Wednesday, 29 July 2015

Annuity. Perpetuity

There are two directions in which the formula for discrete compounding naturally goes.  One towards a limit of continuous compounding, and the other as a geometric series modelling the way that the quantity (for example, money) grows during the observation period.  

The idea of $e$ was implicit by 1614 in Napier's work on natural logs.  The use of $e$ as a constant was due to Euler in 1727 and he was explicitly setting it in the context of taking the compounding formula to the continuous limit.  Just as the idea of $e$ was beginning to be born, in 1613 Richard Witt published the first book dedicated to the maths of (discrete) compounding.

A geometric series has as its sum: $a(\frac{1-r^n}{1-r})$.  Notice here that $r$ is not a rate, but a growth multiplier - the equivalent in present value/future value analysis to $\frac{1}{(1+r)}$, where $r$ in that case is an actual interest rate.  An annuity can be modelled with this formula as long as we set the initial payout amount per period to be $a$ and the common ratio to be a present value formula, being sensitive to compounding. The idea is to realise that an annuity describes a series of evenly spaced cash payments into the future.  To work out how much each is worth, we'd like to present value them with a constant rate and a discounting period counter of $n$.

The annuity thus is a decent model for working our how valuable to you the coupon payments are.  With computers these days doing the hard work of iterating the cash flows, the alternative is just to loop over all the coupon payments, and to make the last payment to include the principal.  But in the days before computers, having mathematics do the hard work was a smart move.

So, for the $n$ periodic coupon payments $c$ on a bond, with the period return being the nominal discount rate $\frac{1}{(1+r)^n}$ , the series of payments $\frac{c}{(1+r)} + \frac{c}{(1+r)^2} + \frac{c}{(1+r)^3} \cdots \frac{c}{(1+r)^n}$ sums to $c(\frac{1-{\frac{1}{(1+r)}}^n}{1-\frac{1}{(1+r)}})$, or, bringing the $n$ down to the denominator, $c(\frac{1-{\frac{1}{(1+r)^n}}}{1-\frac{1}{(1+r)}})$

Finally, there's a limits exposition which shows that, in the case where $r<1$, as it usually is with discount factors, then the sum of a geometric series for an infinite number of steps is the surprising $\frac{a}{1-r}$. So far, again, $r$ is the discount factor and to make it into a rate you replace it with $(1+r)$ - the 1s cancel and you're left with $\frac{a}{r}$, where the $r$ is no longer a geometric series ratio, but a discount rate. This is the present value of a perpetuity.  

Ground rent is an example of a perpetuity.  If someone tells you the ground rent on a freehold is worth 15,000 for annual payments of 100, then they're telling you that the market rate to discount for that infinite flow of 100 payments, forever, is solved by setting $15000 = \frac{100}{r}$ which means $r=\frac{1}{150}$ or about 0.667% annualised.

Note how confusing it is to join together the maths of geometric series (a,r) with that for compounding (P,c,n,r) since in both of these worlds, by tradition, the choice of r can be semantically jarring, as they mean different things.  In geometric series maths, it represents a multiplier, and in compounding it represents the percentage growth (1+r).

Sunday, 19 July 2015

Compounding fictions

Behind the mathematics of compound interest are a series of layers of thought experiment, fiction upon fiction, compounded.  Interest itself as an idea has its modern origin in the act of lending or borrowing money.  In general, it measures the growth (or diminution) of a quantity across an observation period.  

The starting point for the fiction is that you can predict the future.  You can't of course.  But laying out your expectations about the future is a kind of fictional account of what you hope will happen.

Remember also that all interest is compounded, with per year compounding periods of $n=\infty, \frac{1}{365}, \frac{1}{12}, \frac{1}{4}, \frac{1}{2}, 1,\frac{1}{t}$.  So with compounding, you identify a number (1 or more) of regularly occurring, evenly spaced way points along the journey from observation start point (investment initiation, loan beginning) to observation end point (maturity, investment exit).  No cash flow needs to accompany these marker points.  Though cash flows often do.  They can be merely notional.  But at these way points, ownership is transferred from one party to another of some fractional value - cash or something else valuable.

Then, within the boundary of these fictional way points, come even more fine grained way points, where some linear fraction of the sum to be earned in the current way point period is measured out.  This fine grained linear fraction often measures so-called accrued interest.  Again it is often expressed as simple interest on the coupon you're in the process of compounding, but in reality it is often $n=\frac{1}{t}$ compounding within a semi-annual bond coupon compounding $n=2$.  It is done so that when a bond is traded on any day which doesn't fall on a compounding period end date (i.e. on most days), the buyer and seller break up the intermediate value of the current coupon betwen buyer and seller - a bit like Pascal's problem of points, where a game is ended at a point where the rules haven't fleshed out the value.  The presumption of fairness drives a buyer or seller of the bond to agree to apply time-weighted $\frac{1}{t}$ compounding to get a micro read on how most fairly to split up the current coupon.  

When you express $\frac{1}{t}$ compounding for a period, what you're doing is implicitly saying that no further micro structure is worth considering between the two dates - in this case the dates which bound the current coupon.  That being so, then you can use a linear fraction of days (or years, seconds, whatever, as it is a ratio after all) to work out the fair division of the currently in play coupon.  That is, compounding only at the end allows you to treat the rate as a linearly scalable ratio.

The human scale on interest payment doesn't go further than these two levels, namely coupon compounding and accrued interest.  The sums involved in any one deal are usually of a scale that further kinds of compounding don't seem worth it in analysis.

But the mathematics doesn't care about that mere human constraint.  You could keep on carrying on this micro analysis to the sub-day level, in theory to the continuously compounded level.  This might become more relevant in centuries to come when the world of high frequency trading comes to the world of sovereign bond relative value trading.

These layers of fiction exist in different places.  For example, the bi-annual compounding frequency of many bonds exists as a fact of the bond prospectus.  But the accrued interest fiction exists between buyers and sellers of that asset.  It is a form of market practice, that is to say, a sub-prospectus descriptor.  Said differently, the writer of a bond prospectus could imagine a world where a different daily accrual convention could apply.  

Every use of the compounding formula so far in these recent postings concerns discovering something about a trans-temporal analysis of a series of cash flows wrapped up in the conventional legalistic dressing of a bond.   It is a single algorithm which explains how the quantity changes at moments within the observation range (internal period ends and internal period middles and starts).


Saturday, 18 July 2015

Formulae aesthetics: compound interest with a holding period nominal rate or with an annualised rate

$(1+r)^m$ versus $(1+\frac{r}{n})^{nt}$

I've often in books found two formulations for the compound interest rate formula, which, remember, I see as inclusive of the simple and compounded formulae too.  One expression of the compound interest rate - the more messy one - implicitly assumes that $r$ is given to you as an annualised nominal rate, and the other assumes it is simply the nominal rate which applies to the compounding period $m$.  The first has more parts to it, due to the initial step of transforming the annualised rate to the compounding period rate.  The second doesn't have this step.  I prefer this second formula.  

It is more basic.  We annualise for human purposes.  To compare rates across a standardised time horizon.  There's nothing inherently mathematically important about the human centric time horizon of a year.  For the maths, all that matters is the implicit compounding period, and the number of periods you roll the compounding operation.

So in the formulae at the top of this posting read the first formula as saying the following:  Take a dollar and a rate $r$ which has, as all rates do, an implicit compounding basis of some time period.  Lay out $m$ of these time measures end to end.  Walk that dollar forward and at the end of each of those $m$ time periods, grow the value of your dollar pot by $1+r$.    It is a series of $m$ salaries which are paid at the end of some implicit time period, implied in the compounding basis of the nominal rate $r$.  The rate doesn't give you a full picture of itself unless accompanied by the compounding basis, a time measure.   It is also nominal in the sense that it isn't quite the return you will see on your investment.  For example, with a nominal rate of 10% on an annualised compounding basis, run over two years, a dollar becomes 1.21, meaning your holding period return is 21%, not 20% (2 times 10%).  Likewise 10% over four years gets you a holding period return of 46.41%, not 40%.  Think of 'nominal' as meaning a rate which is used internally in the production of your final holding rate return.  It is referenced at $m$ points on this journey.  But when someone asks you how much your journey profited you, the answer only emerges after having run the algorithm, made the journey, evaluated the compounding formula.

Simply, if we prefer to see $r$ quoted on an annualised basis, an initial translation needs to happen to make annualised $r$ into the equivalent compounding period rate.  Compounding periods are almost always less than a year, so you see $m$ expressed as $m=t \times n$, meaning that the $m$ compounding periods in question run $n$ times a year for $t$ years.  So you get to an equivalent compounding period rate by chopping down your annualised rate $r$ to $\frac{r}{n}$

To put this all another way, virtually every time you come to use the compounding formula you'll probably use $(1+\frac{r}{n})^{nt}$ but the mathematics doesn't care about years, and $(1+r)^m$ is the cleaner, clearer formulation.  They both do exactly the same job.

Tuesday, 14 July 2015

Bond puts and Bond calls

In a recent post, I speculated that there might be some differential meaning in the implied vol of puts versus calls embedded in corporate bonds.  I don't think so today.  I looked into it a bit more.  Some things I picked up.  The corporate treasurer is just as likely to consider a refinancing (exercise of his call) based on a significant improvement in the firm's credit rating as he is based on lower interest rates.  That is, the call implied volatility is telling us something about rate expectations together with credit spread changes.  

There are still a minority of corporate bonds even with a call, which is the older historically of the two kinds of embedded option you see in a corporate bond.  Puts are still a rarity with corporates.  It is also likely that the corporates market got the idea for calls form the convertible market, maybe also the holder put too.

Ceteris paribus, the presence of a put is an indication of the market having expected a higher probability of downward future pressure in the bond price during its life.  

Corporate puts and calls are usually both struck at par.  So I imagine if there's an incentive to sell when significantly above the calls price, and a tendency for the bond to become downside sticky around the put strike, I would imagine that this can take a lot of the volatility of the price action of the bond.

Sunday, 12 July 2015

Bonds, sovereign and corporate

An interesting thought experiment to get underneath the seeming disparity between a country's treasury securities and its corporate securities.  I want to find a way to see them all as part of the same model, not as essentially different as they are sometimes characterised in the books.  Bonds are ways to leverage something, future tax receipts or future corporate revenue streams.  So just as credit worthiness of companies worsens when the expected revenues (free cash flow) decreases, so too with nations.  The two ways that expectations on tax receipts may worsen.  Either the effective tax takes get worse (the U.S. takes in 26.9%, the UK 39%, Switzerland 29.4%, Germany 40.6%, Norway 43.6%, Denmark 49% versus Afghanistan 6.4%, China 17%, Equatorial Guinea 1.7%, Egypt 15.8%, Indonesia 12%, Saudi Arabia 5.3%, U.A.E. 1.4%, OECD average 34.8% ), all other things being the same, or else the GDP of the nation deteriorates (smaller population, less productive population, reduced output).  Put another way, just how well developed a capital market can a country expect in the face of very low national tax takes.  Compare in passing how oil rich Saudi Arabia compares with oil rich Norway.  Norway doesn't have an unproductive workforce and a tiny cabal of fabulously wealthy Al Saud princelings, it has an enormous sovereign wealth fund.

Each nation state, especially those in the developed world, with mature capital markets tries to reset the meaning of 'risk free' by considering their country's treasury bonds as risk free.  This only works up to a point.  But it is certainly true that every country with a developed treasury market can and does have market participants whose primary concern is expressed all in that countries national currency.  In other words, FX risk is a bolt-on concern with bonds, at base.  However purchasing power risk is a core risk for all countries everywhere.  As is the risk that interest rates will change (up or down, a primary risk, or up/down more frequently, a kind of volatility risk which crops up with corporate bonds).

There are risks which exist exclusively in the corporate bond world.  But even though the corporate bond world en mass is huge, the market for specific issuers (and more so for issues) is much smaller.  

In a nation there are many people and companies.  Nations are naturally much larger than companies.  So it isn't surprising that liquidity risk is more of a factor for corporates, individually, than for treasuries.  Currently (2015) the world's biggest company is ICBC in China, which is worth 1.8 trillion dollars.  The GDP of the United Kingdom is worth about 2.9 trillion dollars.  It is an open question whether in time any company will become so big and so reliable that the liquidity of its bonds will rival the liquidity of sovereigns.  Part of the reason is that national tax takes in developed countries are much more stable than the revenues of any one particular firm.  Perhaps fairer comparisons are with huge stable companies and medium sized developing countries with nascent treasury markets.

The firm itself is essentially a transient entity, since so many of them go out of business due to competition, poor management, obsolescence.  Very few firms last a long time.  Recent research by Richard Foster at Yale suggests that for S&P 500 companies, the average life has dropped from 67 years in the 1920s to 15 years today.  That is a huge change, but it is also a small number, compared to the expected age of a modern developed country, which is in the hundreds of years.  

So liquidity may always be much more of a risk for corporate bonds than sovereigns.  A second major difference is that, to some degree, nation states are masters of their own interest rate destiny, meaning that interest rates can be adjusted to suit the health of the nation.  Since this is beyond the control of corporates, they try to buy some refinancing protection in their bond issues through contract clauses, provisions, which give them a kind of refinancing right - through owning a long call on the debt.  If rates drop, the issuer might prefer to cancel its debt and issue new cheaper debt.  they do this by constructing call clauses.  Call clauses, being call options, are additionally sensitive to expected interest rate volatility, something which is essentially out of their control.

They are also sensitive to default risk, credit spread risk, and ratings agency downgrade risk, all of which address the very real possibility that the firm might become impaired and fail to meet their repayment obligations.

So out of all this, I can see that the primary differences in corporates and sovereigns must be driven by the fact that corporates are more frequently failing, more often (always) smaller and therefore more often becoming illiquid.  Their response to the exogenous nature of interest rates is to attach refinancing options to their prospectus.  Of course, making a new bond issue is an expensive business, so the timing of their call will be not entirely driven just by interest rate moves.  For the refinancing to be worth it, the cost of the new issue must be factored in too.

One last point.  Corporate bonds can have embedded puts in them too - on a finite set of dates, the holder can ask for his money back at par.  So this clearly works in the opposite direction than for calls.  

Bond calls are primarily a hedge for the issuer so that they can refinance in a favourable new interest rate environment.  If rates, however go up, the call is not exercised, but the price of the call can continue to fall in the market place.  So the call can't really interfere with the read that market participants make on the credit worthiness of the issue (and therefore issuer).  

Bond puts, on the other hand, can be exercised by the holder if interest rates rise dramatically, but also if the credit spread widens.  So the price of the bond doesn't purely reflect the credit worthiness of the issuer, unless you adjust for the stickiness of the price around the fact that there's some put date coming up at which point the holder can offload the bond at par.

Likewise I would imagine the call price is sensitive to the expected interest rate volatility but the put would be sensitive to not only this volatility but also to the credit spread volatility.  If so, then market participants could in theory read off this expected credit spread volatility from an analysis of the respective price of the call and put in an issue.

Bond pricing discounts cash-flows using a rate or set of rates which are a function of the prevailing interest rate environment, but it also assumes reinvestment of cash-flows.  When rates, move, you always gain on one side and lose on the other.  This interesting effect means that there might be some combination of cash flows which 'naturally' is less sensitive to interest rate moves.  The process of finding this out for a portfolio of loans (or cash-flows) is called immunisation.


How to convert between a bank discount yield and a money market yield

The bond books describe the bank discount yield as $(F-P)/F \times 360/D$ and the money market equivalent yield as $(F-P)/PP \times 360/D$.  Let's combine these two equations.  Why?  in doing so, we will arrive at a relationship between the discount yield $y_d$ and the money market equivalent yield $y_m$.  This means you can do a direct yield to yield translation if required.

It turns out that $y_m = \frac{F}{P}y_d$.  Usually $F > P$ so $y_m > y_d$ so the discount yield will always look artificially lower than the money market convention, so you're up scaling the yield in the move from discount to money market.

Also, plugging this newly found relation back into original pair of equations, you see that $y_m = y_d \times \frac{360}{360 - D \times y_d}$ where $D$ is the number of days in the holding period. Again the numerator is greater than the denominator.

Thursday, 9 July 2015

Some yields are meaningful, some aren't, but they can all be used, and serve a purpose

Finally I'm beginning to see a few commonalities in the subject of yields.  First, think of all yields as either economically/financially meaningful, or not quite meaningful.  I suggested historical reasons why these non-financially meaningful yields exist.  

Second, think of simple interest as merely compounding with a frequency of once per holding period.  In that sense, all rates have some kind of time basis, and a compounding frequency.  Separately, those rates can be applied to periods of time different again from the time basis.  The simplest analysis is when the compounding frequency is once per holding period, and the holding period is the same length of time as the usual rate time basis, namely a year.  When the holding period is not a year, then there's always a question of how to convert.   At its most stripped down, the time basis is usually year based and/or day based, with an optional additional month counting algorithm.

Third, there are a number of markets out there with their own quoting conventions (some of which are financially meaningful, some not), so there's a need to use yield translators to facilitate easy comparison across disparate market quoting conventions.

Take the case of the humble six month Treasury bill, with price now $P$ and redemption or face value $F$ and six months left to run, $t=0.5$ or $d=180$.  That market likes to repackage the price as the bank discount basis, being $(F-P)/F \times 360/d$.  This is clearly a non meaningful yield. Notice that the second part of the expression is an attempt at annualising the holding period.  

It is possible to make it comparable to another, somewhat more meaningful yield, the money market equivalent yield, as might be seen with certificates of deposit.  This yield would be $(F-P)/P \times 360/d$.  Two changes make this much more economically meaningful.  First, the denominator is your initial investment, and second, the annualising factor, makes a more realistic assumption about the regularity of years and (implicitly)  months.  In doing such comparisons, you can compare the return more fairly with money market instruments. Ignoring the annualising factors for a moment, you can see that the money market equivalent yield is basically saying that $r = F/P -P/P$ that is, $r = F/P -1$ so that $F/P = 1+r$.  The rate in question is a holding period rate, and we chose one compounding period.  Ie $(1+r)^1 = F/P$; alternatively $P=F/(1+r)^1$.  So we have taken the heart of the money market yield equation and walked it around to look just like the formula for a zero coupon bond.  If we then go the final step and say that the holding period rate $r$ can be expressed in any number $n$ of notional compounding periods within the holding term, then $P=F/(1+r)^n$, which actually is the formula for pricing a zero coupon bond.  If $r$ is understood to be an annualised basis rate, you get to $P=F/(1+\frac{r}{n})^{nt}$.  In the world of bonds, $n=2$ is a hugely significant compounding frequency - namely twice a year.  If I have a rate (e.g. $r=0.2$ for a holding period, say half a year, to find out the equivalent annualised rate $r_e$ I'd use this formula: $r_e = (1+r)^2 -1$, i.e. 0.44.  But there's another of these non-financially meaningful market conventions which says that the equivalent annualised rate is $2 \times 0.2 = 0.4$.  This approach is called the bond equivalent annualised yield method and its main advantage is ease of calculation - you're pretending that this rate linearly scaled up (or down) to a 1 year basis; that is, you're ignoring compounding.




Sunday, 5 July 2015

Three pounds in six months

I walk into my bank and give them £97.  Six months later I walk in again and they tell me I have £100 in there.  What just happened?  Well, I made £3 in 6 months.  But in terms of yield.  Let's tell a number of yield stories.  But before I do, remember they're all tied back to these same basic facts, 97, 6 months, 100.  97, 0.5, 100. 

Story 1.  I got simple interest.  Let's calculate the holding period yield.  Remember the discussion about holding period yield.  The holding period is 0.5 years.  The holding period yield is (100-97)/97 or 3.092783%  The fact that I observed just two cash flows has made this simple interest story seem plausible.  Put the story in reverse. The bank said to me six months ago: give us £97 and we will make you a 3.092783% return on your money at the end of 6 months on a simple interest basis.

Someone might want to know what that return might look like if it was over a year instead of half a year.  Simple interest rates can be considered to scale linearly with time.  In reality things are more complicated.  In reality, you must consider implicit compounding in this re-basing operation.  But simply assuming linear scaling is an acceptable approximation for some circumstances.  So with twice as much time we would assume we would make 6.18556701%  The nominal period of that 6.18556701% rate is now on an annualised basis. Put that story in reverse. The bank said to me: give us £97 and will give you a return of 6.18556701% on an annualised basis, for a term of 6 months.

Story 2. My interest was being monthly compounded.  Well let's leverage off what we found out in story 1 to work out what the monthly compounding rate would be which can make 97 grow to 100 in 6 months.  We say $(1+0.03092783) =  (1+\frac{r}{6})^{6 \times 0.5}$ .  In other words the return is 6.1228718698%

Story 3.  They were rather kindly performing a continuous compounding for me.  The continuously compounded rate, quoted on an annualised basis, is 6.091841496%

All three of these are expressing a financially meaningful return for the 97, 0.5, 100 observed facts of the original thought experiment.

Story 4.  This investment was in US T Bills, and the bank discount rate implied by the move of +£3 over six months (let's say 182 days)  is 3/100 x 360/182, which is 5.934065934%.  

Story 5.  The investment was in money market instruments.  The money market equivalent yield is 3/97 x 360/182 or 6.1175937%

Saturday, 4 July 2015

Savings accounts are soil

The term yield has its origin in gieldan, the Germanic word meaning to repay.  It captures not the percentage element - namely that this measure is described relative to some other reference quantity - but to the fact that a repayment is happening.  So immediately chronology is of relevance.  The repayment amount is expressed as a fraction of some reference entity.  Hence the yield being the additional fraction of the base quantity that constitutes the repayment or reward.  It has clear agricultural meaning too - the yield being a measure of the produce which some quantity of land produces in a period of time.  Notice here that the agricultural yield doesn't naturally get expressed as a percentage.  Whilst it is clear that if you own 5 fields you may expect a correspondingly higher yield of crop, you don't really relate the number of carrots to some reference number of carrots.  Rather, you express your yield as a number of carrots per year, perhaps per field too.  If you imagine that there is a cost to having purchased and farmed that field, and that the crop of carrots likewise have a monetary value, then you can express the yield as a cash amount over a cash amount.  Here the yield is a percentage which captures the move from the beginning of the reference period, with your initial investment in the field to the end of the period, e.g. one year later (agriculture is is clearly a cyclical business, so years are one of the commonly used units of repeating time).  A field costs $P$ and then, a year later, you own some more carrots which are worth $y \times P$.  So in total you are now worth $P + y \times P$.  Assumed, of course, that the field's value has neither gone up nor gone down.  That is, $P(1 + y)$ .  Money itself is like a field.  The essence of the theory of the value of money at different points in the near future is that it can have a yield.  Money naturally makes money.  Money's crop is money.  The assumed easy way to do this is to put it in a savings account.  A savings account is soil.  Implicit soil.  The value of one unit of currency in one year's time is $1+y$, where $y$ in this case is the return you'd expect to get on a savings account in that country, for that sum, in that year.

The simplest financial yield to calculate is one where there are two logical observation points - namely the begin date and the end date.  No other financial action happens other than an implicit purchase at open and a reckoning or settling up at the end of the observation period.  With on intervening cash flows, there is no possibility of compounding.  So the simple interest formula can be used. Whatever happens inside those dates is assumed by hypothesis to happen uniformly or homogeneously.  Check out an earlier posting on the inter-relatedness of the discrete compounding formula and continuous and simple interest.  The difference between the simplest $P(1+y)$ above and the simple interest formula $(1+rt)$ can be explained by the fact that in many circumstances the rate or yield $r$ is expressed on a one year basis.  If you're happy to express the rate as a holding period rate, then $y$ is the same thing as $rt$.

Don't forget, in all discussion of the terms of a loan or description of a return, there are four elements which get applied to the principal or starting value, $P$, and they are {$r_{b_{{u}_1}}$,$t_{u_{2}}$,$n$} where r separately has an implicit time basis (usually a rate described in year units, and often expressing a nominal period of 1 year also, but also, especially when calculating return on an investment $b$ is what's known as a holding period yield); $u_2$ being the units in which your time (of loan or of investment) is measured - this is usually in days or years (or months), and $t_{u_{2}}$ then being the holding period or loan period itself.  Finally, $n$ is a unit-less counter which simply adds regular internal structure to the period under investigation, by adding moments  where an earning is recognised, meaning it officially counts as part of your capital from that moment on.  $n$ makes sense in the holding period window and is assumed to run from as small a value as $\frac{1}{t}$ all the way through to $\infty$. 

 So, when $u_1 = u_2$ namely that the rate basis and the holding period basis is "1.0 means one calendar year" and when $b=t$, namely that the expressed rate the scope of which is identical to the holding period itself (expressed as a number of years) and finally when $n=\frac{1}{t}$ then you have the simplest kind of yield, namely the simple interest one period yield.

If we call the final value $F$ then the equation $F=P(1+y)$ allows you to see clearly , after some re-arrangement, that $F-P=Py$, so $y=\frac{F-P}{P}$.  That is to say, you could express a holding period yield on a holding period ($b=t$) basis and simple interest basis ($n=\frac{1}{t}$) as the money made ($F-P$) expressed as a fraction of the money invested, $P$.

To spell this out a bit.  All you need is a starting value and an end value after a known period of time, and you can begin to enter the world of yields.  Once you have a holding period simple interest yield, you can set about translating this yield into the many other kinds of yield there are.


Why are bond yields usually less than 10% in the developed capital markets?

What is a yield?  It is a percentage.  It is a relative measure.  What do financiers see or think when they see bond yields?  They compare those with others.  What's the meaning of a 5% coupon?  A 5% return on investment?  And why don't bonds pay, say 100% during normal times?  What kind of world would it have to be for that to be a regular occurrence?  The borrower when a business is more often than not borrowing to finance a line of business (or re-financing a previously financed line of business).  If they decided to borrow on those terms, then for the line of business to make a profit, then the expected annual profitability needs to be somewhere north of 100%.  Businesses simply don't mature that rapidly, nor do they remain so profitably too long before competitors arrive in to bring down average profitability for the sector participants.  So capitalism itself, and the natural difficulty business managers have in finding investible lines of business mean that seeking funding in the first place would not be worth it if the lender expects 100% return or anywhere near that.

Yields on bonds are nominal, in the sense of being unadjusted for inflation.  Loans promise to pay back a future fixed cash sum.  An intervening period of high inflation, for example, will erode the real value of the cash flows, including the final payment.

So I ask again, what is a bond yield?  It is made up of a risk free part, a part shared by the population of peer bonds, and a more firm specific part, which measures the market's judgement on how risky this firm (or sector) is.  A macro-economic context and a firm-specific context.  You get, say, 7% for this bond because the prevailing economic climate dictates that even the least risky lender gets, say, 4% and you receive an additional 3% because of the market's judgement on how confident it feels that the firm will honour its debt obligations.  In the jargon of finance, the yield has a risk free component and a credit component.