adjusted for the differencein effective ®nancing periods.     dTdY. dY!T 1  rT1  rYis the difference in net present value of future dividend paymentsuntil expiry.H. FHY!T  FY 
rY rT1  rT   1  rY 
is the difference in the position cost base.This decomposition allows us to extract one more layer of information from the valuation process by breaking the total ®gure into buckets corresponding to different parameters. For example, assumingevery parameter except S remains constant from yesterday to today,the fraction of the valuation exclusively due to a movement in the spotprice is:YP QY  PV  S0Y!T  QY  PV  STVY!T
SY 
Along the same line, the fraction of the valuation attributable todividends only is:YPVY!T
QY  PV  dY!T
Finally, we probably would like to know the impact of interestrates, but this is a little tricky. First of all, r appears everywhere inYP; therefore, there is no single term carrying the informationVY!Twe want. The more important dif®culty, however, is the fact that rcarries the time component. In effect, even if the interest rate is constant, in our formulas rT 6 rY because rY has one more day in the timefraction. To solve this problem, in practice the valuation is run keepingthe rate constant with 1 day difference or with the same number of
294
FINANCIAL VALUATION AND RISK ANALYSIS
days and different rates. This procedure is simple to implement, but wewill not develop the algebraic formulation here because it takes upspace.Instead, let's consider the question from another angle. The cost-ofcarry model relates interest rates and the price of the futures contract;therefore, it is possible to extract the dependence with time from it.Speci®cally, if we express FTH  C  1  r  n=360 d , then25:FTHCrY!T Y!T 360nIn other words, the price of the theoretical futures contract decreaseswith time, by an increment exactly equal to the ®nancing charge. Thisis naturally an approximation because theta here applies only to thefutures price, not to the exact valuation as we de®ned it. However, itgives a very good sense of what is going on with respect to the carry.Along the same lines:FTHCnY!T Y!T 360rwhich indicates that the futures contract is exposed to interest rates onthe full amount of the underlying cash, until expiry. Overall this explanatory valuation proves extremely useful in practice because it allowstraders to track the behavior of their position with respect to eachindividual risk.ForwardsThe key difference between futures and forwards is the absence ofmargin calls, which means that a forward cannot be valued at a marketprice; its valuation is exclusively theoretical. Its expression is exactlyequal to that of a regular theoretical future.26 The ®nal settlement ¯owis therefore always taking place at expiry, and it must be discounted tothe appropriate valuation date.!FTHFHFTHFHTYYYYPVY!T  QY  PV 1  rT1  rY25
Note that n is a number of days; therefore, n  n 1 n  1 from today totomorrow.26Q denotes the number of contracts held. The product Q  PV is often referred to as thenumber of units of the transaction.
CHAPTER 6
V
TD

QT
QY   PV  FTHT 1  rT
P
VALUATION
295
CFT
From a treasury perspective, there is no ¯ow whatsoever at theinitiation of the forward transaction, or during its life. Therefore,there is no overnight cost associated with the payment of dailymargin calls.
Money MarketsFront Of®ceConsider a straight borrowing entered at some point in the past, at arate r. Figure 6-1 shows a representation of all the ¯ows. rT is today'szero coupon to the maturity of the loan. Liquidating the position heremeans entering into another money market transaction, in which wereceive N  1  r on maturity, as shown in Figure 6-2. The valuation ofthe original transaction is therefore equal to:1rrT rNVT  N N 1  rT1  rTThe term (rT r) indicates that we are exposed to a variation in interestrates, which is highly intuitive. If we borrow today at 5 percent and thesame rate goes down to 4 percent tomorrow, we lost 1 percent over theperiod because we are committed to paying 5 percent in a contextwhere the market demands only 4 percent. For example, considerthat we entered into a $1,000,000 borrowing operation 1 monthagoÐthat is, 30 daysÐat 5 percent, for a maturity of six monthsÐthat is, 180 days. Assume that today the 5-month rateÐthat is, 150daysÐis 4.5 percent. The valuation of this borrowing, which we expectto be negative, is:
rT
N
Toda yN * (1 + r)
Figure 6-1
Money Market
296
FINANCIAL VALUATION AND RISK ANALYSISrT
N
N * (1 + r)/(1 + rT)
Figure 6-2
N * (1 + r)
Money Market
 150  4:5%180  5%360360VT  $1,000,000 150  4:5%1360
$6,134:97
YPWe can now express VY!T, which is simply:rT r rY rrrY   1  rYPVY!T  N N T1  rT 1  rY1  rT   1  rY 
As for V TD , it can be computed according to the same principle. If wecall rDT the rate at which a transaction was executed today:!rT rDTTDV  N 1  rTNote that this formula applies to an individual transaction. In contrastto stocks and futures that we might trade very actively during a givenday, in general, money markets are traded occasionally. Therefore,each operation is likely to have unique characteristics and should betreated independently.The above formula provides a very clear illustration of the impactof the bid-ask spread on the valuation. If we use midmarket prices forthe zero-coupon yield curve, the difference rT rDT is probably close to0.06 percentÐassuming rDT is an offer priceÐwhich indicatesV TD < 0. By evaluating all our position at a midmarket price, we book aloss equal to half the spread whenever we trade, regardless of the direction.Conversely, if we chose rT to be an offer price, we would not recordany loss, but for every lending transaction, we would then have torecognize a loss of 0.125 percent.From a treasury perspective, a money market operation has a wellde®ned impact because when a transaction has been made, no more
CHAPTER 6
VALUATION
297
¯ows occur until maturity. In other words, CT  CY  N on the day themoney market is entered into, CT  CY N  1  r when it expires,and CT  CY in between.To illustrate the implications of this situation, consider an activitywith a treasury balance equal to 0: CY  0. Assume the trader enterstoday into a lending operation for a nominal N at a rate r : CT  N.For simplicity we consider that rT  r, so that V TD  0.YPis given by27:The next day, VY!TYPVY!T
N
rT rY   1  r1  rT   1  rY 
N  rT
rY  
N  rT360
under the simplifying assumptions 1  rY   1  rT   1  r  1,and rT  rY . The difference rT rY  can then be simpli®ed into rT =360 because there is exactly 1 day difference between yesterday andtoday.28 The total valuation includes the overnight and is thus equal to:!N  rTN  rT N  rOrT rOYYO CY  IY   rY NVY!T 360360360360This interesting result shows that the exposure created by the transaction is a differential risk, between the rate remaining to maturity andthe overnight. In other words, this risk is of the second order. In contrast, if we buy a stock outright, the risk created is of the ®rst order,directly on the price of the stock.ApplicationsAs a numerical application, consider the same money market operationwe used above. Its valuation today is VT  $6; 134:97. Let's imaginenow that the next day the market rate has changed to 4.6 percent.YPis equal to:VY!T 149  4:6% 150  4:5%180  5% 1360360360YP VY!T $1,000,000 149  4:6%150  4:5% 11360360 $285:2327
Note that we value a lending transaction; therefore, the formula has been adaptedfrom N to ±N.28Naturally if there were a banking holiday or a weekend between yesterday andtoday, the difference would be more than 1 day.
298
FINANCIAL VALUATION AND RISK ANALYSIS
A direct calculation gives VT of $285.23.
$5,849.74Ðthat is, an equal difference
Explanatory ValuationsThe above difference of $285.23 between yesterday and today is theresult of two effects, change in time and in interest rates. As we did fora theoretical future, it is useful here to introduce again an explanatoryvaluation and try to break the overall ®gure into several boxes. If ndenotes the number of days between today and the money marketexpiry, we have29:n  rm  rTrr360 VT  N  TNn  rT1  rT1360where m stands for the total number of days accruing over the entirelife of the loan. Therefore:n  rn  1  rY m  rm  rT360YP360 NVY!TNn  rTn 1  rY11360360assuming for simpli®cation that the actual number of days until expirywas (n  1) yesterday. If we want to isolate the effect of time, we keepthe interest rate constant equal to rY :n  rn  1  rY m  rm  rY360YP360 NVY!TNn  rYn  1  rY11360360m  rr 1Y 360 N:n  rYn  1  rY3601: 1360360This quantity is called the theta () of the position by analogy to thetheta of an option. It represents the variation in valuation due exclusively to the passage of time. In the example above, r  5 percent onm  180 days and rY  4:5 percent on n  150 days. Therefore:29
Note that we used a shortcut: n is in fact the number of days from today's moneymarket settlement date to the money market settlement date of the loan.
CHAPTER 6
VALUATION
299

Y!T 
4:5%$1,000,000 360
180  5%360 149  4:5%150  4:5%1 13603601
$123:47
If we want now to evaluate the effect of r, we have to keep the timeconstant:n  1  rT m  rn  1  rY m  r360360YPVY!TNNn  1  rTn  1  rY11360360m  rr1rY 360 N  n  1  Tn  1  rTn  1  rY3601 1360360This quantity represents the variation in valuation due to a change ininterest rate, and it is used to compute the rho of the position:Y!T 
YPVY!TrT rY
Numerically:
4:6% 4:5%YP $1,000,000  150 VY!T360180  5%1360   $411:34150  4:6%150  4:5%1 1360360
which leads to30:Y!T 
$411:34 $411,3400:10%
YPFrom there we can reconstruct VY!T Y!T  rT rY  Y!T  $123:47  $411:34  $287:87. This value is slightly different
30
For back-of-the-envelope estimates, theta and rho can be approximated to the ®rstorder:r n1Y!T  N Y!T  N  Y360360
300
FINANCIAL VALUATION AND RISK ANALYSIS
from the one we computed before, $285.23, because the methodologywe followed is an approximation of more complex differential calculus.Considering that theta and rho need to be informative and not absolutely exact here, these results are in fact acceptable.Finally, it is important to note the respective sign of theta and rho.In our example, theta is negative, which is consistent with the fact thatwe carry a loan, and each day we owe one more day of interest. Rho,on the other hand, is positive because the rate to maturity increasesfrom 4.5 percent to 4.6 percent, which means that the loan we carry ismore valuable.Back Of®ceBecause of its simplicity and unambiguous formalization, there is notheoretical valuation on a money market transaction, but there is adistinct back-of®ce presentation as we mentioned earlier. The principleis derived from an accounting perspective, based on cost. To illustratethe point, consider Figure 6-3. The total loan lasts m days, of which phave lapsed already and there remains n until expiry (m  p  n).Considering that we have to pay m  r  N=360 in interest at maturity,the accrued cost as of today is p  r  N=360. The valuation is therefore:VTBO 
prN360
Naturally on expiration both valuations come together because there isno discounting anymore in VT :VT  N
N  1  r 
rN
The back-of®ce valuation is ¯awed from a front-of®ce perspectivebecause it does not incorporate the market. It simply considers thatthe operation is carried to maturity.
n days, rT
Np days
Toda y
Figure 6-3
N * (1 + r)
Money Market
CHAPTER 6
VALUATION
301
Interest Rate SwapsFront Of®ceConsider a 6M two-year annual swap against LIBOR, a notional N,traded at rSW , as shown in Figure 6-4. Both ¯ows rSW are known todaybecause they are ®xed. rV1 , rV2 are also known, but only rV1 has beenpaid. As we did earlier when pricing a swap, we operate a transformation of the two unknown variable ¯ows rV3 and rV4 , as shown in Figure6-5. In this last structure there is no more unknown ¯ow, so the liquidation is simply a discounting of all future payments at the currentmarket rate. The valuation is then equal to:rSW rV2 1rSW  1NN  rV1VT  N 1  rT11  rT2The ®rst bracket is the present value of the ¯ows on the next reset,whereas the second bracket is the present value of the last reset. Thisexpression can be generalized and written as: next nX1rV11NNVT  N  rSW 1  rT11  rTi1  rT1 1  rTni1XCFpastwhere:. i  1 . . . n is used to index all future ®xed ¯ows and the corresponding zero-coupon market rates.. N  rSW 
nP
1Ti1 1  ri
is the valuation of the remaining ®xed ¯ows. next rV.N1  rT1rv1
Toda y
rv3
rv2
r1rSW
rv4
rSWr2
Figure 6-4 SwapNote: The diagram's convention is that we receive rSW Ðthat is, we soldÐthe swap.
302
FINANCIAL VALUATION AND RISK ANALYSISrv1
rv2, N
r1Toda y
rSW, N
rSW
Figure 6-5
r2
Swap
is the valuation of the next variable ¯ow, which is the only known¯ow on the variable side.11.N1  rT1 1  rTnis the valuation of the variable side, excluding the next ¯ow.P.CFpast is the algebraic sum of all ¯ows paid to date, ®xed orvariable.31YPFrom there it is easy to express VY!T:
YPVY!T N  rSW 

n Xi1
1N1  rT1
11  rTi
11  rYi

N  rnextV
11NY1  rTn1  r1

11  rT1
11  rYn
11  rY1


which can be simpli®ed to:YPVY!T N  rSW 
N
31
nXi1
rYirTi1  rTi   1  rYi 
N
rnext 1  rY1 rT1 V1  rT1   1  rY1 
rYn rTn1  rTn   1  rYn 
The inclusion of past ¯ows may seem relatively surprising for a valuation todaybased on the liquidation of the position. The reason that they must be included iscontinuity. If we perform a valuation over a period during which payments areexchanged, we have to include them after they have been paid because they wereestimated before.
CHAPTER 6
VALUATION
303
As for today's deals, V TD is given by the pricing formula weintroduced in the part dedicated to swaps in the money marketchapter32:"#nX1rTnTDV  N  rSW 1  rTn1  rTii1where rSW is the ®xed priced at which the swap has been traded, and rTidenotes the mid-market zero coupon until the ith ®xed payment date.If we call rTSW the fair market price of the swap today, we know that:!nTX1r1nwithrTSW nTn1  rn1  rTii1which leads to:V TD  N  rSW  n
rTSW  n  N  n  rSW
rTSW

In other words, the valuation is directly proportional to the difference (rSW rTSW . This difference is usually equal to half the marketspread because rTSW is a midmarket price, between 0.01 and 0.02percent.From a treasury perspective, nothing happens when a swap istraded because there is no exchange of notional. Therefore, the swapdoes not have any overnight ®nancing associated until ¯ows areactually received or paid on each side. On each payment date,CT  CY   CF, and the overnight charge can be assessed on thetotal balance CT .ApplicationsAs a numerical application, consider that we want to value the swapshown in Figure 6-6. We assume rV1  4:7 percent, rV2  5:1 percent,rSW  5 percent, r1  4:8 percent, r2  5:5 percent, and N  $1,000,000.Based on these parameters, the ¯ows rV1 , rV2 , and rSW are known:rV 1  $1,000,000  180/360  4.7 percent  $23,500, rV2 $1,000,000  180/360  5.1 percent  $25,500, and rSW $1,000,000  360/360  5 percent  $50,000.3332
Note that the valuation process has to be carried out swap by swap as for moneymarket operations, and a given swap is either preexisting or traded today.33We sell the swapÐthat is, receive the ®xed and pay the variable sideÐand forsimplicity all variable ¯ows are separated by 180 days.
304
FINANCIAL VALUATION AND RISK ANALYSIS
180 daysrv1
rv3
rv2
rv4
r1
30 days
rSW
Figure 6-6
rSWr2
Swap Valuation
The different terms in VT are respectively equal to:X N  rSW i
11  rTi


rnextVN1  rT1

1N1  rT12

$50,000$50,000  $95,405:38150  4:8%510  5:5%11360360
$25,500150  4:8%13601 $1,000,0001  rT2

$25,000
3
7611764150  4:8%510  5:5% 511360360XCF  N  rV1  $23,500
$52,676:66
Finally:VT  $95,405:38
$25,000
$52,676:66
$23,500 
$5,771:28
This ®gure is negative because we traded at 5 percent when the currentlong-term market rate is 5.5 percent. We receive the ®xed side, which isless than the market.Now consider that we value the same swap tomorrow, withslightly different parameters: r1  4:85 percent and r2  5:45 percent.YPis then the sum of three terms:VY!T
N  rSW 
Xi
where:
rYirTi v1  v21  rTi   1  rYi 
CHAPTER 6
VALUATION
305
 150  4:8%149  4:85%360360   $3:54 v1  $50,000  150  4:8%149  4:85%1 1360360 510  5:5%509  5:45%360360   $37:03 v2  $50,000  510  5:5%509  5:45%1 1360360Y1  rnextrT1 V   r1 $1,000,0001  rT1   1  rY1  180  5:1%150  4:8% 149  4:85 %1360360360  $72:55150  4:8%149  4:85% 11360360 510  5:5%509  5:45%rY2 rT2360360  N $1,000,000  509  5:45%510  5:5%1  rT2   1  rY2 1 1360360
N
 $740:52which ®nally leads to:YPVY!T
$3:54  $37:03  $72:55  $740:52  $846:56
A direct calculation gives VT  $4,924.72Ðthat is, an identical difference of $846.56. The P&L is positive here because the market rate wentdown from 5.5 to 5.45 percent.Explanatory ValuationsThe breakdown of a swap valuation into distinct components istedious, so we will keep the algebra to a minimum. Recall the expresYP:sion of VY!TYP N  rSW VY!T
Xi
N
rYirTi1  rTi   1  rYi 
rYn rTn1  rTn   1  rYn 
N
rnext 1  rY1 rT1 V1  rT1   1  rY1 
306
FINANCIAL VALUATION AND RISK ANALYSIS
for which we want to keep alternatively the time and interest ratesconstant. Assuming ®rst that rates are constantÐthat is, rYi  rTi ÐtherTi  and 1  rYi   1  rTi  can be written as34:terms rYirYi
1  rYi
rTi 
ni  1  rYi360
  1  rTi 
ni  rYi
n  1  rYi1 i360
!

rYi360n  rY1 i i360
!
assuming ni denotes the number of days between today and the ith®xed ¯ow.If we take the example we used before, rY1  4:8 percent, rY2  5:5percent, n1  149, and n2  509, we have:   150  4:8%149  4:8%YT 1 1:040261  r1  1  r1  1 360360
   510  5:5%509  5:5%1  rY2  1  rT2  1  1 1:16174360360
and:

N  rSW 
Xi
rYirTi1  rTi   1  rYi 
4:8%5:5%$50,000 3603601:0