How to Use JIBAR Futures to Hedge Against Interest Rate Risk

Transcription

1 How to Use JIBAR Futures to Hedge Against Interest Rate Risk Introduction A JIBAR future carries information regarding the market s consensus of the level of the 3-month JIBAR rate, at a future point in time. A strip of JIBAR futures can be used to construct a short-term yield curve, which can in turn be used to measure short-term interest rate risk, as well as to determine appropriate hedge ratios. In this paper we describe a simple bootstrap than can be used to construct the JIBAR futures yield curve, and illustrate by way of an example, how this curve can be used to determine appropriate hedge ratios. Note, a more rigorous bootstrap exists, but is beyond the scope of this document. 2 The JIBAR Futures Yield Curve Suppose we want to construct the JIBAR futures yield curve on any arbitrary date, t 0. Consider the JIBAR future that expires on d, and let y denote the yield on this particular future. If d 2 represents the date that is three months Modified Following from d, then the forward discount factor for the period from d to d 2 can be calculated as: Z(t 0 ; d, d 2 ) = + yα(d, d 2 ), () where α(d, d 2 ) = (d 2 d )/365. Note that if Z(t 0, d ) is known, then in the absence of arbitrage we must have that: Z(t 0, d 2 ) = Z(t 0, d )Z(t 0 ; d, d 2 ). (2) Now, consider a strip of JIBAR futures, with expiration dates t, t 2,..., t n, and yields y, y 2,..., y n. Define t n+ as the date that is three months Modified Following from t n, and assume that t i+ is three months Modified Following from t i, for i =, 2,..., n. The following algorithm can then be used to construct the JIBAR futures yield curve: The JIBAR futures construction algorithm.. Estimate Z(t 0, t ), i.e. estimate the zero-coupon spot rate corresponding to t, i.e. r(t ). 2. Set i =. 3. Plug y i, t i and t i+ into equation (), and estimate Z(t 0 ; t i, t i+ ). 4. Plug Z(t 0, t i ) and Z(t 0 ; t i, t i+ ) into equation (2), and estimate Z(t 0, t i+ ). 5. Set i = i Repeat steps 3 to 5 until i = n +. Johannesburg Stock Exchange October 20

2 The assumption that t i+ is three months Modified Following from t i is naive, and will only be remotely viable if we consider a strip of quarterly expiring futures, i.e. if we only consider futures in the March/June/Sep/Dec expiration cycle. Notwithstanding, this assumption provides the foundation for an extremely simple curve construction algorithm. Note, in order to perform the first step of our construction algorithm, i.e. in order to estimate Z(t 0, t ), we will require the, and 3-month JIBAR rates. Let τ denote the date that is one month Modified Following from t 0, and similarly, let τ 2 denote the date that is three months Modified Following from t 0. Define r(τ ) and r(τ 2 ) as the respective, and 3-month JIBAR rates. We need to consider three distinct scenarios: Estimation methods for r(t 0 ).. t τ. Here we set r(t ) = r(τ ). 2. τ t τ 2. Here we have to interpolate between r(τ ) and r(τ 2 ). A powerful method for interpolating yield curve data involves modelling r(t)t as a straight line, for τ t τ 2, resulting in the following interpolation formula: t τ τ r(t ) = 2 τ r(τ 2 )α(τ 2, t 0 ) + τ2 t τ 2 τ r(τ )α(τ, t 0 ). (3) α(t 0, t ) 3. t > τ 2. Here we set r(t ) = r(τ 2 ). The price of the zero-coupon bond maturing on t can then be calculated as: Z(t 0, t ) = + r(t )α(t, t 0 ). (4) Note that the simple zero-coupon spot rate, r(t i ), corresponding to the discount factor Z(t 0, t i ), can be obtained as follows: ( ) r(t i ) = Z(t 0, t i ) /α, (5) for i =, 2,..., n +. When combining the set of zero-coupon spot rates, r(t 0, t i ), for i =, 2,..., n +, with an appropriate interpolation algorithm, we can obtain the zero-coupon spot rate corresponding to any arbitrary date t, where t t n+. 3 An Example of the JIBAR Futures Yield Curve Consider the strip of quarterly expiring JIBAR futures on 8 August 20, given in Table. Note that for this particular example, t 0 = 8 August 20, n = 8 and t n+ = 9 September 203. Furthermore, on 8 August 20, the -month JIBAR rate was 5.495, whilst the 3-month JIBAR rate was For this particular example, we have that τ = 20/09/08, whilst τ 2 = 20//08, implying that τ t τ 2. It follows that r(t ) can be estimated through the use of equation (3): r(t ) = = , Z(t 0, t ) = / ( ( )) = Johannesburg Stock Exchange October 20 2

3 Expiration Yield (%) Z(t 0 ; t i, t i+ ) Z(t 0, t i ) r(t 0, t i ) (%) 2/09/ /2/ /03/ /06/ /09/ /2/ /03/ /06/ /09/ Table : The structure of JIBAR futures yields on 8 August r(t) Sep-20 Dec-20 Mar-202 Jun-202 Sep-202 Dec-202 Mar-203 Jun-203 Sep-203 Figure : The JIBAR futures yield curve on 8 August The Hedging Strategy Any cash flow to be received at a future point in time is valued off a yield curve. A portfolio comprising a number of future cash flows is thus subject to a substantial degree of yield curve risk. JIBAR futures can be used as an effective tool for managing yield curve risk in the nought to two year region of the curve. Consider hedging a particular portfolio as follows:. Bootstrap the JIBAR futures yield curve, and use this curve to find V ; the value of the portfolio under consideration. 2. For i = to n, bump the yield on the i th JIBAR future up or down by one basis point (leaving all other yields unchanged), and re-bootstrap the JIBAR futures yield curve. Find i ; the absolute change in value of the portfolio under consideration, under the i th curve, for i =, 2,..., n. 3. Let δ i denote the number of futures expiring on t i, needed to hedge the particular portfolio. The value of δ i can then be estimated as follows: for i =, 2,..., n. δ i = i 2.5, (6) Johannesburg Stock Exchange October 20 3

4 4. The Mathematics Underlying the Hedge Ratio Define Λ as the matrix such that Λ(i, j) represents the absolute change in value of the j th JIBAR future, under the i th JIBAR futures yield curve. The i th diagonal element of Λ will be equal to 2.5, whilst all other elements will be equal to zero. We want to construct a portfolio that is immune to adverse yield curve shifts. One possibility would be to assume that the short-end of the yield curve is defined by the strip of quarterly expiring JIBAR futures, and to assume that the yields on any the individual futures change on a one by one basis. We can obtain a vector, δ, such that from where it follows that Λδ = 0, (7) δ = Λ. (8) From equation (8) the ratio in equation (6) is immediately evident. Note, we are only interested in hedging against adverse changes in V. Only one of the scenarios; bumping y i up or down will have an adverse effect on V. If bumping y i up results in an adverse change in V, then V should be hedged with a short position in the future expiring on t i. Conversely, if bumping y i down results in an adverse change in V, then V should be hedged with a long position in the future expiring on t i. 5 An Example Portfolio Consider hedging the following portfolio on 8 August 20: Date Cash Flow Discount Factor Present Value 20/2/2 R 50, R 48, /03/2 R 50, R 48, /06/20 R 50, R 47, /09/20 R 50, R 46, /2/9 R 50, R 46, /03/9 R 50, R 45, /06/20 R 50, R 44, /09/9 R,050, R 925, R,253, Table 2: Portfolio to be hedged on 8 August 20. Our portfolio consists of a strip of cash flows to be received at future points in time. Note that for simplicity, we have constructed a portfolio where each cash flow date corresponds exactly to one of the dates in Table. If this were not the case we would have to rely on an appropriate method of interpolation, in order to find the discount factor applicable to each cash flow date. Note that on 8 August 20, the total value of our portfolio was R,253, Furthermore, note that decreasing any of the discount factors in Table 2 decreases the value of our portfolio, i.e. our portfolio is adversely affected by upward shifts in the yield curve. As such, our portfolio should be hedged with short JIBAR futures positions. Consider bumping the yield on the 20/09/2 JIBAR future from 5.6% to 5.6%. Table 3 shows that under this particular scenario, the value of our portfolio changes to R,253,402.62, a decrease of R In order to hedge the value of our portfolio against changes in the yield of the 20/09/2 JIBAR future, we thus need to short twelve of the 20/09/2 JIBAR futures. Note, the actual hedge ratio is calculated as 30.82/2.5 = 2.38, however, we cannot buy/sell a fraction of a contract. Thus, we round to zero decimal places. Johannesburg Stock Exchange October 20 4

5 Date Cash Flow Discount Factor Present Value 20/2/2 R 50, R 48, /03/2 R 50, R 48, /06/20 R 50, R 47, /09/20 R 50, R 46, /2/9 R 50, R 46, /03/9 R 50, R 45, /06/20 R 50, R 44, /09/9 R,050, R 924,978.3 R,253, Table 3: The value of the portfolio in Table 2 after bumping the yield on the 20/09/2 JIBAR future from 5.6% to 5.6%. Table 4 shows the number of futures, for each expiry, needed to hedge the portfolio in Table 2. The value of this particular portfolio is adversely affected by bumping any of the JIBAR futures yields in Table. As such, the portfolio is hedged with a strip of short JIBAR futures positions. t i V i i δ i 20/2/2 R,253, R /03/2 R,253,404.4 R /06/20 R,253,404.7 R /09/20 R,253, R /2/9 R,253, R /03/9 R,253,408.5 R /06/20 R,253, R /09/9 R,253,40.5 R Table 4: The appropriate hedge ratios for the portfolio in Table 2. 6 Caveat When constructing the JIBAR futures yield curve we ignored the concept of a convexity bias, which refers to the difference that typically exists between the rates implied by short-term interest rate futures, and those implied by Forward Rate Agreements (FRAs). Before bootstrapping, we should ideally subtract the convexity bias from the futures rate. Note, the value of the bias can be excessive (more than 50 basis points), however, for maturities less than 2 years, the effect of the bias should be limited (typically less than 2 basis points). In a follow up article, we will investigate methods for estimating the convexity bias, and illustrate how to accommodate for the convexity bias when constructing the JIBAR futures yield curve. One of the assumptions underlying our hedging strategy is that the short-end of the yield curve is defined by a strip of quarterly expiring JIBAR futures. Furthermore, we assume that changes in the yields of any of the JIBAR futures occur on a one by one basis. Thus, the hedged portfolio will not be immune to more complicated yield curve changes. Lastly, the hedging strategy is dynamic, implying that hedge ratios should be recalculated on a frequent basis. Notes A Business Day Convention whereby payment days that fall on a holiday or Saturday or a Sunday, roll forward to the next business day, unless that day falls in the next calendar month, in which case the payment day rolls backward to the immediately preceding business day. A one basis point change in the yield of a JIBAR future translates to a R 2.5 change in contract value. In a follow up article we will investigate methods for interpolating yield curve data. Johannesburg Stock Exchange October 20 5