Case 098Rates and hedgingWarm up

A bank holds Rs 5,000 crore of government bonds with a modified duration of 5 in a mark-to-market book. Yields rise 60 basis points. What is the loss, and what does it do to a Rs 800 crore capital buffer?

The situation

Amarvani Bank, a mid-sized private bank, holds Rs 5,000 crore of government bonds with a modified duration of about 5. They are held in a book that is valued at market prices, so price changes flow into its reported capital.

The bank's capital above its regulatory minimum, its buffer, is Rs 800 crore. Government bond yields rise 60 basis points over a quarter. The accounting and regulatory treatment of bond losses in banks is set by the regulator and has changed over time; confirm the current rules before relying on how a loss reaches capital.

Your task

How much does the bank lose, what is left of the buffer, and what would you do about the exposure?

Quick check

Roughly how much do the bonds lose when yields rise 60 basis points?

Worked solution

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

30-second answerThe answer to give first

The bonds lose about Rs 150 crore, 3% of their value, taking the buffer from Rs 800 crore to Rs 650 crore, a fall of 18.8%. Duration of 5 means each percentage point of yield moves price about 5%. Government bonds carry almost no default risk but full rate risk. A rise of about 320 basis points would use up the whole buffer, so the bank should size, limit or hedge the exposure.

Step 1How do you get from duration to a rupee loss?

Modified duration is the percentage price change for a one point move in yield. Five times 0.60 points is a 3.0% price fall, and 3% of Rs 5,000 crore is Rs 150 crore. A fixed deposit locked at 7% for five years is worth less the day new deposits start paying 7.6%: nobody would buy yours at face value when a better one is on offer. Put another way, each basis point costs the bank about Rs 2.5 crore.

The relationship
ΔP≈−Dmod×Δy×P=−5×0.006×5,000=−150\Delta P \approx -D_{mod} \times \Delta y \times P = -5 \times 0.006 \times 5{,}000 = -150
D_modmodified duration, 5
0.006the 60 basis point rise in yield
5,000market value of the bond book, Rs crore
What it says in wordsA 60 basis point rise takes about Rs 150 crore off the value of the book; convexity would trim that by only about Rs 3 crore.
The capital buffer after a rise in yields, Rs crore800Buffer today-150+60 bp loss650Buffer afterLoss at bigger shocks+60 bp150+100 bp250+200 bp500+320 bp800all of the buffer
A 60 basis point rise costs Amarvani about Rs 150 crore and takes its capital buffer from Rs 800 crore to Rs 650 crore, while a 200 basis point rise would cost Rs 500 crore and about 320 basis points would use the buffer up entirely.
Step 2Why does a safe bond threaten capital?

Because safe describes the chance of being repaid, not the price along the way. In a book valued at market, a rate rise shows up as a loss in capital immediately, even though every bond will be repaid in full at maturity. A bank that must sell to meet withdrawals turns that paper loss into a real one. That is how banks holding only government bonds have run into trouble when rates rose fast: the assets were never going to default, but they were worth less than the deposits they funded on the day money was needed.

Step 3What would you do about the exposure?

First, measure it in the language of the buffer: Rs 2.5 crore per basis point against Rs 800 crore of room means about 320 basis points to zero. Then decide how much of the buffer rate risk may use, say a quarter, and cut the book's duration or hedge until a plausible shock stays inside it. A pay-fixed interest rate swap gains when rates rise and offsets part of the loss; shortening the book into bonds of two or three years cuts duration directly but gives up some yield. The limitation: this uses a parallel shift, and a real move might hit the long end harder than the short.

Where candidates lose it

The common error is saying government bonds are risk-free and stopping. They are close to free of default risk, not of price risk, and in a book valued at market the price risk hits capital straight away.

The second is an arithmetic slip with basis points: 60 basis points is 0.6%, not 6%, so the price fall is 3%, not 30%.

What the interviewer asks next

  • What swap notional would halve the loss, if the swap has a duration of about 4.3?
  • How does convexity change the answer for a 200 basis point move?
  • Why might a bank move bonds into a book held to maturity, and what does that give up?

Company names and figures are illustrative.