Bond portfolios residing on either the asset or liability side of the balance sheet can be restructured internally or externally. I think of internal risk management as rebalancing via bond purchases and sales, for instance, as above, changing the average duration or convexity of the portfolio to conform to a particular rate view. The idea of “internal” is that the transactions are routine for the portfolio manager – buying and selling bonds. Often, external risk management accomplishes the same end. “External” here means using derivatives to transform some particular aspect of the asset or liability portfolio (i.e., its currency mix, credit exposure, or interest rate risk statistics). These are called derivatives overlay strategies in that the underlying portfolio is left intact while futures, options, or swaps modify its risk-return profile.

Now suppose that you are a passive-aggressive manager of an investment- grade corporate bond portfolio. You benchmark the quarterly returns on the portfolio against a widely used index that is produced and updated by an investment bank. You are essentially passive (i.e., a “closet indexer”) in that the portfolio tracks the index. Therefore, it has a very similar distribution of bonds by credit quality and industry, and an average yield, duration, and convexity that are close to those of the benchmark. Note that it's important to know how those summary statistics are calculated for the index – probably they are market-value-weighted averages.

Your aggressive side comes out in overweighting and underweighting credit components that you expect to outperform and underperform recent trends and expectations from market commentators. Also, when you have a strong view on the Treasury yield curve, you extend or contract duration and convexity. Your problem is that it is costly to manage toward your particular rate view. Corporate bonds are not nearly as liquid as Treasuries, and there is a limited supply of certain maturities, especially long-term securities. You decide to manage the portfolio with a focus on getting the credit risk where you want it and then use interest rate swap overlays to reflect your view on the Treasury yield curve. Fortunately, interest rate swaps can be entered at low transactions costs and unwound expeditiously when your rate view happens to change.

Currently, your bond portfolio has a market value of $245 million and a market-value-weighted average modified yield duration of 6.50, matching the index used to benchmark your performance. The basis point value (BPV) of your portfolio is $159,250 (= $245,000,000 * 6.50 * 0.0001). Your rate view for the next several weeks is a bull flattener in which the long-term end of the yield curve drops more than the short-term end as inflationary fears dissipate. You decide to extend your modified duration out to about 7.50 using an interest rate swap. That restructuring corresponds to an increase in the BPV to about $183,750 (= $245,000,000 * 7.50 * 0.0001).

Thankfully, this is an anticipated yield curve shift for which average duration does serve as an indicator for the extent of the aggressive move. Suppose that 5-year and 10-year, quarterly settlement, fixed-versus-3-month- TIBOR, interest rate swaps are available to you from a swap dealer. Do you enter the swap as the fixed-rate payer or the fixed-rate receiver? Can you increase the average modified duration of the portfolio to 7.50 from 6.50 using a 5-year swap?

The answer to the first question is that you enter a receive-fixed swap that has positive duration. We saw in Chapter 8 that a receive-fixed/pay- LIBOR swap can be interpreted as a long position in a (high-duration) fixed- rate bond and a short position in a (very low-duration) floating-rate note paying LIBOR. The modified duration of the swap is the difference in the modified durations of the constituent bonds. In the same manner, the swap BPV is the difference in the BPVs of the implicit bonds. Suppose that the 5-year receive-fixed swap has a fixed rate of 1.75%, a modified duration of 4.56, and a BPV of 0.0456 per 100 of notional principal: 0.0456 = 100 * 4.56 * 0.0001. Also, a 10-year swap has a fixed rate of 3.00%, a modified duration of 8.36, and a BPV of 0.0836. Pay-fixed swaps have negative duration. Surely they do not reflect your bull flattener view.

The second question is harder and requires a deep understanding of swaps. It is tempting to conclude that entering a 5-year receive-fixed swap lowers the duration of the portfolio. After all, if the swap is like buying a 5-year fixed-rate bond financed by issuing a floater, then the duration of the portfolio that starts at 6.50 must go down if you add in lower duration assets. It would seem that only by entering a 10-year, receive-fixed swap can you lift the portfolio duration up to the 7.50 target.

This line of thinking, while compelling, misses a key idea – the interest rate swap does not change market value. Swaps simply modify, or transform, the duration of a segment of the portfolio. A receive-fixed swap, regardless of its time frame, adds duration to that segment. Equation 10.1 captures how to determine the notional principal (NP) needed to change the duration of a portfolio given its market value (MV) to the target duration.

(10.1)

This equation is based on two ideas. First, the duration of a segmented portfolio is estimated by a weighted average of the durations of the segments, using shares of total market value as the weights. Second, entering a plain vanilla interest rate swap transforms the duration of a segment of the portfolio but does not change the overall market value.

In equation 10.1, the segment of the original portfolio unaffected by the interest rate swap has a market-value weight of (MV – NP)/MV. The affected segment has the remaining weight, NP/MV, because the swap itself does not add nor subtract value at origination. It does, however, make the duration of the affected segment the sum of Swap Duration and the original Portfolio Duration. Therefore, a received-fixed swap increases, and a pay- fixed swap reduces, the duration of assets. In applying this formula, it is useful to isolate the NP term by algebraic rearrangement.

(10.2)

Using the 10-year receive-fixed swap, you need a notional principal of about $29.3 million to increase the average duration of the portfolio from 6.50 to 7.50.

For the 5-year swap, you need more notional principal, about $53.7 million, because the each unit of the derivative is less powerful.

In general, the higher the duration of the swap, less notional principal is required. It's like curing a headache – you can take a few low-dosage pills or just one gigantic pill providing a really big dose.

The hedge ratio for the swap, that is, the notional principal needed to achieve the new target risk measure, also can be calculated with the BPVs. The idea is to add the BPV of the swap to the original portfolio BPV to equal the target BPV. This is summarized in equation 10.3.

(10.3)

In this problem, the portfolio BPV is $159,250 and the target BPV is $183,750. Given the swap BPVs of 0.0456 and 0.0836 per 100 of notional principal for the 5-year and 10-year swaps, the same results are obtained.

This reminds us that modified duration and BPV contain the same information.

I've used the word “about” in setting the requisite notional principal judiciously because this is hardly an exact, tightly constructed calculation from a technical bond math perspective. The looseness arises in how average portfolio duration itself is calculated. I suggested in Chapter 9 that for corporate bond portfolios that inevitably contain callables, aggregating curve durations is more appropriate than yield durations to maintain apples with apples. But then what do you do if the benchmark index reports only the market-value-weighted average of yield durations?

How is the modified duration of those two swaps calculated? Is it yield duration or curve duration? Is your bond portfolio duration based on a periodicity of 2 (meaning the price sensitivity to changes in yields quoted on a semiannual bond basis) and your quarterly settlement swap on a periodicity of 4? If your rate view is based on Treasuries, how will the LIBOR swap curve shift if your bull flattener expectation prevails?

It's important not to fall for the illusion of precision just because your technology can deliver a very accurate-looking number. Sometimes you should round off your result to signal that it is measured with model error.

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