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.