worries.