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.

neo = classical + Keynes + Fisher

An interesting short cut way of thinking about neoclassical economics and how it derives form classical economics is to imagine the following formula:

necoclassical = classical + Keynes + Fisher

The classical story is one where a market's supply and demand characteristics drive the equilibrium price of a good (including commodities, manufactured goods, services, and the price of labour, namely average wages).

Keynes pointed out that there are nominal price rigidities which prevent certain markets form clearing, a point in case being the labour market.  Nominal prices are sticky upwards - they don't like down adjustments.

Fisher pointed out that debt was also a rigidity in certain points during the economic cycle.  The debt is expressed in nominal terms, but during deflationary periods, for example, the borrower may find it increasingly hard to make their debt payments.  

They're both therefore, pushing forward the classical economic view by highlighting the need for more structure in the too simplistic classical view of a market driven economy.

I think that some time in the next one hundred years, real (as opposed to nominal) debt contracts will be  the norm.  This will take the sting out of so-called debt-driven deflationary periods.  I.e. the nominal rigidity of debt will become a solvable economic problem.

Whether the inflexibility of labour to deflationary periods has a rational expectations description (e.g. it is fairer for the average employee to take an inflation-induced wage cut rather than some subset of struggling companies going to the wall) or a behavioural economics one (we find thinking about real economic variables a system 2, slow brain activity and prefer the 'what you see is all there is' fast brain fallacy), or an unintended side effect of the historical fact of  'great moderation' hastened  by the arrival of semi-apolitical effective central banks (an institutional explanation) will help determine the prognosis of this current economic reality.


In today's highly politicised world of academic economics, I'd be obliged to refer to the classical + Keynes + Fisher as 'post-Keynesian'.

Certificates of Deposit : credit spread?

In a previous post, I mentioned that one possible explanation for why two identically termed CDs available from two companies might have different yields.  I'm not so sure any more.  In the US anyway, there is the FDIC system, which covers short dated CDs up to a quarter of a million dollars per person per bank.  It ought in principle to be possible for one person to pay an intermediary to spread around any arbitrary sum across N banks such that no one bank gets more than a quarter of a million dollars of his money.  In other words, it ought to be possible to put much larger sums in CDs, if you so desire, and receive the US government's full backing.  Maybe the net effect of this possibility is to drop off the credit risk associated with you having lent an institution some of your cash.  If your CD was with Lehman Brothers in 2009 versus Wells Fargo, then if it wasn't for deposit insurance then you'd expect to get a higher yield for leaving your cash with Lehman.  This yield differential $y_{L}$ versus $y_{WF}$ is a kind of credit spread.  The credit spread is the little bit extra you expect to get for leaving your cash with an institution which could go bust.  But perhaps FDIC pushes both of these rates down towards a common $y$, since the deposit insurance takes virtually all of the (domestic currency valued) risk out of the saving.  This therefore must be a constraint from above on CDs.  You can offer no more than $y_h$ since any higher is effectively ignoring the FDIC factor.  

And the constraint from below must be bounded by inflation: since if the CD supplier offers a nominal rate which is lower than expected inflation $y_{l}$, then capital is being constantly eroded.  It does not make sense for an economic agent to lend his money and get back less in real terms than he lent.  This window $y_l \leq y \leq y_h$ is probably quite narrow, meaning the product is more or less going to keep your cash safe from inflation, probably.    The 'probably' comes from many factors, not least of which is the fact that no-one can predict future inflation, even over a short time period, with any certainty, so there'll be a prediction error in the offered near-inflation rate.

CD providers like to throw in extra terms and conditions on the purchase of a CD, all if which can affect the valuation.  This makes it harder for customers to do side-by-side comparisons.  This is a familiar trick.  The CD I have been describing in the last couple of postings has been as simplistic a schematic CD as you could imagine.

All of this is a bit too vague.  I'd like to continue thinking about the rate implied in CDs, but before diving into some detail on CD rates, I'd like to pull back and ask a general question: what goes into a rate in the first place?  What factors make up a rate?  What risk are you taking in lending your cash to someone else for a while?

Certificates of Deposit : market price = fair value

If the simple CD is tradeable, when what is its price?  Well, that's two questions:  what is its fair price at any moment during its life, and second, what is its market price.  Well, remember, it is a two step process - first you get the future value of that single cash flow you'll get on expiry (par plus whatever interest you earned) for some given principal amount $P$.  There's nothing really in that which could vary over time.  The value at expiry is the same regardless of how close to expiry you are.

What might change is the yield you'd use to discount this singular known future value.  Why? Well, imagine you bought a 1 year CD for principal $P$ and a nominal yield $y$, which was the going rate when you entered the market.  The market being that set of institutions from that country who are also offering 1 year CDs on principals the size of $P$.  Now where does this number come from?  Well, as you'd imagine, it is sensitive to short term interest rates.  So Imagine you just bought this new CD and got a rate of $y$ when that very moment the domestic central bank raised the short term policy rate by a whole percentage point.  Well, the CD market would adjust and offer the marginal next customer a higher rate of return $y^\prime$.  So when you come to present value the same fixed future value $P(1+y)$ you get $P \times \frac{1+y}{1+y^\prime}$ since the fraction is less than 1, which results in some amount less than $P$.  That is, the fair value of your security, this instrument which was going to give you $P(1+y)$ in a year, is now worth less than $P$.  Another way of saying this is that the value of the CD is sensitive to fluctuations in interest rates in the economy.  It has interest rate risk.  That new value, call it $P^\prime$, is the new fair value of the CD.

The only moving part here is $y^\prime$, the single prevailing rate you discount your future payment.  This, in a sense, is also the market price.  Now this is unlike more complex securities in a number of ways.  Often other securities have more moving parts, but you'll always just have a single market price.  But for now, enjoy the simplicity of the relationship.  Regardless of how the market actually quotes this rate, whether they tell you it as the current value $P^\prime$, whether it is quoted as $y^\prime$ itself, whether it is $100-y^\prime$ or any other transformation, the bottom line is, that market quote can be transformed into $y^\prime$.  Now imagine I had two CDs, each with different nominal yields $y_1$ and $y_2$, on identical principals $P$ and expiry 1 (year).   Clearly they'll be worth different amounts in any given prevailing market environment $y^\prime$ and time to expiry $t$.

Just for now, let's pretend the market quotes the market yield as the current cash value of an invested principal $P=1$.  That is, pretend the market price of a CD is expressed as $P^\prime = \frac{1+y}{1+y^\prime}$.  This market price is then synonymous with the fair value of the instrument, which is also $P^\prime$.  That identity relationship doesn't often happen with other financial instruments.  With other instruments, there's a gap between the market price and the fair value.

In the next post I'd like to introduce you to the second of the great risks in finance, already present in this simplified product. 

Certificates of Deposit : first pass at a quantitative analysis

CDs are pretty simple money market instruments, or so the story goes.  Well, what better place to start looking at issues in the quantitative analysis of fixed income products.  In my mind, the end goal is the analysis of the queen of all securities - the convertible bond - that fiendishly complex hybrid of fixed income and equity derivatives.  But one step at a time.

In this blog, I pretend that there's even less complexity in CD instruments than there is.  But first, I'd like to make a clarification on the difference between CDs and your local bank account, something I touched on in the last post.  CDs are bearer instruments - whoever has the legal ownership of them could in theory sell them on (or buy them) in a secondary market.  Now, while this may not happen with every CD, it certainly cannot happen to your own bank current or savings account. 

On a first pass, taking the easiest fake CD I can think of, you would interpret the nominal annualised yield of the CD, call it $y%$, as  the yield on some principal $P$ on a simple interest basis, for a CD which lasted one year.  You expect to receive  $P \times (1+y) \times 1$.  That is, the future value of the single cash flow as seen from a valuation date of day 1 of the life of the CD.  As a last step you'd look to find the present value of that future cash flow.  Well, the discount rate to use for the future cash flow $P \times (1+y) \times 1$ arriving in your bank account would be the effective yield of other CDs just like this out there in the world.  I.e. $y$ (on the relatively modest assumption that when the original deal was struck and $y$ was the first CD's annualised yield, it was so since that's what the prevailing rate in the market for CDs just like this was trading at, and the period of time you'd discount it for would be the distance between the valuation date (which is still day 1 of the instrument) until that future cash payment (1 year).  This present valuing back to the first day of the CD leads you back to a PV of $P$.  This makes intuitive sense, since in a way you just bought it and paid $P$ for it, so it isn't too surprising $P$ is the fair value.

Under the normal course of events, as time passes, the present value of the CD moves from $P$ towards the value of $P \times (1+y)$.  That is to say, on the last day, regardless of prevailing interest rates, you will indeed receive $P \times (1+y)$ and that cash flow would then be an immediate cashflow which didn't require further present valuing.  Another way of thinking of this, of making it more natural, is to realise that, the closer you get to the payoff day, the more it'll cost you to buy an instrument which will be worth $P \times (1+y)$.  It starts off costing you $P$ and day by day you find purchasing that same cashflow moving monotonically from $P$ to $P \times (1+y)$.

In future postings, this all gets more complex along a couple of dimensions: day count conventions vary from the implicit ACT/ACT in the above analysis; intervening coupons may be paid out, with compounding effects, the product itself may be contain more structure in the terms and conditions, the PV discounting factor can change on a moment by moment basis through the life of the CD.  But for now, the above represents the simplest possible CD analysis on the simplest possible CD.  It would be good to see how this ground-zero CD ticks over time.

Certificates of Deposit - the one night stand

A certificate of deposit is like a vomited savings account.  Imagine that set of financial institutions in an economy with a reasonably well developed financial industry, which offer their own customers a savings account.  The rate the customer gets is partly by virtue of them having an ongoing relationship with that financial institution.  It is also a function of the funding cost for that financial institution.  They'll put in a profit margin on top of their own funding cost and call that the savings rate.  This they then pay you.

Time deposits are the placing of your money with a financial institution for a fixed period of time.  You're not supposed to withdraw your money before a certain agreed maturity, and if you do, you face financial penalties which will probably wipe out most of your anticipated returns.  Again, your local friendly bank will be only too happy to take your money off you for a fixed period of time (and will reward you with a slightly higher rate of return for the discipline you show in not asking for it back before maturity).

This financial institution is probably signed up to some kind of deposit insurance scheme run by a central banking authority on behalf of the domestic government of the economic region, which means that some of the money it could have lent to you has to go to pay this government bill.

Also, the saver will get his money plus interest back at expiry, and not before (unless he presses a big red button and invokes a get-out clause which will probably cost him the majority of his accrued interest, up to 6 months worth for longer dated CDs).  The issuer can also set up a regular CD which pays interest on a schedule.  Now, a CD is just something quite like one of those bank savings accounts except you don't need to be a customer of that bank in the same way as you do with the savings account.  If the bank savings account is a bit like a steady relationship, then a certificate of deposit is a one night stand.  (Or one month, or three months, or half a year or a year, and on up to five years).  Another simile is to consider a regular savings account as some food digesting in a human belly, and the CD as food vomited up into a sick bag.  There's a degree of distance implied in having a CD with a financial institution.  Whereas the internal bank savings account is a bit of a black box, CDs are on their way to being an external market.  You can certainly shop around, but it is possible to make ready comparisons of various CD rates.  Ceteris paribis, you will expect to get a better rate with smaller, riskier CD issuers, and when you lend larger sums, and when you are lending on a five year window over the shorter expiries.   By the way, these days you don't often get  a certificate, just a book entry change in your account.

By tradition, the CD rates tend to compensate you at approximately the current rate of inflation, approximately, meaning you are not actually growing your real wealth, merely allowing it to keep up with inflation.  But this is only based on periods of healthy shaped yield curves.  In general you're lending to a bank.  These days, that's considered a lot more risky than it used to. So there ought to be a risk free and a credit component.  In the days pre-crisis, the credit spread would have been quite minimal and so you had a risk free 

As you might expect, the usual rules on calculating simple or compound interest apply.  Part of what it means to compound is the implicit understanding that you re-invest your interest payments, as soon as you get them.