Concave Shape of the Yield Curve and No Arbitrage
aa r X i v : . [ q -f i n . M F ] A ug Concave Shape of the Yield Curve and NoArbitrage
Jian SunEconomics School of Fudan UniversityAugust 13, 2018
In fixed income sector, the yield curve is probably the most observedindicator by the market for trading and financing purposes. A yield curveplots interest rates across different contract maturities from short end toas long as 30 years. For each currency, the corresponding curve shows therelation between the level of the interest rates (or cost of borrowing) andthe time to maturity. For example, the U.S. dollar interest rates paid onU.S. Treasury securities for various maturities are plotted as the US treasurycurve. For the same currency, if the swap market is used, we could also plotthe swap rates across the tenors which would be called the swap curve.The shape of the yield curve gives an idea of future interest rate changesand economic activity. There are three main types of yield curve shapes:normal, inverted and flat (or humped). A normal yield curve is one in whichlonger maturity bonds have a higher yield compared to shorter-term bondsdue to the risks associated with time. An inverted yield curve is one inwhich the shorter-term yields are higher than the longer-term yields, whichcan be a sign of upcoming recession. In a flat or humped yield curve, theshorter- and longer-term yields are very close to each other, which is also apredictor of an economic transition.A normal or up-sloped yield curve indicates yields on longer-term bondsmay continue to rise, responding to periods of economic expansion. Wheninvestors expect longer-maturity bond yields to become even higher in thefuture, many would temporarily park their funds in shorter-term securitiesin hopes of purchasing longer-term bonds later for higher yields.In a rising interest rate environment, it is risky to have investments tiedup in longer-term bonds when their value has yet to decline as a result1f higher yields over time. The increasing temporary demand for shorter-term securities pushes their yields even lower, setting in motion a steeperup-sloped normal yield curve.An inverted or down-sloped yield curve suggests yields on longer-termbonds may continue to fall, corresponding to periods of economic recession.When investors expect longer-maturity bond yields to become even lowerin the future, many would purchase longer-maturity bonds to lock in yieldsbefore they decrease further. The increasing onset of demand for longer-maturity bonds and the lack of demand for shorter-term securities lead tohigher prices but lower yields on longer-maturity bonds, and lower pricesbut higher yields on shorter-term securities, further inverting a down-slopedyield curve.A flat yield curve may arise from normal or inverted yield curve, depend-ing on changing economic conditions. When the economy is transitioningfrom expansion to slower development or even recession, yields on longer-maturity bonds tend to fall and yields on shorter-term securities likely rise,inverting a normal yield curve into a flat yield curve. When the economyis transitioning from recession to recovery and potential expansion, yieldson longer-maturity bonds are set to rise and yields on shorter-maturity se-curities are sure to fall, tilting an inverted yield curve towards a flat yieldcurve.According to Salomon Brothers working paper [7], It is commonly knownthere are three main influences on the treasury yield curve shape :1. Yield curve shape reflects the market’s expectations of future ratechange.2. Yield curve shape reflects the bond risk premia (expected return dif-ferentials across different maturities)3. Yield curve shape reflects the convexity benefit of bonds of differenttenors.Even the yield curve can be flat, upward or downward (inverted), how-ever, yield curve is generally concave. There is a lack of explanation of theconcavity of the yield curve shape from economics theory. We offer in thisarticle an explanation of the concavity shape of the yield curve from tradingperspectives.Our main argument is to construct an investment portfolio consistingfixed income instruments and demonstrate that if the yield curve is not2oncave, an arbitrage will emerge. Our results also depend on an assumptionthat yield curve moves up and down in parallel. This assumption is notprecisely true but approximately acceptable in reality.
We first discuss the relations between different notions of interest rates. Inthe discussion below, certain simplifications will be made. We mainly discussthree types of rates which are often used in industryzero rates, forward ratesand par rates. Correspondingly,there are three types of instruments: zerocoupon bonds, forward rate agreements and swaps (or par bonds). In all thecalculations, we ignore the day count and business conventions. For swapswe assume the exchange of payment occurs once a yearand for forward rateagreements we also assume that the contract expire in integer years and spanfor one year.Let the current time be 0, and future annual years be 1 , , · · · , n . Thezeros rates are yield to maturity of the zero coupon bonds maturing at thesetimes. Zero rates and the zero coupon bond prices have the relationship p i = 1(1 + y i ) i (1)where the p i is the zero coupon bond price and y i is the yield to maturity.The forward rates for time interval ( i, i + 1) are the rates specified in theforward rate agreements (FRA) to lockin the interest rates at the futuretime i for the one year period. This rate f i can be calculated as f i = p i p i +1 − y i +1 ) i +1 (1 + y i ) i − s i are calculated as s i = 1 − p i p + p + · · · + p i (3)It is known that as long as f i >
0, there is no arbitrage between thesetrading instruments. Hence in theory, the zero curve can have any shapeas long as f i >
0. For example, the curve might be upward or might bedownward. The curve might be convex or might be concave.3ut in reality the curves are usually concavely upward shaped. In thisarticle we show that the zero curve and the swap curve has to be concave ifthe following conditions are true: there is no arbitrage; yield curve moves inparallel. The first assumption is realistic as many trading desks around theworld are watching the yield curve and try to take arbitrage opportunities atany time. The second assumption is not realistic, but close to reality. In arising rates environment, the rates of different tenors all increase in generaland in a falling rates environment, rates of different tenors all decrease.
Our main results depend on the following crucial well known results:
Convexity Inequality
The function f ( x ) , x ∈ R is a convex function, thenby definition it should satisfy the following inequality. For any λ > , λ > λ + λ = 1 and a, b ∈ R, we should have f ( λ a + λ b ) ≤ λ f ( a ) + λ f ( b )We now set our securities. We have three zero coupon bonds, callingthem B , B , B corresponds to three maturities T < T < T with theiryields be y , y , y . So far we impose no conditions on these yields as longas the implied forward is positive. We now construct a trading portfolio bypurchasing λ dollar amount of B , λ dollar amount of B and short λ dollar amount of B . We choose quantities λ > , λ > , λ > λ + λ = λ and λ T + λ T = λ T . We notice that by combining the two equations, we have λ ( T − T ) = λ ( T − T )In fact by linear algebra, the solutions of λ i is unique up to a scalar λ = T − T , λ = T − T , λ = T − T
4s a consequence λ λ + λ λ = 1We now claim that portfolio we have constructed has zero cost. Zerocost is obvious given the rule that λ + λ + λ = 0. Theorem 1.
If yields y , y , y as a function of time to maturity T , T , T is convex, i.e. ( T − T ) y + ( T − T ) y ≥ ( T − T ) y (4) the portfolio we constructed admits an arbitrage.Proof. Now we assume yields move by the same amount a and time movesforward by t , therefore our portfolios new value becomes P ( a, t ) = λ e − a ( T − t )+ y t + λ e − a ( T − t )+ y t − λ e − a ( T − t )+ y t We want to show that this quantity is positive i.e. λ λ e − a ( T − t )+ y t + λ λ e − a ( T − t )+ y t ≥ e − a ( T − t )+ y t By the convexity inequality we have λ λ e − a ( T − t )+ y t + λ λ e − a ( T − t )+ y t ≥ e − a λ λ ( T − t ) − a λ λ ( T − t )+ λ λ y t + λ λ y t = e − a ( T − t ) e λ λ y t + λ λ y t But if the yield y i are convex, by definition we have λ λ y + λ λ y ≥ y therefore λ e − a ( T − t )+ y t + λ e − a ( T − t )+ y t ≥ λ e − a ( T − t )+ y t it true.We have completed our argument that arbitrage exists by constructiona zero cost portfolio consisting of three zero coupon bonds. The argumentis valid for any three maturities as long as corresponding yields are convexand the yields move by the same amount. The entire argument is based onthe convexity inequality. We have proved so far:5. Under parallel movement in yields, we can construct zero cost portfolioand achieve positive profit instantaneously.2. If Yields are convex with respect to time, we construct zero cost port-folio and achieve positive profit at any future time. We now extend the results in the previous sections to nonparallel movement.We now try to refine the argument above. For nownot only we assumethat yields y i are different, the movement a i can also be different. We firstcheck the instantaneous result. After the yield movement the portfolio valuebecomes P ( a , a , a ) = λ e − a T + λ e − a T − λ e − a T Once again, we require λ + λ = λ thus the bond portfolio is a zero costportfolio. Secondly, we investigate the constraints on a i so that the portfoliovalue is always positive. By the convexity inequality λ λ e − a T + λ λ e − a T ≥ e − λ λ a T − λ λ a T As a consequence, we would need to set − λ λ a T − λ λ a T ≥ − a T which is equivalent to λ λ a T + λ λ a T ≤ a T If we require that ratios between a , a , a are all fixed, the above equalityimplies that λ λ T + λ λ T = T As a consequence, we have λ = a T − a T , λ = a T − a T , λ = a T − a T . 6econdly, we investigate the requirement on a i to achieve positive port-folio value at any future time. For this purpose, let time march forward byamount of t . Therefore the new portfolio becomes λ e − a ( T − t )+ y t + λ e − a ( T − t )+ y t − λ e − a ( T − t )+ y t and we hope to have λ λ e − a ( T − t )+ y t + λ λ e − a ( T − t )+ y t ≥ e − a ( T − t )+ y t Again by applying the convex inequality, λ λ e − a ( T − t )+ y t + λ λ e − a ( T − t )+ y t ≥ e − λ λ a ( T − t )+ λ λ y t − λ λ a ( T − t )+ λ λ y t For our purpose, we should have three requirements λ λ ( y + a ) + λ λ ( y + a ) ≥ y + a (5)The first inequality says that y i + a i should be concave which we have exam-ined in the first section. As long as a i are in this range, the portfolio valueis always positive at any future time. We now turn to the convexity trading strategy using swaps. Swaps are oneof the most liquid OTC fixed income instruments. In general, swaps aretrading within only as small as 0.25 basis point bid offer spread. FixedIncome desk often uses swaps to express views on interest rate curves andto hedge interest rate risks, in particular the duration risks.However, swap curve mathematics involves much more than those in zerocoupon bond yield curves due to the complexity of bootstrap procedures.This additional complexity makes the arbitrage argument more difficult andlengthier.An interest rate swap is a contract to exchange certain cash flows atfuture time. Initially there is an exchange of notional amount and subse-quently, there is an exchange of coupon payments, one leg is fixed couponpayment and another leg is floating coupon payment. At maturity there7s another exchange of the notional amount. The floating leg usually hasLIBOR as the coupon rate, therefore the floating leg always prices back tobe par value. As a consequence, the fixed leg at the inception, also pricesback to par value since swap has zero value at inception. As a consequencethe swap rates represent the par bond coupon rates.In real world, the payment frequency for the fixed leg and the floating legmight be different. The fixed leg usually has 6 month as the payment fre-quency and the floating leg has 3 month as the payment frequency. Also eachpayment day should adjust for weekend and the holiday. However, in ouranalysis, we just ignore these technicalities without loss of any generalities.To establish our results, we use annual payment frequencies at time1 , , . . . . The swap rates correspondingly are x , x , . . . . Each swap rate x n corresponds to the fixed leg payment rate of a n year swap. We also use p , p , . . . as the discount factors i.e. zero coupon bond prices corresponds toyear 1 , , . . . . Since we are using annual payment frequency, we can explicitlywrite down the bootstrap procedure of converting swaps rates to discountfactors.The first year swap rate is simply the compounding rate for the first yearhence p = 11 + x Starting from the second year, the conversion is getting complex. Therecursive identity can be written in general as p n = 1 − x n P n − i =1 p i x n (6)or equivalently x n = 1 − p n P ni =1 p i (7)Because zero bonds’ prices are decreasing in order to prevent arbitrage,we should impose p = 1 , p n > p n +1 , p n > , for all n = 1 , , . . . , This fact would impose conditions on swap rate itself. We don’t wantto explore the necessary and sufficient conditions on swap rates, but thefollowing fundamental fact is very interesting and explains the general shapeof swap curve. Let us first introduce more notations. The fixed leg of swap8as one principal payment at the final maturity and all coupon payments inbetween. The coupon payment has discounted value x n ( p + p + · · · + p n )and professionals always call the sum P n = p + p + · · · + p n the annuity factor. We now state the first necessary condition on swap rates. Theorem 2.
The following limit exits lim n →∞ x n = x ∞ ≥ . If the limit x ∞ > , we must have lim n →∞ p n = 0 . Proof.
In the identity x n = 1 − p n P ni =1 p i = 1 − p n P n the numerator has limit as n → ∞ because discount factors are positiveand decreasing. The Denominator also has limit because P n is positive andincreasing. Therefore x n must have a finite limit. If the limit is positive, theinfinite series P ∞ n =1 p n must be finite hence p n → We now introduce the swap curve shifting. If the swap curve changes andthe new swap curve for time n becomes x n + y n , the bootstrapped discountcurve will also changes. In order to signify the dependence of the changes we9se p n ( y ) as the new discount factor. Notice that y here is not necessarilythe constant but the entire vector. Correspondingly the annuity factor alsochanges from P n to P n ( y ). We have to explore the properties of the newdiscount factors and the old discount factors in order to explore the profitand loss of any trading strategy. These properties are very interesting ontheir own. Lemma 1.
If the changes in swap curve y n ≥ for n > , we must have P n ( y ) ≤ P n , n > If the changes in swap curve y n ≤ for n > , we have P n ( y ) ≥ P n , n > Proof.
We prove the y n ≥ P n = 1 − ( P n − P n − ) x n therefore we see that P n = 1 + P n − x n On the other hand it is obvious that P ( y ) = 11 + x + y ≤
11 + x = P We can now use induction method. If we already have P n − ( y ) ≤ P n − , it isclear that P n ( y ) = 1 + P n − ( y )1 + x n + y n ≤ P n − ( y )1 + x n ≤ P n − x n = P n Its financial meaning is obvious. If the rates are moving higher, theaverage (or total) discount factors should be lower. The result is intuitivebut certainly not obvious. We cannot stretch it to make the statement thateach discount factor is lower i.e. p n ( y ) ≤ p n , for all n > Lemma 2.
If the swap rates change y n = y > is constant, for each n > we should have < p n P n − p n ( y ) P n ( y ) ≤ y < P n ( y ) − P n ≤ y Proof.
These two inequalities are not obvious at all. We notice by (7), wehave 1 P n − p n P n = x n , P n ( y ) − p n ( y ) P n ( y ) = x n + y Subtracting, we have (cid:20) p n P n − p n ( y ) P n ( y ) (cid:21) + (cid:20) P n ( y ) − P n (cid:21) = y We have already proved the second bracket is always positive we just needto prove the first bracket is also positive. We notice that1 P ( y ) − P = y therefore p P − p ( y ) P ( y ) = 0Now if there is n such that p n P n − p n ( y ) P n ( y ) ≥ p n +1 P n +1 − p n +1 ( y ) P n +1 ( y ) < P n ( y ) − P n < y < P n +1 ( y ) − P n +1 (cid:20) P n +1 ( y ) − P n +1 (cid:21) − (cid:20) P n ( y ) − P n (cid:21) > p n +1 P n +1 P n ≥ p n +1 ( y ) P n +1 ( y ) P n ( y )But by the condition again p n +1 P n +1 P n > p n +1 ( y ) P n +1 ( y ) P n ( y ) > p n +1 P n ( y ) P n +1 > p n +1 P n P n +1 which is a contradiction.With this lemma, we see that we have the following interesting ordering p n ( y ) P n ( y ) < p n P n < P n < P n ( y )for any constant movement y > Lemma 3.
When the parallel movement y > the quantity p n ( y ) < p n forall n > .Proof. Because in lemma 2,we have proved that p n P n − p n ( y ) P n ( y ) ≥ p n ( y ) p n < P n ( y ) P n < . Lemma 4.
When the parallel movement y > , the quantity P n ( y ) /P n ismonotonically decreasing. roof. In order to prove the monotonicity, we just need to show P n ( y ) P n − P n +1 ( y ) P n +1 > P n ( y ) P n +1 ( y ) > P n P n +1 But this is true because p n +1 ( y ) P n +1 ( y ) < p n +1 P n +1 by the previous lemma 3.Now with all the build-ups in our knowledge, finally we are able to provethe following Lemma 5.
When the spread y > and when swap curve increasing, theratio of two discount factors p n ( y ) /p n is also decreasing.Proof. We want to prove that p n ( y ) p n − ≤ p n − ( y ) p n for all n >
1. But this inequality is equivalent to p n ( y )( p n − − p n ) ≤ p n ( p n − ( y ) − p n ( y ))Due to the fact that p n = 1 − x n P n , p n − = 1 − x n − P n − we have p n − − p n = x n P n − x n − P n − = ( x n − x n − ) P n + x n − p n Similarly, p n − ( y ) − p n ( y ) = x n P n ( y ) − x n − P n − ( y ) = ( x n − x n − ) P n ( y )+( x n − + y ) p n ( y )Therefore what we want to prove is equivalent to − ( x n − x n − ) (cid:18) p n P n − p n ( y ) P n ( y ) (cid:19) ≤ y p n P n p n ( y ) P n ( y )which is obvious under our assumptions.13inally, we need a technical lemma. Lemma 6.
For three integers n < m < k , the three points on the plane (cid:0) P n , P n ( y ) (cid:1) , (cid:0) P m , P m ( y ) (cid:1) , (cid:0) P k , P k ( y ) (cid:1) is concave (convex) if and only if p n ( y ) /p n is decreasing (increasing) withrespect to n .Proof. Let three points n < m < k be the maturity of the swaps, Theconcavity inequality is( P k − P n ) P m ( y ) ≥ ( P m − P n ) P k ( y ) + ( P k − P m ) P n ( y )which is equivalent to( k X i = n +1 p i )( m X j = n +1 p j ( y )) ≥ ( k X i = n +1 p i ( y ))( m X j = n +1 p j )After cancelation, we need to prove k X i = m +1 p i m X j = n +1 p j ( y ) ≥ k X i = m +1 p i ( y ) m X j = n +1 p j But by monotonicity, we know for each i > j , we have p i p j ( y ) ≥ p j p i ( y )therefore proved our claim.Combining lemma 5 and lemma6, we have Lemma 7.
When the spread y > and when swap curve increasing, thethree points on the plane (cid:0) P n , P n ( y ) (cid:1) , (cid:0) P m , P m ( y ) (cid:1) , (cid:0) P k , P k ( y ) (cid:1) is concave. When the spread y >< and when swap curve increasing, thethree points on the plane (cid:0) P n , P n ( y ) (cid:1) , (cid:0) P m , P m ( y ) (cid:1) , (cid:0) P k , P k ( y ) (cid:1) is convex. Theorem 3.
If the swap curve is upward, plot each swap rate against itsduration, the curve should be concave under parallel movement assumption,otherwise we can construct an arbitrage.Proof.
We now construct a portfolio consisting with three swaps maturingat time T , T , T . The corresponding swap rates are x , x , x . The notionalof these three swaps are λ , λ , λ . In particular we need λ = P − P λ = P − P λ = P − P According to the condition, we have x < x < x . We assume by contradic-tion that ( P , x ) , ( P , x ) , ( P , x ) is convex, we will construct an arbitrageportfolio. We long the fixed leg in the first and third swap while short thefixed leg in the second swap. After the parallel movement by an amount y >
0, our profit and loss comes from two components, L = L + L where L is the interest accrual and L is the mark to market profit andloss. The interest accrual is L = λ x + λ x − λ x > L component is the market to market L = − yλ ˜ P ( y ) − yλ ˜ P ( y ) + yλ ˜ P ( y ) (8)where ˜ P i ( y ) is the annuity factor from time t the final maturity year i . Wewant to show that if y >
0, we have λ ˜ P ( y ) + λ ˜ P ( y ) ≤ λ ˜ P ( y )which is equivalent to L ≥
0. First by lemma 5 we see know that( P , P ( y )) , ( P , P ( y )) , ( P , P ( y ))is concave, given the fact that P i ( y ) − ˜ P i ( y ) is a constant regardless of i , wehave ( P , ˜ P ( y )) , ( P , ˜ P ( y )) , ( P , ˜ P ( y ))15ust be concave. This is equivalent to saying λ ˜ P ( y ) + λ ˜ P ( y ) ≤ λ ˜ P ( y )which proved the theorem.In summary, technically we have shown that under the parallel movementassumption if the swap curves are increasing, the curve must be of concaveshaped. 16 eferences [1] Carr P., and M. Schroder, 2003, “On the valuation of arithmetic Aver-age Asian options: the GemanYor Laplace transform revisited”, NYUworking paper.[2] Dothan, U, 1978, “On the term structure of interest rates”, Journal ofFinancial Economics , , 59-69.[3] Cox, J., J. Ingersoll, and S. Ross, 1985, “A Theory of the Term Struc-ture of Interest Rates”, Econometrica , , 385–408.[4] Heston, S., 1993, “Closed-Form Solution for Options with StochasticVolatility, with Application to Bond and Currency Options”, Review ofFinancial Studies , , 327–343.[5] Peter Carr, Jian Sun, 2014, “Implied Remaining Variances and Deriva-tive Pricing”, Journal of Fixed Income , 19–32.[6] Jian Sun, Qiankun Niu, Shinan Cao, Peter Carr, 2016, “Implied Re-maining Variances under Bachelier Model”, Journal of Fixed Income , 78–95.[7] Antti Limanen, “Overview of Forward Rate Analysis:Understanding theYield Curve: Part 1”, Salomon Brothers .[8] Peter Carr, Jian Sun, 2007, “A new approach for option pricing understochastic volatility”,
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