Accrual valuation and mark to market adjustment
AAccrual valuation and mark to market adjustment
Alexey [email protected] 22, 2016
Abstract
This paper provides intuition on the relationship of accrual and mark-to-marketvaluation for cash and forward interest rate trades. Discounted cash flow valua-tion is compared to spread-based valuation for forward trades, which explains thetrader’s view on valuation. This is followed by Taylor series approximation for cashtrades, uncovering simple intuition behind accrual valuation and mark-to-marketadjustment. It is followed by the PNL example modelled in R. Within the Taylorapproximation framework, theta and delta are explained. The concept of deferral isexplained taking Forward Rate Agreement (FRA) as an example.
Financial accounting requires that valuation of a trade must at least reflect trade eco-nomics. At best, the market value is either known and can be applied directly, or canbe derived via certain proxies like comparables or hedges that replicate the trade. Themost basic way to value an interest flow receivable in the future is to linearly accrue itsvalue over the waiting time. Consider 1 of notional we lend for a T amount of time ata fixed interest rate of r. We will denote the year fraction since the start of the tradeuntil time t as ∆ t , e.g. t . Likewise, the year fraction from the start of the trade untilthe interest payment at time T is ∆ T . We will accrue interest linearlly. Given thatthe payment amount is r ∆ T and the linear portion of the period to date t is ∆ t ∆ T , theaccrued portion of interest may be written as ∆ t ∆ T ∗ r ∆ T or simply r ∆ t . The accrual-based PV in that case is: P V accr = 1 + r ∆ t (1)Since market conditions at the moment of pricing will be different to those when thetrade was made, the market will price the trade differently to accrual-based valuation.Let’s assume that we do discounting with known market rates. The discounting factorfor time t as of t = 0 can be written as (2) where z ,t is the market zero rate. DF t = 11 + z t ∆ t (2)1 a r X i v : . [ q -f i n . P R ] F e b .2 Cash trades and forward trades Forward trades are trades that start in the future. Continuing with the example above,let’s assume that we lend 1 of notional on some future date t and get it back with fixedrate rebate r on another future date t ( t = 0 < t < t ). We will drop t from notationfor the sake of brevity, so the interest paid on date t (accrued over the period t to t )is written as r ∆ , . In this case, discounted cash flow valuation may be written as: P V
DCF = − ∗ DF , + (1 + r ∆ , ) ∗ DF , (3)Extending (1) for a forward-starting case, we can define accrual-based valuation as: P V accr = t < t r ∆ t ,t if t < t < t t > = t (4)where ∆ t ,t = t − t and r ∆ t ,t is the accrued amount.This piecewise definition of accrual-based valuation follows from a simple economic prin-ciple: if the trade starts in the future ( t < t ), we have not yet started to accrue interest,hence, accrual valuation will be giving 0 accrued value by definition. This means thatuntil the accrual period starts, accrual-based valuation will yield zero PNL and flat valu-ation at P V accr = 1. However, this is not the case with the DCF-based apporach. It willbe subject to time decay (theta) and revaluation at market rates (delta) from the tradebooking date. It will be explained more clearly in due course of this paper. Anotherimportant observation is that accrual-based valuation does not depend on market rates:it is driven only by trade parameters (accrual period length and rate) and the relativeposition of valuation date with respect to accrual period.Interest is assumed to be paid on trade maturity date, so accrual goes to 0 at this mo-ment. Since the principal is also paid at maturity, entire accrual valuation (1) will go to0 at and past the maturity date. This is equivalent to transforming a security asset tocash, so we don’t have the security on the balance anymore, just cash. DCF valuationalso goes to zero since we assume that the discounting factor on the maturity date is 0( DF t = T = 0). Below is the depiction of the accrued amount in time. T ime
Forward trade Cash trade
Accrualr ∆ .2.2 ”Cash” trades ”Cash” trades are the trades that are already started ( t < = t ). Accrual-based valuationis in line with (1) and (4). We may re-write DCF valuation as: P V
DCF = (1 + r ∆ , ) ∗ DF t, (5) Following the fixed rate example above, let’s prove a simple lemma that introduces thenotion of equivalence of DCF and spread-based valuations. LHS is a DCF valuationformula (3), RHS is a spread-based valuation formula
P V spread − based = [ r − z ] ∗ ∆ , ∗ DF , Lemma 1.
DCF valuation and spread-based valuations are equivalent: − ∗ DF , + (1 + r ∆ , ) ∗ DF , ≡ [ r − z ] ∗ ∆ , ∗ DF , (6) Proof.
Let z be the forward rate from the start date to maturity. By definition: DF , = 11 + z ∆ (7)At the same time, DF , = DF , DF , . From this follows that 1 + z ∆ = DF , DF , . Substitut-ing DF , in (3) we get: P V
DCF = − ∗ (1 + z ∆ ) ∗ DF , + [1 + r ∆ ] ∗ DF , = [ r − z ] ∗ ∆ , ∗ DF , (8) Taylor approximation for x = 0 can be written as:11 + x ∼ − x + o ( x ) (9)Hence, the discounting factor may be written as: DF t = 11 + z t ∆ t ∼ − z t ∆ t + o ( z t ∆ t ) (10)Applying Taylor approximation for the RHS of (6) we get: P V spread − based = [ r − z ] ∗ ∆ , ∗ DF , ∼ [ r − z ] ∗ ∆ , + o ( z ∆) (11)3 umerical example 1. Trader’s view of forward pricing
Consider a tomorrow-next (TN) interest at maturity trade with the nominal of1000000 done at 500bps. Let the TN market rate be 300bps, ON market rate be 200bps. What is the value?We would use Taylor-approximation of the spead based valuation that we explained in(11):
P V spread − based ∼ [ r − z ] ∗ ∆ , (12)Spread-based (following (11)): 1000000 ∗ (0 . − . ∗ /
360 = 55 . DF , = 1 / (1+0 . ∗ / . DF , = 1 / (1 + 0 . ∗ / . DF , = DF , ∗ DF , =0 . P V
DCF = − ∗ . ∗ (1 + 0 . ∗ / ∗ . . Let’s now see what happens with DCF valuation when the trade stops being forwardand becomes ”cash” so we begin to accrue interest. We will continue with the same fixedrate one-period trade example.
Lemma 2.
Within the first-order Taylor series approximation, DCF valuation may bepresented as the sum of the accrual-based PV and the mark-to-market adjustment.
P V
DCF = (1 + r ∆ ,T ) ∗ DF t,T ∼ r ∆ ,t + ( r − z t,T )∆ t,T + o ( z ) (14) Proof.
Let’s denote the valuation date as t , the maturity date as T , the trade rate as r and the zero rate from t to T z t,T as z .1 + r ∆ ,T z ∆ t,T = 1 + r (∆ ,t + ∆ t,T )1 + z ∆ t,T = r ∆ ,t (1 + z ∆ t,T ) − r ∆ ,t z ∆ t,T + 1 + r ∆ t,T z ∆ t,T = r ∆ ,t (cid:124) (cid:123)(cid:122) (cid:125) Accrual + 1 + r ∆ t,T − r ∆ ,t z ∆ t,T z ∆ t,T (cid:124) (cid:123)(cid:122) (cid:125) MtM Adjustment (15)Let’s now apply Taylor approximation (9) to the MtM component, removing the second-order components. Note that rz ∆ ,t ∆ t,T is also a second-order component. r ∆ ,t + 1 + r ∆ t,T − r ∆ ,t z ∆ t,T z ∆ t,T ∼ r ∆ ,t + (1 + r ∆ t,T − rz ∆ ,t ∆ t,T )(1 − z ∆ t,T ) ∼ r ∆ ,t + 1 + r ∆ t,T − rz ∆ ,t ∆ t,T − z ∆ t,T − rz (∆ t,T ) + rz ∆ ,t (∆ t,T ) ∼ r ∆ ,t (cid:124) (cid:123)(cid:122) (cid:125) P V accr (1) + ( r − z )∆ t,T (cid:124) (cid:123)(cid:122) (cid:125) MtM Adjustment (11) + o ( z ∆) (16)4 tM Adjustment ∼ ( r − z )∆ t,T (17)As shown above, DCF valuation is equivalent to the sum of accrual-based valuation(1) and mark-to-market adjustment (17). Note that the spread-based valuation (11)applied from the valuation date t until the expiration date T here represents mark-to-market adjustment (17) that brings the accrual-based valuation in line with themarket rate z .If the market rate z is equal to the trade rate r , the MtM adjustment component can-cels out. This draws some parallels to bond markets vocabulary. From the clean priceperspective, the trade is priced at par in this case. From the dirty price perspective, thetrade is priced at par plus accrued. Once the trade rate is different to the market rate,the clean price becomes affected by the mark-to-market adjustment. Based on (16) let’s derive the basic Greeks that define the PNL. Recollect that ∆ ,t = t − t , ∆ t,T = T − t ϑ = ∂P V∂t ∼ r − r
360 + z ∼ z (cid:37) = ∂P V∂z ∼ − ∆ t,T (18) • ϑ (”theta”) addresses sensitivity to time decay t . First-order dependency is on themarket rate. It shows the market cost of carrying 1 day of notional over to thenext day. We may as well re-write ϑ as the sum of accrued and mtm components: ϑ ∼ z ∼ r (cid:124)(cid:123)(cid:122)(cid:125) Accrued + ( z − r
360 ) (cid:124) (cid:123)(cid:122) (cid:125)
MtM adjustment (19) • (cid:37) (”rho”) addresses sensitivity to market rates z . We have an inverse relationshipon the remaining duration of the trade. This means that the less time is left untilexpiry, the less sensitive is the price to changes in the market rate. The minus signtells us that PV decreases as the market rate increases. R is a scientific language popular in many areas. One of its advantages is the easinessof working with vectors. We will populate an array of increasing daycount in the days object, decreasing daycount in the daysRemaining object and will then calculate:5 iscFact - a vector of daily discounting factors for each day in daysRemaining object accrued - a vector of daily accrued amounts PV - a vector of DCF-based PVs (one PV per day) mtmAdj - a vector of mark-to-market adjustments (11) PVTaylor - a vector of Taylor-approximated PVs, see (16) unexplained - a vector of the difference between DCF PV and Taylor-approx PVWe will assume the market rate of 700 bps which is intentionally higher than the traderate of 500 bps (1M of notional for 10 days) to get the negative mtm adjustment (it ismore instructive this way). This market rate of 700 bps will be constant each day forthe sake of simplicity. Below is the R code that calculates valuation for the above.days < − seq ( 1 , 1 0 )daysRemaining < − seq ( 9 , 0 , − < − days / < − daysRemaining / < − / (1+ 0 . 0 7 ∗ daysRemaining / < − ∗ ∗ a c c r u a l F r a c t i o n sPV < − ∗ ( 1 + 0 . 0 5 ∗ / ∗ d i s c F a c tmtmAdj < − ∗ ( 0 . 0 5 − ∗ f r a c t i o n s R e m a i n i n gPV T a y l o r = 1000000 + a c c r u e d + mtmAdju n e x p l a i n e d < − PV − PV T a y l o rPVs < − cbind ( days , PV, a c c r u e d , mtmAdj , PV Taylor , u n e x p l a i n e d )PVsExecuting this code in the R environment yields the following PNL simulation:This example provides some nice intuition: • Accrual linearly increasing which reflects the coupon amount being linearly recog-nized • Negative MtM adjustment reflects the fact that the market rate of 700bps is higherthan the trade rate of 500 bps. It decreases with time as the valuation date t gets6loser to the expiration date T , since the mtm component is scaled by the durationmultiple of ∆ t,T . This is in line with sensitivity (cid:37) (duration) to the market rate z which is exactly that: − ∆ t,T • Since the market rate is not changing, there is no (cid:37) attribution. The entire dailyPnL is explained by time decay ϑ ∼ ∗ . / ∼ .
44 (18). Thishighlights a very important point. While PNL attribution is purely theta-driven,it might be split into the accrual component and the mark-to-market componentwhich depend on the trade rate and the curve rate, respectively. Following (19)we get the following theta components: ϑ accr ∼ r ∼ ∗ . / ∼ . ϑ MtM ∼ z − r ∼ ∗ (0 . − . / ∼ .
56 (20)
Let’s now assume the valuation of a single-period forward-starting floating trade. In theformula (8) we need to replace the fixed trade rate r with a floating rate f and a fixedspread s . Taylor approximation will be written as: P V
DCF = [ f + s − z ] ∗ ∆ , ∗ DF , ∼ [ f + s − z ] ∗ ∆ , (21)If forward rates f and z are forecasted from the same yield curve (e.g. LIBOR) theywill cancel out and valuation is approximated as simply as P V
DCF = s ∆ , (22)In the case of zero spread s = 0, so the valuation of such forward period is 0. This iseasily understood, as we both forecast and discount over the period with the same rate.The amount we pay at time 1 (notional amount) will be equal to the discounted (to time1) value of the flow we receive at point 2, so − z ∆ , z ∆ , =0.In the case of LIBOR forecasting and OIS discounting we will get PV depending onLIBOR to OIS spread f − z Once the rate is fixed f = r , valuation will be equivalent to (14). Intuition on accrual valuation and mark-to-market adjustment provided in this paperapplies to a wide range of interest rate products such as FRA, repo (considering justthe cash side of the transaction), interest at maturity, as well as bonds and interestrate swaps. An extreme case are the products where we assume zero duration, e.g. callaccounts. For those, the most conservative valuation would be purely accrual-based (if7e assume that the money can be withdrawn on the valuation date) with no mark-to-market adjustment.Another extreme case is forward rate agreements (FRAs). Those are forward tradesthat pay at the beginning of the period. While they stay forward, valuation is drivenby the mark-to-market adjustment. However, once a FRA pays, the MtM adjustmentcomponent disappears entirelly. Since we received the interest at the beginning of theperiod, we need to defer it. The explaination of that is provided in the next paragraph.
Forward rate agreement (FRA) is a financial contract where the lender (seller) agreesto receive the fixed ”contractual” interest rate r in the future (a fixed rate is locked-innow) in exchange for a floating payment in the future (becomes known at the futurepoint t ). A special property of FRA is that it pays at the beginning of the forwardperiod. Let’s again denote the beginning of the forward period t as 1 and the end t as2. For the sake of simplicity, let’s assume that both the fixing and the payment happenat t . Since the payment is at t , there is a need to lock-in the discounting rate from 2to 1. The convention is to assume that it is the same forward rate f that the period isforecasted (and later fixed) with. Thus, PV of a FRA at point t is: P V t F RA = [ r − f ] ∗ ∆ , f ∆ , (23)Discounting to time 0 we get: P V
F RA = [ r − f ] ∗ ∆ , f ∆ , ∗ DF , (24)Applying Taylor approximation (9) to this formula gives us the same approximationas in (12), which tells us that FRA valuation is a purely mark-to-market adjustment -driven (17): P V
F RA ∼ [ r − f ] ∗ ∆ , (25) To better understand the concept of deferral, we need to introduce the notion of NPVfirst. NPV is the sum of the outstanding cash position and the present value of thetrade. The graph below shows what accrual-based PV, Cash position and NPV look likefor the one-period example that we used in (4).8 ime
P V accr
11 + r ∆ T ime
CashP os -1 r ∆ T ime
N P Vr ∆ As can be seen, NPV gives the notion of linarly increasing interest income that remainsconstant after the coupon payment date is achived.
With FRA we would like to linearly recognize the income over the FRA period. Thismeans that we need to offset the interest payment received at the beginning of the periodwith the deferral amount so that the net of those would equate to linearly increasingaccrual. We can define deferral as:
Def erral = t < t − r ∆ t,T if t < t < t t > = t (26)where − r ∆ t,T is − r T − t .It can be seen that the sum of the coupon and deferral would form the profile equivalentto linear accrual. Def erral + Coupon = Accrual (27)9he graph below depicts that.
T ime
Def erral - r ∆ T ime
CashP osr ∆ T ime
Incomer ∆ Using Taylor approximation helps to bridge the gap between accrual and mark-to-marketvaluation. We can see that PV may be split into accrual and mark-to-market adjust-ment components. Similarly, theta can be viewed as the sum of the accrued and MtMcomponent.We see that the forward valuation formula (6) turns into the mark-to-market adjustmentfor trades that began to accrue interest. The less time remains until trade expiration,the less effect mark-to-market effect has on valuation. This makes sense since the risk(duration) decreases with time to maturity. • Options, futures, and other derivatives / John C. Hull ••