Case 004Hedging a bookHard

A home finance company hedged its fixed-rate mortgage book with payer swaps of matching DV01. Rates then fall 150 basis points and prepayments surge. Why does the hedge lose money, and how much?

The situation

Grihavel Home Finance holds a Rs 5,000 crore book of fixed-rate home loans funded with floating-rate borrowing. The book's expected duration, allowing for normal prepayments, is 4. To protect against rising rates, treasury entered payer interest rate swaps, paying fixed and receiving floating, with a DV01 equal to the book's.

Rates then fall 150 basis points. Borrowers refinance, prepayments surge and the book's expected duration drops to 2.5. Assume the book's duration moved in a straight line between the two points as rates fell, and treat the swaps as having constant DV01.

Your task

What is the DV01 mismatch now, how much of the swap loss do the mortgages fail to offset, and what would you do about it?

Quick check

Rates fall 150 basis points. Which statement fits a matched DV01 hedge on this book?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

The swaps lose Rs 300 crore while the mortgage book gains only about Rs 243.75 crore, leaving Rs 56.25 crore unhedged. Prepayments cut the book's DV01 from Rs 2.00 crore to Rs 1.25 crore a basis point, so 37.5% of the swaps are now surplus. The book is negatively convex: a DV01 hedge must be rebalanced as rates move, or paired with bought options.

Step 1Why does a matched DV01 stop being matched?

Imagine lending a friend money at a fixed 9% with the right to repay whenever they like. If rates fall to 7%, they borrow elsewhere and repay you, just when your 9% loan had become valuable. If rates rise, they keep your cheap loan for years. A prepayable mortgage gives the borrower that option, so the book shortens when rates fall and lengthens when they rise, always the wrong way for the lender. That is negative convexityWhen an asset gains less from a fall in rates than it loses from an equal rise, because its duration moves against the holder..

DV01 at the start: Rs 5,000 crore times a duration of 4 times one basis point is Rs 2.00 crore. The swaps were sized to that. After the fall the book's DV01 is Rs 5,000 crore times 2.5 times one basis point, Rs 1.25 crore. The swaps still carry Rs 2.00 crore per basis point, so the hedge is now 60% bigger than the thing it hedges.

Before the move the book and the payer swaps each carried Rs 2.00 crore of DV01; after rates fall 150 basis points the book carries Rs 1.25 crore while the swaps still carry Rs 2.00 crore, so 37.5% of the hedge is surplus.

Step 2How much does the hedge lose over the whole move?

The swaps lose on a straight line: Rs 2.00 crore times 150 basis points is Rs 300 crore. The book's gain has to be added up as its duration shrinks. With duration falling steadily from 4 to 2.5, the average over the move is 3.25, so the gain is Rs 5,000 crore times 3.25 times 1.5%, about Rs 243.75 crore. The hedge that was meant to net to zero loses Rs 56.25 crore.

The relationship

\Delta V_{\text{net}} = -\frac{V}{100}\cdot\frac{s}{2}\,(\Delta y)^2 = -\frac{5{,}000}{100}\cdot\frac{1}{2}\cdot 1.5^2 = -56.25

V	the book, Rs 5,000 crore
s	how much duration changes per percentage point of rates, here 1.0
\Delta y	the rate move in percentage points, 1.5

What it says in wordsThe straight swap line and the bent mortgage curve differ by a term in the square of the rate move, so the unhedged loss grows fast as the move gets bigger, and it is a loss whichever way rates go.

The payer swaps pay for a straight line with a DV01 of Rs 2 crore, while the mortgage book bends below it on both sides; at a 150 basis point fall the swaps lose Rs 300 crore and the book gains only Rs 243.75 crore.

Rate move	Swaps	Mortgage book	Net
-150 bp	-300.00	+243.75	-56.25
-100 bp	-200.00	+175.00	-25.00
-50 bp	-100.00	+93.75	-6.25
+50 bp	+100.00	-106.25	-6.25
+100 bp	+200.00	-225.00	-25.00
+150 bp	+300.00	-356.25	-56.25

Rs crore. The net hedge result is a loss for every move, small for 50 basis points and nine times larger for 150, because the gap grows with the square of the move; rising rates lose too, as the book lengthens.

Step 3What would you do about it?

Two answers, one tactical and one structural. Tactically, rebalance: unwind 37.5% of the payer swaps, Rs 0.75 crore of DV01, so the hedge matches the shorter book. Structurally, a DV01 hedge will always lag a book whose duration moves, and rebalancing locks in the loss each time. Buying receiver swaptions, which gain more as rates fall, adds the positive convexity the mortgages lack, at the cost of an upfront premium. The limitation to state: the straight-line duration path is an assumption, and real prepayment behaviour lurches when a refinancing wave starts.

Where candidates lose it

Candidates say the hedge was DV01 matched, so it cannot lose. DV01 is a local measure: it matches the first basis point, not the hundred and fiftieth, and prepayments are exactly what makes the book's DV01 move.

The second miss is thinking the problem only appears when rates fall. The same curvature costs money when rates rise and the book extends, which is why the net is a loss both ways.

What the interviewer asks next

Rates rise 150 basis points instead. What is the net result, and why is it also a loss?
How would you price the receiver swaptions you would need, roughly?
Why might a home finance company's accounts not show this loss even though the economic value has fallen?
What data would you want on the borrowers to forecast prepayments better?

Company names and figures are illustrative.