Joe Pmbley, unpublshed, 005. Yeld Curve Calculatons Background: Everythng s dscount factors Yeld curve calculatons nclude valuaton of forward rate agreements (FRAs), swaps, nterest rate optons, and forward rates. The most mportant component of all these calculatons s the determnaton of ero coupon dscount factors (or, ust dscount factors ). Ths note focuses on the problem of computng the forward rate between any two future dates. Ths ordeal effectvely ends when we get the dscount factors for the two dates snce the remanng labor for the forward rate s straghtforward. Dscount factors are synonymous wth present value Each forward date has an assocated dscount factor that represents the value today of a hypothetcal payment that one would receve on the forward date expressed as a fracton of the hypothetcal payment. * For example, f we expect to receve $000 n sx months, then ts present value mght be $989. Thus, the dscount factor for sx months would be 0.989. The queston then becomes how to derve the $989 of ths example. Before we begn, let s remark that dscount factors are all about nterest rates. Unless stated otherwse (very unlkely n ths artcle!), we deal here wth the LIBOR/swap curve. The meanng of ths term wll emerge n the followng sectons. Begn wth the smplest calculatons for dscount factors The dscount factor for a forward pont n tme s the value today (also known as present value ) of a payment to be receved at the forward tme expressed as a fracton. By defnton, then, the dscount factor for today s.0 snce a payment today has present value of precsely the payment value. In the example of the precedng secton, we made up the number $989 as the present value of a payment of $000 to be receved n sx months. The connecton between these two amounts s nvestment wth nterest. If we have $989 today, we can nvest ths amount for sx months. If n sx months tme our orgnal prncpal and nterest totals $000, then we are ndfferent to havng $989 now or $000 n sx months. Let L be the nterest rate (quoted annually) at whch we can nvest for sx months. Then the nterest earned wll be $ 989* L * where we multply * Please don t take the word today too lterally. Due to market conventon, today n some contexts means two busness days forward from today. Ths s spot settlement and we choose not to elevate the mportance of ths dstncton here.
by one-half to denote sx months (half a year). Addng the prncpal and nterest and settng the sum to the $000 value at sx months shows $989 + $ 989* L * = $000. Re-arrangng terms and expressng as a fracton of ultmate payment gves 0.989 = L The equaton above s somewhat backward snce we have a constant value on the left-hand sde and what appears to be a varable ( L ) on the rght-hand sde. In real lfe, though, we wll know L and wll need to determne the dscount factor whch, n ths example, we stpulated to be 0.989. We ve over-smplfed. We need to take nto account the daycount conventon of the nterest rates we use. The L - whch we wll shortly descrbe as sx-month LIBOR has the daycount of actual/0 whch means we multply by the number of days n the perod and dvde by 0 (as f each year has 0 days). In a sx-month tme perod, the actual daycount s more lkely to be 8 than 80. Our conventon wll be to use t - often wth a subscrpt to ndcate the tme perod n years between two dates measured by the governng daycount conventon. To denote both aspects that the dscount factor s the value to be calculated and that that we need to determne the precse t as per the daycount conventon, we re-wrte the prevous equaton as Dscount factor =. L t Let s revew the avalable market nformaton To compute dscount factors, we begn wth market nterest rate nformaton. The market rates of nterest to us are LIBOR (the London nterbank offered rate ) for maturtes of twelve months and less and swap rates for maturtes of two years to thrty years. (See Bloomberg USSWAP.) The most promnent LIBOR maturtes are month, months, months, and months, though there are also settngs for every other month, overnght, week and weeks. (See Bloomberg BBAM.) We gnore Eurodollar futures contracts prmarly for convenence. They complcate the analyss sgnfcantly. In prncple, the swap data we use wll gve us almost dentcal results to what we d expect wth futures. * All LIBOR settngs mply smple nterest wth actual/0 daycount. Swaps are not so accommodatng. In a swap, one sde pays a fxed rate wth 0/0 daycount sem-annually. The other sde pays -month LIBOR wth. * The dfference n fnal results s convexty and s mportant to professonal swap dealers.
actual/0 daycount quarterly. A quoted swap rate of, say, 4% for a 5-year maturty means that a 5-year swap n whch the fxed-rate payer pays 4% semannually and the floatng-rate payer pays -month LIBOR quarterly wll have a ero value (.e., the swap s at-market). Re-vst the smplest calculatons for dscount factors The earler secton that dscussed smplest calculatons for dscount factors treated only the case wth a -month maturty. The calculaton s nearly dentcal for any maturty that matches that of an avalable LIBOR maturty. For example, the dscount factor at months s ust Dscount factor = L t n whch the t s the actual number of days to the forward date (lkely 90, 9, or 9) dvded by 0. Ths method works for any LIBOR maturty. But such maturtes do not extend beyond months and t s ths pont at whch the dscount factor calculaton becomes more challengng. Lnear nterpolaton for yelds Before tacklng the challengng problem of dscount factors for maturtes beyond months, let s magne we want the dscount factor for.5 months. There s no.5-month LIBOR, so we apply the expresson Dscount factor = L t Here we desgnate L as the lnear nterpolaton of the -month and 7-month LIBOR settngs. Generally speakng, lnear nterpolaton works well for yelds but s not approprate for dscount factors. Based on the mathematcal nature of how the dscount factor depends on yeld, lnear nterpolaton of yeld suggests logarthmc nterpolaton for dscount factors. Hence, we vew the latter as the proper choce for dscount factor nterpolaton when necessary. It s not uncommon to hear others promote (cubc or hgher order) splnes for yeld nterpolaton. Our vew s that splnes do not provde greater accuracy and, n fact, ntroduce numercal rsk. Splnes may be more exotc, but that doesn t make them better. Bootstrappng for dscount factors beyond months Beyond months we must rely on swap data. Swaps behave much dfferently than smple borrowng and lendng at LIBOR. As we noted prevously, the fxed-rate payer pays the swap (fxed) rate sem-annually and receves -month LIBOR quarterly (from the floatng-rate payer). So the swap rate s, roughly speakng, an average of -month LIBOR from now untl the swap contract maturty..
To compute dscount factors we adopt a recursve procedure known as bootstrappng. Our goal s to compute dscount factors at sx-month ntervals. We know the dscount factors at sx and twelve months from the smple-nterest LIBOR calculatons. For the dscount factor at eghteen months, we use both the (fxed) swap rate R and the two pror dscount factors and to compute the 8-month dscount factor. Remember that the dscount factor for today s 0. Let f be the forward LIBOR rate for the nterval t to t. As we ll note later, forward LIBOR s the expected future value of LIBOR gven today s market. Then, we wrte the market value of the swap as the present value of expected future cashflows. Snce a swap at-market has ero value, we get R tˆ f t 0 (). (To get, the dscount factor at 8 months, we set n ths equaton.) We defne t t t wth the actual/0 daycount conventon and t ˆ t t wth the 0/0 daycount conventon. Before proceedng, we must make a remarkable observaton. The second term n the summaton of equaton () smplfes drastcally to f t 0 (). Let s call ths the algebrac wonder of fnance and t explans why LIBOR floaters reset to par wth each (quarterly) rate settng. Of course, the math doesn t really drve the fnance. Rather, t s the underlyng physcs of LIBOR floaters that manfests tself n the wonder of equaton (). The curous reader may verfy equaton () gven the forward rate f that emerges from an arbtrage argument: f t (). Combnng equatons () and () and realng that 0 and t ˆ, we can solve for as R R (4). Ths s bootstrappng! Equaton (4) gves us, for example, when we know,, and R. Once we do have, we d then compute 4 and so on.
There are three loose ends here. Frst, we need to have swap rates R at all -month maturty ponts begnnng at 8 months. The market does not quote ths many swap rates. Where necessary, then, we nterpolate lnearly to fll n mssng swap rates. Second, we get dscount factors only at these -month maturty ponts. When we need a dscount factor at any other tme pont, we nterpolate logarthmcally as we dscussed earler. Fnally, our equaton () seems to have forgotten that the floatng rate payments n a swap are quarterly. We smplfed the problem by makng the floatng rate payments sem-annual. When we treat the payment frequency correctly, we do get the same results n the ensung equatons. I removed a layer of complexty by the approxmate method of equaton (). Arbtrage argument for the forward rate Equaton () relates dscount factors to forward rates and s central to the new bond math. * How does one derve equaton ()? Let s thnk of the forward rate f between two future dates as the expected future rate lnkng the dates. For example, f I agree now to lend you $ sx months from today and you agree to re-pay me $ plus nterest nne months from today, then I am makng a -month loan months forward. If you and I knew today what - month LIBOR would be n months, then we would use ths future LIBOR value as the nterest rate for the loan. But we don t know what -month LIBOR wll be! Thus, we stpulate the forward rate as the nterest rate for the forward loan. One would thnk that you and I should negotate the forward rate based on what we beleve -month LIBOR wll be n months. But that s not rght! Instead, the market tells us forward LIBOR by a clever argument. Instead of makng the forward loan of the pror paragraph to you, I wll go to a bank and lend for 9 months today at 9-month LIBOR and borrow for months today at - month LIBOR. Instead of a forward loan, I have two smple, old-fashoned loans (wth one long and one short ). Both loans are n the amount of $ Lt. # Snce the loans are of the same amount, I have no net payment today. My loan cancels my borrowng. But I must re-pay precsely $ n months. In 9 months I wll receve L9t9 Lt as repayment for the 9- month loan. Wth my two smple loans I have created the (synthetc) forward loan. Now I can compute the effectve nterest rate of ths forward loan wth L t L t f t t (5). 9 9 9 * In contrast, the old bond math s comprsed of yeld-to-maturty and expressons of the form n y to compute forward values. # I hope the notaton sn t confusng. The subscrpts and 9 refer to sx and nne months, respectvely. The t s, once agan, tme as fracton of a year n the actual/0 daycount conventon.
Solvng equaton (5) for f wth t t 9 t and relatng dscount factors to the LIBOR values gves equaton (). Ths argument apples for any two dates rather than ust those less than months for whch we can use LIBOR values as we ve done here. For the dates t and t, we would create the synthetc forward loan by borrowng an amount for tme t and lendng ths same amount for tme t. At tme t we d then pay $ whle at tme t we d receve. Equaton (5) would then become Meanng of the forward rate f t t (). It s not uncommon to hear market players dsparage forward rates wth a comment such as forward rates are not good predctors of future rates. When you hear ths statement, brace yourself! You re about to be assaled wth an heroc market story n whch the speaker-protagonst shrewdly predcted some past market move. If the speaker s your boss, then nod apprecatvely. If not, smrk dersvely. The forward rate s the market s expectaton of the future rate. It s not a predcton n the sense that any of us should beleve that LIBOR wll be ths value. Rather, the forward rate s the consensus average value of potental future outcomes. For llustraton, f the forward rate for a partcular perod s 4% and you beleve rates wll, n realty, be hgher (lower) at ths future tme, you can place a bet (wth a Eurodollar futures contract). If LIBOR n the future s hgher (lower) than 4%, you wn. Conversely, f LIBOR n the future s lower (hgher) than 4%, you lose. If LIBOR actually does come n at 4%, you break even. The forward rate, then, s not ust a calculaton and t s not the output of some econometrc model. It s real! The forward rate for a defned future perod s ust as much a market varable as -month LIBOR or the IBM stock prce. The current value of any market varable s the market consensus. It s nether rght nor wrong, t s the market!