Guest Post: Legerdemath II: Anatomy Of A Banking Trick

Submitted by Omer Rosen of Legedermath, who will gladly answer any reader questions related to the matter presented in the comment section.

Legerdemath II: Anatomy of a Banking Trick

In my previous article, “Legerdemath: Tricks of the Banking Trade,” I made brief mention of Treasury-rate locks:

Most brazenly, we taught clients phony math that involved
settling Treasury-rate locks by referencing Treasury yields rather than
prices.

A number of readers expressed a doubt that using a settlement method
based on Treasury prices was appropriate. What follows is as good an
explanation of Treasury-rate lock settlements as 2,000 words will allow.
I have simplified some of the bond math and concepts and will end with
an analogy that I hope will elucidate what the math did not. However, as
this post hardly qualifies as an easy read, feel free to ask questions
in the comments section. Confession: I fudged the word count a few
sentences ago to increase the likelihood of you reading on.

Forget for a moment, everything you have heard or think you know
about Treasury bonds. Taken in isolation, the purchase of a Treasury
bond is nothing more than the purchase of a fixed set of future cash
flows. If you find the term “cash flows” confusing, think instead of the
predetermined dates in the future.

In this column I will be referencing a 10-year Treasury bond paying a
coupon of 5.00%, with a notional amount of \$100. For convenience, I
will christen this bond “Bondie.” Sans jargon, the fixed set of cash
flows received when purchasing Bondie would be \$2.50 every 6 months for
10 years and an additional \$100 at the end of the 10th year.

There are two basic ways to describe the value of this fixed set of
cash flows, either by price or by yield. Price answers a simple
question: How much would it cost you to purchase this fixed set of cash
flows? This price will change over time, in much the same way that the
price of a stock changes over time. Yield expresses the return earned by
purchasing these cash flows at a certain price.

If you had to pay \$100 in order to receive the fixed set of cash
flows I described above, then your yield would be 5.00%. If you had to
pay more to purchase these same cash flows, say \$105, then the return
you would be earning (the yield) would be lower than 5.00% – it would be
4.3772%. Intuitively this should make sense – the more you have to pay
for a given set of cash flows the lower your return will be. Or, more
simply, when prices go up, yields come down. Conversely, if you had to
pay only \$95 for these same cash flows, the yield earned would be higher
than 5.00% – it would be 5.6617%.

Algebraically speaking, price and yield are linked by an equation
where all the other variables are known. Therefore, if you know the
yield of a given bond you can calculate the price of that bond and vice
versa. In plain terms, saying you are willing to pay \$100 for Bondie is
the same as saying you are willing to buy Bondie at a yield of 5.00%
(i.e. at a price that will allow you to earn a return of 5.00%). It is
similar to how one can describe the speed of a car either by the number
of miles per hour it is traveling at or by the time it takes it to
travel one mile – if you know one you can solve for the other, and if
one goes up the other comes down.

To belabor the point, if a car is traveling around a 1-mile track at
an average speed of 1 mph then it is easy to solve for the time needed
to complete a single lap: 60 minutes. Either “1 mph” or “a 60-minute
car during that lap. And, if the car’s speed were to increase, the time
it would take to complete another lap would decrease (At 2 mph a mile
would only take 30 minutes). The same inverse relationship holds true
between prices and yields.

Now back to Treasury-rate locks. When a company puts on a
Treasury-rate lock, it is doing nothing more than taking a short
position in a Treasury bond. A short position is a bet that will pay off
for the company if Treasury prices go down and go against them if
prices go up. Why would they do this? That is a subject for another
column and I ask that you accept as an article of faith that sometimes
this bet, rather than being a gamble, reduces risk and uncertainty for a
company.

The short position can be viewed as an agreement under which the
client will sell the bank Treasury bonds at a certain price on a set
date in the future. This price is determined based on current market
conditions. For example, let us say, that based on what current market
conditions dictate, the client agrees to sell Bondie to the bank at \$95
one month hence. A month passes and Bondie is now trading at \$100. The
client will have to go into the market, buy Bondie at the current price
of \$100, and then sell it at a loss of \$5 to the bank at the previously
agreed upon price of \$95. For expediency’s sake, the client just pays
the bank the \$5 it has lost and the bank takes care of all the buying
and selling behind the scenes. The calculation of \$5 in the above manner
– subtraction – is an example of the price-settlement method of
Treasury-rate locks.

However, when it comes to bonds, corporate clients do not think in
terms of price; they think in terms of yield because yield is expressed
in the language of interest rates, the same language companies are
familiar with from business concepts such as rates of return and
borrowing costs. In theory, this should add only a simple step to the
settlement process. The company locks in a sale of Bondie at the same
level as before, \$95, but rather than quoting them that price the bank
quotes them the corresponding yield of 5.6617%. We can refer to this
yield as the locked-in yield.

A month passes and the Treasury rate lock is settled. Rather than
telling the client that Bondie is now trading at \$100, the bank tells
them that the yield is now 5.00%, having fallen by 0.6617%. But 0.6617%
is not a dollar value that can be paid out as a settlement. To calculate
the settlement, both yields, 5.6617% and 5.00%, need to first be
converted back to their respective corresponding prices, \$95 and \$100.
Taking the difference between the two prices results in the same
settlement value we calculated before: \$5.

But the client is never shown how to settle based on prices. Instead
they are introduced to a nonsensical and more complicated method called
yield settlement. The sole purpose of this settlement method is to trick
the client into allowing the bank extra profit.

Whereas price settlement asks the question, “By how much did Treasury
prices change?” yield settlement asks, “By how much did Treasury yields
change?” As mentioned in the previous paragraph, the yield decreased by
0.6617%. But how does one convert 0.6617% into a dollar value that can
be paid out?

First, a unit conversion is necessary. For clarity and convenience,
finance makes use of a unit called a basis point. Each basis point is
equal to 0.01%. Using this new unit, the above decrease of 0.6617% can
be expressed as 66.17 basis points. Of course, this solves nothing, only
modifying our most recent question slightly: now we ask, how much is
each of the 66.17 basis points worth in dollar terms?

At this point the client is introduced to a concept called DV01
(Dollar Value of One Basis Point). DV01 is defined as the change in
price of a bond for a one basis-point change in yield. For example, if
the yield on a bond changes from 5.00% to 5.01% or from 5.00% to 4.99%,
by how much would the corresponding price of that bond change? This
change in price is the DV01. If yields shifted by 66.17 basis points,
DV01 will answer the question of how much each of these basis points is
worth.

The starting point for this calculation is the yield at the time of
settlement. In our example, the yield at the time of settlement is
5.00%. At this yield, the corresponding price of Bondie is \$100. If the
yield were to rise by one basis point to 5.01%, the corresponding price
of the bond would fall to \$99.922091, a decrease of 7.7909 cents. If
instead the yield were to decrease by one basis point to 4.99%, the
corresponding price would rise to \$100.077983, an increase of 7.7983
cents. By convention, the average of these two changes in bond prices is
taken to be the DV01. So, at a yield of 5.00%, the DV01 would be 7.7946
cents per one basis-point move ((7.7983 7.7909) ÷ 2). If the yield
changes by one basis point, price is said to move by 7.7946 cents. Or,
in more plain terms, each basis point has been assigned a value of
7.7946 cents.

The DV01 is then multiplied by the difference between the current
yield and the locked-in yield. In our example the difference between
5.00% and 5.6617% is 66.17 basis points. From the previous paragraph we
know that each of these 66.17 basis points is worth 7.7946 cents.
Multiplying 66.17 by 7.7946 cents we arrive at a settlement value of
\$5.1577. This is the yield-settlement method of Treasury-rate locks.

Apart from being confusing, the yield-settlement method has resulted
in a settlement value that is greater than the \$5 calculated using the
price-settlement methodology. For a good-sized rate lock, say \$500
million dollars worth of 10-year Treasuries, the client would pay the
bank an extra \$788,500 (500 million x (5.1577 – 5.00) ÷ 100) when
settling using the yield-based methodology. This “extra” is profit for
the bank.

I ask that you stop reading here for a moment. I have stated from the
beginning that yield settlement is incorrect. However, when reading the
explanation of yield settlement, did you find yourself agreeing with
the logic? At what point, if any, did you spot the flaw? And can you
guess what happens if prices had gone the other way? If prices had gone
down instead of up, say to \$90, the bank would have owed the client
money. However, yield settlement would have allowed the bank to earn a
profit by paying the client less than it actually owed them. No matter
what happens to prices, yield settlement allows the bank to earn extra
profit.

Now picture yourself as a client receiving a tutorial on
Treasury-rate locks. You are being instructed by a banker on a matter
without any warning that something might be amiss. You are led through
the yield-based settlement process and taught how the DV01 is
calculated. If you have access to a Bloomberg terminal you are shown
where the DV01 can be found on the relevant Treasury bond’s profile
page. Perhaps presentation materials are sent over detailing the
mechanics of rate locks and different possible outcomes depending on
various possible market movements. And all this is part of a larger
interaction, a relationship even, during which the banker is nothing but
genuinely friendly and informative. Furthermore, there is a good chance
that someone from a different part of the bank, someone who has advised
you before, was the one that introduced the two of you in the first
place. Would you question your banker?

Clients, among them some of the largest corporations in the world,
never did. Confident in the tools provided them and blinded by specious
logic, the client never even thinks to question the underlying
methodology. And, especially since the client is never made aware of
price settlement, the methodology does sound logical: Check to see by
how many basis points Treasury yields moved. Calculate the dollar value
of each basis point. Multiply the two and arrive at a settlement value.

However, this methodology is an approximation that always works out
in the bank’s favor. Why? Because each of the 66.17 basis points has
erroneously been assigned the same value of 7.7946 cents. The DV01
calculated at a certain yield is only valid for a one basis-point move
away from that yield. Therefore, while the first basis-point shift away
from 5.00% is indeed worth 7.7946 cents, successive ones are not.

Put another way, DV01 at 5.00% is different than DV01 at 5.01% is
different than DV01 at 5.02% is different than DV01 at every other
yield. And so the value of the basis-point change from 5.00% to 5.01% is
different than the value of the basis-point change from 5.01% to 5.02%
is different than the value of all successive basis-point changes. In
fact, even the original DV01 is inaccurate because it was taken to be an
average of two different movements. Multiplying the 66.17 basis-point
change by a single DV01 ignores all this and assumes that the
relationship between changes in yield and changes in price is constant –
that each one basis-point move results in a fixed change in price no
matter what the yield. Yield settlement takes the graphical
representation of the relationship between prices and yields – a curve –
and flattens it into a straight line.

Admittedly, all this can be a bit confusing. After all, if price and
yield are both valid ways of expressing the value of a bond, shouldn’t
you also be able to measure the change in value of a bond by looking at
either the change in its price or the change in its yield? The math says
no. Resorting to hyperbole, teaching the client yield-based settlement
is akin to selling them on time travel.

Return for a moment to the example of a car driving along a 1-mile
track (a conceptual, though not mathematical, equivalent to rate lock
settlements). In this analogy, “mph” will play the role of “yield” and
“travel time” will play the role of “price.” Assume the car is traveling
at a speed of 1 mph. If the car speeds up to 2 mph, the time required
to travel a mile decreases from 60 minutes to only 30 minutes – a
30-minute decrease in travel time. This 30-minute change plays the role
of “DV01″.

Now assume that the car is traveling at a speed of 120 mph. If again
the car’s speed increases by 1 mph, here to 121 miles per hour, does the
time needed to travel a mile again decrease by 30 minutes? Since a mile
only takes 30 seconds to complete at a speed of 120 miles per hour,
short of a DeLorean and some lightning, reducing the completion time by
30 minutes would be impossible. The actual reduction in travel time –
the “DV01″ – would be only a fraction of a second at this high speed.
“DV01″ is not a constant in this analogy either.

To extend the analogy, calculating a rate lock settlement would be
akin to calculating the difference in travel times for each of two laps.
If lap 1 were completed at a speed of 120 mph and lap 2 at a speed of 1
mph, how would you calculate the difference in travel time between the
first and the second lap? Would you take the difference between 120 mph
and 1 mph and multiply that difference by the 30-minute “DV01″
calculated above? Doing so would imply an impossibly high difference
between the two lap times: 3,570 minutes ((120 – 1) x 30). This
calculation is the parallel of the yield-settlement method.

For makes and models without a flux capacitor, you would simply look
at the difference between the times the car took to complete each lap.
If a stopwatch is not handy, the following quick math provides the
answer: a 120-mph lap takes 30 seconds to complete and a 1-mph lap takes
60 minutes to complete. The difference in travel time between the two
laps is therefore 59.5 minutes. This calculation is the parallel of the
price-settlement method. As you can see, the 3,570 minutes calculated
using the other method is far off the mark.

In price/yield relationships the same problem exists – that problem
being the realities of math. Yet banks I encountered almost always
instructed clients to use the yield-based settlement method. And so a
product that is meant to return the difference between two Treasury
prices, a matter of elementary subtraction, is perverted for profit.

If yields change by very little, this profit does not amount to much.
Fortunately, depending on one’s point of view, banks have other tricks
for profiting from rate locks and do not rely solely on yield-based
settlement. In fact, miseducating clients with yield-based settlement is
almost an afterthought, just a bonus that pays off with large movements
in yield. Because as yields move by more and more basis points two
things happen: First, there are more basis points to infect with an
erroneously constant DV01. Second, the constant DV01 becomes an even
worse approximation for the proper DV01 of each basis point.

In behavior that might be considered yet more sinister, sometimes
banks had to implicitly agree with one another to use yield settlement.
This transpired if a client decided to divvy up a single rate-lock
transaction, with each bank getting a piece of the deal and each bank
knowing that settlement of the rate lock would have to be a coordinated
affair.

All this mathiness is hidden in plain sight. Some examples of yield
settlement can be found online. Or you can just ask a company that put
on a rate lock to dig up some trade confirmations and see what
settlement methodology was used. There are hundreds, if not thousands,
such documents in corporate offices around the country, each one part of
an unwarranted transfer of millions of dollars from clients to banks.

Comment viewing options

youtube theweeklytelegram referenced fed funds rate setting/correlating with a yield on a short term bond (months)

related perhaps.

The latest financial trick -blaming the weather- is now being employed in the upcoming complete defeat of the NATO forces in Libya.

Watch as Joint Chiefs Chairman Mike Mullen deflects the incompetence of the "coalition", and blames the weather for Gaddafi's victory:

Now that is what I call useful financial information. Good effort with a very dry subject...

none of this bullshit there eh...

TLDR.

I'm just gonna heckle from the bleachers.

Basis points are NOT hard to amortise.

correct. also, bonds (govies) trade on yield basis but settle up on price t+1 because accrued interest isn't factored into yield.

Thank you O.R.

++ Double plus good.

This is an area of weakness for me. Thanks for the analogy.

Ban exponents NOW!

This reminds of those test questions about constructing arbitrage free zero coupon curves (each cash flow discounted at the implied forward rate for the yield/price to be arbitrage free).

I don't get this. Then again, bloomberg swap calculators tend to solve these problems for me.

I was told there would be no math.

Hey, I actually understood that!

Kidding, right?

I understood it too, buy gold.

No yield = easy...

but then gold strips the bullshit from any transaction, except the am/pm fix...

What if rates go up? Then banks would lose extra money in this example by overestimating the settlement price change.

Nope. The bank makes money either way. If prices go up (rates go down) the client overpays the bank.  If prices go down (rates go up) the bank underpays the client. I touched on this latter scenario briefly when I discussed what would have happened if price had decreased to \$90.  Because you are taking the DV01 on the settlement yield, in one direction the DV01 used is too high, and in another it is too low. In either case, the house wins.

(I'll see if I can post a graph that shows why this is if you like...)

Also, the bank effectively has two opposite positions on, one with the client where the client is short and the bank is long and one matching position the bank puts on via the market as a hedge where the bank is short Treasuries.  The profit from this trick is the difference between the amount the client pays/receives using the incorrect yield-based formula and the amount the bank pays/receives on its hedge, which settles using the correct price-based formula (i.e. subtraction).

ahhh.  i re-read that part 3 times, even pulled out the calculator.  would have been helpful if you had illustrated the change in DV01 based on SETTLEMENT prices.  once i realized that it was based on 90, instead of 100, it made total sense.

Yeah, I would have wanted to do that, but the damn article is too long as it is...and some people are put off by numbers and stop reading the more numbers they see...anyway, if you scroll down I just posted a scenario analysis with different DV01's in different scenarios in response to someone else. Glad you got it...will still try to get a graph up later for others.

Thanks, this info is all much appreciated.

They believed each bp was consistently worth the same!  This is so painful!  There is a point where taking advantage of another's stupidity is no longer unethical.

Really good explanation

Thanks...though I would say that it was less that they believed each bp was worth the same and more that it just never occurred to them that that might not be the case. Perhaps some of them even misinterpreted it as some sort of average DV01...though that is speculation. As best I can tell, they never had even a passing thought about the validity of a constant DV01 - it was simply accepted passively, like one accepts without even thinking that it is night when it is dark outside. Damn. I'm not happy with the night analogy I just used...but I got nothing better coming to mind now so it will have to stand.

Article is appreciated. Thanks. Basically, we're trying to replace a curved function with a linear one. It's close to reality for small yield changes, but breaks down when dealing with larger ones.

Tsy bonds do have positive convexity but I don't think that's what the author is talking about..

Yep...I have often thought that the math in this article could be replaced by two words: convexity flattening. But what fun would that be?

Tsk, tsk.  Very sneaky.  Of course, this is a good illustration of why people should learn math.  Great explanation, though.

Thanks...though I would say that knowing math is one thing, knowing how to apply that math in a new and unfamiliar situation is another thing, and knowing that something is wrong with a certain way of doing things by being to access all the math you know at any given point (i.e. knowing that you have to apply the math you know how to apply) is quite another thing altogether.

Frankly, I wouldn't be surprised if people with an intimate understanding of convexity would fall for this simply because of how it is explained and because the context is new and because they don't even think to put their math thinking caps on as they just passively accept what they think is procedural.

Omer,

you are three words short.

Or, it it six counts...and I am three word long?

I asked in another thread if anyone here at ZH still dream that government can tell the truth?

I would ask here if anyone here at ZH still dream that bankers can tell the truth?

Thanks for the profitable sharing. I am going to tell my CFO and CFA friends to take this 1997 word lesson.

Excellant how to discussion.

Hopefully you can share many more tricks of the trade.

I actually think your situational description of how a client's CONfidence gets played was more important then the math lesson.

You might be right on that account. However, I do think specific and detailed examples are still a necessary component of the discussion...they make arguments harder to dismiss.

This sounds like a Carine trick. Why would anyone want any part of this. The bond is a contract / IOU. The flow is agreed to. So it's \$5 per year, paid in two payments of \$2.50 per year. Plus you get the principle back (\$100) at the end of the agreement. That's it! No more no less. I had a car salesman come up with so crap about how I could save money by financing the new car, and not pay cash?? But telling us that this stuff works with large corporations (CFO's)??

Just to be clear, the bond is not really an investment in this case. It is used here as a reference of sorts for a bet of sorts. The company is taking this bet because it will soon be issuing its own bond and is afraid that its borrowing costs will rise by the time it actually issues its bond. If this were to come to pass, then the hedge would compensate them. The downside to putting on this hedge is that if rates go down and the company's borrowing costs go down with them, it does not get to benefit from the situation because in this scenario it would be paying a cash settlement to the bank.

very nice article....a fucktard always wants a linear world because non-linear worlds are too hard...why does the client accept this? i hate to invoke caveat emptor but it seems fitting.

A client accepts this because he doesn't know any better (perhaps even doesn't know that he doesn't know any better)...and because a banker does his best to make sure the client doesn't know any better, whether by miseducation or by omission. If a client ever has his guard up at all, then he will be very vigilant when it comes time to actually put on the hedge - he'll want to be as close to market prices as possible (or as close to what he thinks market prices are - again, this faulty knowledge would either stem from his own lack of knowledge or because of banker miseducation) plus any agreed upon bank profit. But, as best I can tell, clients think that these other issues, these details, are just procedural. And it doesn't occur to them that someone might lie about these types of things...and lie about them so openly. Furthermore, a client has a whole other job to do that doesn't necessarily leave time for them to become experts in all the possible ins and outs of a derivative trade.

I can see an order taker or clerk being sucked in this way but don't these companies have expert accountants and CFOs who can see through the tricks?

excellent article thx for the lesson

The question from the banks to the clients is 'Do you know the relationship between "x" and "1/x?"'

As in, when "x" goes to zero, what does "1/x" do?

Its self-preservation instincts kick in and it lobbies Congress to keep "x" from ever reaching zero?

For those who want more...

When googling rate lock settlements, I've only ever seen references to settling based on changes in yields - settling based on prices is never mentioned. Here are some old bank marketing materials I found on the internet…you’ll find the incorrect formula on page 3. Something else that's interesting is that the formula is shown again on page 4...but with a different DV01.

http://www.interestrateswaps.info/LOCKING%20IN%20TREASURY%20RATES%20WITH%20TREASURY%20LOCKS.doc

This is extremely illuminating.  Based on personal experience, I am quite sure that many municipalities are in the process of being screwed in an analogous way on interest rate swaps entered into in "synthetic fixed rate" transactions that were fashionable in 2003-2007 and that are now very expensive to get out of now that rates have fallen.  Quotes for settlement of these swaps are also expressed in DV01.  This leads me to believe that this type of chicanery--using linear functions to represent curves--is pervasive and not limited to treasury rate locks.  However, some intuitive math part of my brain insists that there must have been situations where yield-settlement methods cut against the bank.  For instance, taking the example in the original post, if the client went long instead of short, thereby trading places with the bank in the example, would the bank still insist on yield settlement, and, if so, would that not cut in favor of the client rather than the bank?   Or would the banks just choose the settlement method that cut in their favor, in that case, price settlement rather than yield?

So a few things. Going long Treasuries would involve a simple purchase and wouldn't require a derivative (in theory a short doesn't require a derivative either, just borrowing Treasuries, selling them into the market, depositing the cash from the sale with the lender of the Treasuries, receiving a finance rate on the cash you deposited, paying the lender of the securities the Treasury Coupons he would have received if he had not lent out the Treasuries, then buying the Treasuries in the market and returning them to the lender and receiving back your cash. The derivative sort of houses all this in an instrument for the client while the bank does all the other stuff I just mentioned behind the scenes). Also, within the context that I dealt with them, it would not be normal for a client to ask us to help it go long Treasuries (perhaps this might occur as part of a complicated black-box type derivative but that is a different matter). In short, no, there was never a situation where we somehow had to use yield settlement in a trade in the opposite direction and therefore lose money.

As for swaps, they're a bit different than Treasuries...are you sure the swaps you are discussing settle on DV01 and not PV01? When it comes to swaps the two mean different things.

While various people use these terms in different ways, the two underlying concepts in the following should make sense even to people who use the terms differently: For a swap, DV01 is the change in the value of the swap for a 1 basis point shift in the yield curve. The PV01, on the other hand, is the present value of an annuity of 1 basis point of the swap's notional amount (though probably broken up into parts as per the swap details) over the remaining life of the swap. While using the DV01 to settle the swap would not be correct, settling based on PV01 would be mathematically correct.

If you think the swaps you are talking about were settling based on DV01...let me know and we can talk about it more.

Thank you for responding.  Technically, they are supposed to be settled using the ISDA second method/market quote but when the clients ask for an indication of what it would cost to terminate, they would typically get a "value of a basis point" chart from their dealer/counterparty that showed sensitivity of settlement cost to the prevailing rate for the fixed rate leg, which is what I thought DV01 is.  So while this might have overstated the estimated cost, it is quite possible the market mechanism in these cases protected the clients that actually unwound, assuming the market was in fact competitive.

But my main conceptual question is how can the yield settlement method always cut in favor of the bank, regardless of which way the trade moved after execution?   I guess I'd like to see the graph you referred to in your reply to babylon15, above.

Sure. I'm not sure if I can upload graphics here so I'll edit this reply with a link later today.

Banking is all phony math. Trying to make it sound scientifical by applying physics velocity equations to money velocity. LOL

If you are an accountant you can never understand the banking system it's entirely populated by no-account liars and thieves.

superb piece

Thanks for this informative article, I learned from it. As with so much here at ZH I almost wish I hadn't but in the end its better to know what the insignificant men behind that curtain are really up to however uncomfortable such realizations are.

I'm sorry, but in your example you showed yields moving from 5.6617 to 5% instantaneously resulting in a price move of \$5.00 and a yield settlement of the lock of \$5.1577.  Advantage to bank of \$0.1577.

In the scenario where the bond goes to \$90, instantaneously, the equivalent yield would be 6.36722%, for a move of 70.55393 basis points.  Using the yield settlement math you showed before the bank would then pay \$5.5020 (70.55393 bps times 7.7983 cent DV01).  This is a disadvantage to the bank of \$0.5020.

What am I missing here?

If you use an equivalent basis point move down, so the bond goes to 6.3234% and an equivalent price of 90.3015.  Yield settlement is \$5.16 to the client versus bond movement of \$4.70.

You wouldn't use the same DV01. You always calculate the DV01 off of the settlement yield, whatever that is. I'll edit this reply in short order with all the numbers needed to settle at \$90

so, first off, the 7.7983 DV01 you used isn't really applicable to any of the scenarios since it is only one of the two numbers used to arrive via averaging at the DV01 of 7.7946 at a yield of 5.00% (though averaging is techinically wrong, that's the standard calculation for DV01). That said, if it were correct, it would only have been used in the specific scenario in which yield at settlement was 5.00% (in accordance with yield-settlement instructions). Here are the various settlement calculations for the scenarios you described:

locked-in yield: 5.6617%
locked-in price: \$95

scenario 1 at settlement:
yield: 5.00%
price: \$100
dv01: 7.7946
price settlement: client pays \$5.00
yield settlement:
change in yield: 66.17 basis points
client pays 7.7946 * 66.17 basis points = \$5.1577

scenario 2 at settlement
yield: 6.3672%
price: \$90
dv01: 6.8554
yield settlement:
change in yield: 70.55 basis points
client receives 70.55 * 6.8554 = \$4.8365

scenario 3 at settlement:
yield: 6.3234%
price: 90.3012
dv01: 6.8835