Hedging from economic value perspective implies stabilization of the net present value (NPV) of the product or portfolio in question and thus mitigating its volatility. As interest rate change the net stream of cash flows must diverge from the stream initially planned in terms of its total amount, its timing, or both, if the net present value of the portfolio is to remain unchanged. It is necessary to calculate the change in the present value of the portfolio for a small change in interest rates. This change in net present value must be offset by an opposite change in the net present value of the hedging instrument.
For the purpose of stabilizing the NPV and reducing its volatility BPV hedging could be applied. For illustration, lets consider the same bullet loan with a notional of 100, fixed coupon rate of 5% with 5 years to maturity. The coupon of the bullet loan consists of 3% risk free and 2% commercial margin. The NPV of the bullet loan is exposed to value volatility in case the interest rate change. The BPV is calculated as 1bps change in the NPV of the product or portfolio in question. This can be shown in Table 5.1. The BPV of the bullet loan is(4.76+4.53+4.32+4.11+82.23)−(4.76+4.54+4.32+4.11+82.27) =−0.043. This means that the loan loses -0.043 EURO in value whenever the interest rate moves up with 1 bps. This negative BPV is intuitive, because a higher discount rate result in a lower present value of the asset.
T (in years) 1 2 3 4 5
Total CF bullet loan (in EURO) 5 5 5 5 105 Discount factor (with 5%) 0.952 0.907 0.864 0.823 0.784
Total DCF (in EURO) 4.76 4.54 4.32 4.11 82.27
Total CF bullet loan (in EURO) 5 5 5 5 105 Discount factor (with 5% + 1bps) 0.952 0.907 0.864 0.822 0.783
Total DCF (in EURO) 4.76 4.53 4.32 4.11 82.23
Table 5.1: Computing the BPV for the bullet loan
The NPV of the bullet loan could be hedged with a swap that offsets the BPV profile of the bullet loan. The BPV of the swap transaction is the sum of the BPV of its fixed and floating legs. The characteristics of the fixed leg would be 100 notional, fixed coupon
of 100 and floating coupon that reprices annually. The floating leg of the swap from the economic value perspective is seen as depositing money on a bank account for a period of 1 year and receiving interest rate and then unwinding the position. Att=0
the interest rate to be received at t = 1 for depositing money for one year is known. At t=0 the interest risk free interest rate at t=i+1, where i ={1..n}is not known. However, regardless what the value of the rate is the NPV of the deposited amount will equal the notional amount (notice that this view changed after the introduction of Overnight Index Swap discounting, but this will not be taken into account in this research). Therefore, given the characteristics of the fixed leg and assuming 3% for the floating leg att=1the BPV of the fixed leg is -0.0458 and the BPV of the floating leg is -0.0094. The total BPV is then -0.0094 - -0.0458 (BPV asset leg - BPV liability leg) = 0.036 (with a notional value of 100). In order to match the required BPV profile of the original transaction the notional amounts of the swap legs need to be adjusted. The needed notional amounts are obtained with the following computation: 100 × −0.043
0.036 .
The required notional amounts for these two legs would be 120. The notional of 120 for both swap legs would result in a BPV of 0.043 (with a notional value of 120 the BPV fixed leg is -0.054 and BPV floating leg is -0.011). In case of changes in interest rates the change in NPV of the original transaction would be offset by the change in the NPV of the swap transaction. If the interest rates increase by 1% the change in NPV of the original transaction of -4.3 will be offset by change in the NPV of 4.3 in the swap transaction. Such a hedge would enable to stabilize the value to certain extent. The illustrative representation of the swap transaction is depicted in Figure 5.2.
Figure 5.2: Hedging from economic value perspective: BPV hedge
5.5 End of chapter summary
In this chapter the following research questions have been answered:
Which hedging techniques can be used to hedge IRRBB?
Two main hedging techniques are used in practice to hedge IRRBB: notional and BPV hedging. Notional hedging consists of assigning a notional to the swap transaction such that it equals the notional of the original transaction. BPV hedging consists of assigning a notional to the swap transaction such that the BPV profile of the swap transaction matches the BPV profile of the original transaction. The drawback of economic value hedging is that it does not take into second order effects such as that earnings may change as well.
Notional hedging seem more adequate to hedge IRRBB from earnings perspective thus stabilizing the NII. BPV hedging seem more applicable to hedge IRRBB from the eco- nomic value perspective thus stabilizing the EVE. As a consequence, the economic value
is that it does not take into account economic value. Another drawback is that when interest rates do not go against the bank, the hedge is unnecessary. For instance, when interest rates go up, the bank should receive more coupon, but because of the hedge this coupon is not received.
An algorithm must be seen to be believed.
Donald Knuth
6
Model
The goal of this research is to investigate the relationship between EVE and NII with and without hedging. For this purpose a model is constructed in the MATLAB envi- ronment using Object Oriented Programming. The model is able to output EVE and NII metrics given a portfolio of a bank as well as automatically generate a BPV and notional hedge. The un-hedged and hedged portfolio can then be assessed with respect to the six scenarios mentioned in chapter 3, while taking into account the possibility of prepayments. As mentioned in chapter 3 all assumptions made for EVE and NII are based onBCBS(2016).
In this chapter, all components of the model will be outlined. The chapter starts with a delineation of the term structure, discount curve, prepayment, outflow and repricing models. Afterwards, the cash flow generating algorithm will be elaborated. Finally, the BPV and notional hedging models will be explained. A stylized illustration of the conceptual model is given in figure 6.1.
Figure 6.1: Stylized representation of the conceptual model
Before going into more depth on the modeling of the retail products, some support- ing models are required. These models include the term structure model, prepayment model, the outflow model and the repricing model. These models support the modeling of the cash flows of the retail products elaborated later on.
is sold at a discount from its face value. The zero coupon curve represents the yield to maturity of hypothetical zero coupon bonds. The zero coupon bonds are not directly observable in the market and thus needs to be estimated from existing zero coupon bonds and fixed coupon bonds prices or yield.
The interest rates data to construct the yield curve for this research has been retrieved from the European Central Bank (ECB). The data set contains AAA-rates and other euro area central government bonds. The missing points on the term structure have been calibrated with the Nelson-Siegel-Svensson (NSS) model. The NSS model is a parametric model that specifies a functional form for the spot rate and can be expressed with the following formula:
y(T) =β0+β1 [ 1−e(−Tτ1) −T τ1 ] +β2 [ 1−e(−Tτ1 ) −T τ1 −e(−Tτ1 ) ] +β3 [ 1−e(−Tτ2) −T τ2 −e(−Tτ2) ] (6.1)
where T denotes the term to maturity and βi and τi are parameters to be estimated. The term to maturity ranges from 1 month up to and including 30 years of residual maturity. The estimated parameters are shown in Table 6.1. The current yield curve is illustrated in Figure 6.2.
Parameter β0 β1 β2 β3 τ1 τ2
Value 2.762834 -3.316999 37.887917 -42.725487 1.702520 1.772731
Table 6.1: NSS parameters Source: European Central Bank
Figure 6.2: Term structure of interest rates