In the previous section, we explored different methodologies of constructing yield curves, in particular, through the use of different interpolation rules. In this section, we consider how movements in the yield curve lead to price changes in a portfolio consisting of underlying instruments that depend upon our constructed yield curve.
Consider a portfolio of securities with value , where is a function of the yield curve . The securities in are typically not in the benchmark set and could contain IR options, swaps that are no longer at par, etc. Since the yield curve, , is itself a function of the benchmark set
our portfolio, is implicitly a function of the and explicitly a function of a vector , containing model parameters used to price the underlying instruments within the portfolio (for example, volatilities). Hence, we may write
Clearly, is dependent on the benchmark securities via a curve-construction algorithm, so this dependence will change if we were to use say natual-cubic splines as opposed to piecewise flat forward-rate bootstrapping.
Taking the differential of and employing the chain-rule, we find that
Clearly, this is a mathematical construct that has no practical use in a live risk-management system. However, for non-infinitesimal moves in each parameter, equation (1) holds to second-order accuracy, i.e. for small moves , , , , we have
This equation is useful, and allows the attribution of moves in a portfolio to moves in the underlying parameters, or risk-factors, in a process known as PnL attribution. At most institutions managing portfolios of fixed income securities, this process is run overnight and generates a report known as a PnL explain report. This report allows the managers of said portfolio to understand exactly where moves in their positions came from. In particularly volatile market conditions, the first order moves captured in equation (2) above may not be enough to adequately capture all of the PnL moves in a book. In this case, higher-order terms including second and cross-derivative terms may need to be included to properly understand where all movements in PnL came from.
For the purpose of managing first-order risks in the yield curve, equation (2) suggests that the collection of derivatives , - often referred to as bucketed interest-rate deltas - constitute a natural metric for calculating portfolio risk. If all these derivatives were zero, then our portfolio, to first-order, would be immune to any moves in the yield curve consistent with the chosen curve construction algorithm. More explicitly, our portfolio would be immune to small moves in the underlying benchmark instruments utilised to construct said yield curve.
On the other hand, if we were to construct a portfolio of positions and find that some of these derivatives were non-zero, we could manage our risk by constructing a hedge portfolio of benchmark securities with exposures for . This hedge portfolio could be constructed out of the benchmark securities themselves, or out of another set of liquid instruments providing exposure to these factors. The key idea is simply that we can hedge our original portfolio without having to use the original portfolio's securities and by utilising a set of liquid securities with low-transaction costs. It is important to emphasise that the hedge resulting from using such a hedging portfolio would not be model-consistent - most interest rate models assume that yield curve risk arises from a few stochastic yield curve factors that move the curve smoothly, in a generally parallel fashion. Hence, in theory, a bucketed immunisation against each term is overkill in the sense that our resulting portfolio would be hedged against too many risk factors, as each factor is theoretically driven by a common, smaller set of fundamental factors. However, this approach is still the industry standard and works well in practice.
Perhaps the simplest approach to computation of the delta involves a manual bump to by a fixed amount followed by a reconstruction of the yield curve and a subsequent repricing of the portfolio . Then, the delta can be approximated by the finite-difference
This procedure is known as the par-point approach and the resulting derivatives are known as the par-point deltas. This approach is attractive in that it fits into an existing yield curve construction and portfolio valuation framework with no extra work on behalf of the implementor. However, for it to work effectively, it is important that whatever yield curve construction algorithm is in place is fast and produces local perturbations of the yield curve when the benchmark prices are shifted. For example, suppose our benchmark set contains a short-dated FRA. Perturbing the price of this FRA should not cause noticeable shifts in long-term yields. Doing so could lead us to the conclusion that, for example, a 30 year swap could be perfectly hedged using a 1 month FRA, a clearly erroneous assumption. As we alluded to in the previous post, simple bootstrapping methods and Hermite splines exhibit good perturbation locality due to the dependence of the yield curve on benchmark prices only directly adjacent to the point in question. Cubic splines, on the other hand, involve the inversion of a system of linear equations that consequently leads to the bleeding of a shift in any benchmark price across the whole yield curve (see the final figure of the previous post).
An alternative approach to perturbing the benchmark security prices directly is to apply perturbations directly to the discount curve, hence eliminating idiosyncrasies arising from the yield curve construction algorithm itself. In practice, this technique involves applying bumps to the forward curve , to which we apply functional shifts , . Expressing the direct dependence of the portfolio on the yield (or forward curve) by writing , we then compute the functional derivatives for as
Some standard choices for are:
A common choice is to set the three months apart, with dates corresponding to Eurodollar futures maturities. This clearly gives rise to a large number of deltas, , and the derivatives thus give a fairly detailed description of where the portfolio risk is concentrated on the forward curve.
Since FRAs and Eurodollar futures are liquid only up to around the 4 year maturity, the forward rate deltas do not directly reflect hedging instruments for the longer end of the yield curve. However, by considering the dependence of the benchmark instrument on the discretised forward curve points , via the chain-rule we see that
so that forward rate deltas can be translated to hedging notionals through the knowledge of the values . These sensitivities form the Jacobian describing how changes in the forward rates relate to changes in the underlying benchmark instruments. This idea shall be explored further in the next section.
As described in the previous section, a collection of forward rate deltas is useful only in its ability to be translated into actual hedge transactions. To do so requires some basic linear algebra as we shall now see.
Suppose we have available a set of hedging instruments with values
This set may or may not correspond with the benchmark set used for curve construction. Using equation (3), we denote the sensitivities of the hedging instruments to the shifts by , , . If the th hedging instrument is held in our hedging portfolio with notional weight , and we define the vector of weights as
then our hedge portfolio has value given by
Hence, the th perturbation to the hedge portfolio is given by
where
In general, we would like to choose the weights in such a way as to offset as much of as possible through our hedge portfolio sensitivity , for . Let be the relative importance of the th sensitivity and let represent the relative reluctance to use the th hedging instrument (a function of the instruments liquidity, for example). Then, the optimal hedging weights satisfy the condition
By defining the Jacobian matrix to have columns , the vector by
the matrix to be a diagonal matrix with , and the matrix to be the diagonal matrix with , we find that equation (4) can be recast in matrix-vector form as
This problem can be solved analytically, with solution given as the solution to the linear system
When solving equation (5), it is important to consider the relative dimensions of the matrices involved.
Firstly, if we use fewer hedging instruments than shifts, i.e. , then it is generally not possible to offset all risks. In this case, the weights gain importance because they allow the user to focus hedging towards risk buckets deemed to be more important to immunise. On the other hand, the weights become less important, because we have less instruments in which to transact.
Conversely, if there are more hedging instruments than risk buckets, i.e. , then there are infinitely many hedging portfolios that perfectly offset all the risks. In this case, the weights can generally be ignored (set to 1), whereas the weights matrix becomes critical because it allows the user to decide which of the possible hedge portfolios is most desirable from a cost perspective.
Finally, if then there normally exists a single portfolio that hedges all risks. Both and are relatively unimportant in this case, though one may still wish to use non-zero values for to dampen or reduce oscillations in the computed solutions as yields vary. In the case that , and , the solution to the optimisation problem reduces to the analytic solution
The above methodology for producing hedge portfolios from arbitrary shocks to the forward curve via the matrix problem (5) is known as the Jacobian method. Combined with forward rate deltas, the Jacobian method allows the aggregation of fine-grained risks with respect to forward buckets into hedges in a set of hedging instruments.
Note that the par-point method can be seen as a special case of the Jacobian method. In this case, the hedging set is chosen to correspond with the benchmark set and the s are set to exactly represent shifts of the forward curve that bump the benchmark securities by a fixed bump size. In this instance, the Jacobian matrix and the original par-point deltas are recovered.
The Jacobian approach has two primary benefits over the other two methods. The first is that it decouples the risk/ sensitivity calculations from the particular curve construction algorithm used. This allows for combining of smooth curves with localised risks. It is also a useful approach when the underlying yield curve must be reconstructed very quickly - this is particularly apt when using a Libor or Treasury curve, since the underlying instruments trade very liquidly and their prices hence change very frequently. In this case, the Jacobian approach allows changed in benchmark prices to be translated into changes of the forward curve via a simple matrix multiplication, rather than through a full curve reconstruction upon every benchmark price update, i.e. since
we find that
where
Roughly speaking, swap rates represent a weighted average of forward Libor rates across the tenor structure of the swap. For example, an year swap with annual tenor structure can be written as
where . The weightings will be roughly of size , so that we find that
Inverting this relationship for shows that
Hence, a naive bump to the forward rate by an amount while leaving and unchanged propagates into the Libor forward rate as a shift of size and into the forward rate as a shift of size . As a concrete example, if we were to shift the 10 year swap rate by 1bp while keeping the 9 and 11 year points flat, then the forward Libor rate would move by 10bp and the forward rate would move by -10bp. Now suppose that our portfolio contained a spread option on the difference . We see that the underlying of this option just moved by 20bp despite the swap rate moving just 1bp!
This situation helps highlight the importance of applying shifts to the forward curve that reflect realistic moves in interest rates. It is highly unlikely that the forward curve would move in a way that caused the 10 year swap rate to move by a basis point while the 9 and 11 year swap rates stayed constant.
A tweak to the par-point approach that leads to more realistic shifts in the forward curve is the cumulative par-point approach. The idea here is simple. The shift to the th benchmark security is retained while calculating the derivative with respect to the th security, i.e. the th derivative is calculated as
Notice that, by Taylor expansion about the point , we find that
which clearly converges to the standard par-point delta
in the limit as for . The forward curve moves implied by the cumulative par-point approach are less extreme than those implied by the ordinary par-point approach, making te cumulative method attractive in practice. Another advantage of the cumulative approach is the fact that the sum of the deltas calculated by the method are always exactly equal to the parallel delta, by construction. Due to second-order effects, the same is only true of the ordinary method in the limit as the shifts tend to zero.
It is easy to enact the cumulative approach in the Jacobian framework laid out above. Clearly, the th cumulative shift roughly corresponds to a piecewise flat move in the forward curve between the maturities of the th and th benchmark. Hence, we define
In fact, performing the cumulative shifts in forward space is often a more attractive approach than doing so in price space. By shifting in forward space, forward curve shocks are similarly scaled, in contrast to the basic cumulative method where the size of the forward curve shocks grow linearly with maturity.
A final point to note is that most yield curve risk-management systems compute deltas as the average of a delta first computed using a positive bump and then computed using a negative bump. This approach helps to improve accuracy and stability of the computed deltas and applies to all of the delta calculation methods explored above.