PV01 and Value at Risk in a Bank Treasury
Rupees per basis point, and the loss you should not exceed on 99 days in 100.
PV01 tells a treasury how many rupees it gains or loses when yields move by one basis point (0.01%). Value at Risk (VaR) tells it how much it could lose over a set period at a set confidence level. Mid offices use both: PV01 to size and limit rate exposure, VaR to put interest rate, forex and other risks on one scale.
In the Certified Treasury Professional (CTP) syllabus, PV01 sits in Module III with duration and convexity, and again in Module V as PV01 of swaps. RBI's foreign exchange risk rules also let banks use tenor-wise PV01 and VaR limits in place of the aggregate gap limit.
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The Formulas
PV01 per ₹100
Formula
Modified duration x price / 10,000
Unit
₹ per ₹100 face, per 1 bp
PV01 of a position
Formula
Face value x price / 100 x modified duration x 0.0001
Unit
₹ per 1 bp
Portfolio PV01
Formula
Sum of position PV01s (longs positive, shorts negative)
Unit
₹ per 1 bp
Hedge size
Formula
PV01 to hedge / PV01 per unit of the hedge instrument
Unit
Face value or notional
Parametric VaR (rates)
Formula
PV01 x daily yield volatility in bp x z
Unit
₹ over 1 day
Parametric VaR (price)
Formula
Position value x daily price volatility x z
Unit
₹ over 1 day
Scaling to n days
Formula
1-day VaR x square root of n
Unit
₹ over n days
| Measure | Formula | Unit |
|---|---|---|
| PV01 per ₹100 | Modified duration x price / 10,000 | ₹ per ₹100 face, per 1 bp |
| PV01 of a position | Face value x price / 100 x modified duration x 0.0001 | ₹ per 1 bp |
| Portfolio PV01 | Sum of position PV01s (longs positive, shorts negative) | ₹ per 1 bp |
| Hedge size | PV01 to hedge / PV01 per unit of the hedge instrument | Face value or notional |
| Parametric VaR (rates) | PV01 x daily yield volatility in bp x z | ₹ over 1 day |
| Parametric VaR (price) | Position value x daily price volatility x z | ₹ over 1 day |
| Scaling to n days | 1-day VaR x square root of n | ₹ over n days |
Worked Example 1: PV01 of a Two-Bond Book
Illustrative holdings. Price per ₹100 and modified duration (MD) are given.
Bond A: ₹50 crore face, price 98.40, MD 3.6
Working
50,00,00,000 x 0.9840 x 3.6 x 0.0001
PV01
₹1,77,120
Bond B: ₹30 crore face, price 101.20, MD 7.8
Working
30,00,00,000 x 1.0120 x 7.8 x 0.0001
PV01
₹2,36,808
Book
Working
1,77,120 + 2,36,808
PV01
₹4,13,928 per bp
| Holding | Working | PV01 |
|---|---|---|
| Bond A: ₹50 crore face, price 98.40, MD 3.6 | 50,00,00,000 x 0.9840 x 3.6 x 0.0001 | ₹1,77,120 |
| Bond B: ₹30 crore face, price 101.20, MD 7.8 | 30,00,00,000 x 1.0120 x 7.8 x 0.0001 | ₹2,36,808 |
| Book | 1,77,120 + 2,36,808 | ₹4,13,928 per bp |
Bond B is the smaller holding but carries more risk, because its duration is more than twice Bond A's. A 10 basis point rise in yields costs the book about 10 x 4,13,928 = ₹41.39 lakh.
Worked Example 2: Hedging With PV01
Illustrative. The desk wants to cut the book's PV01 to zero by selling Bond C: price 99.50, MD 6.5.
- 1
PV01 of ₹1 crore of Bond C
1,00,00,000 x 0.9950 x 6.5 x 0.0001 = ₹6,467.50.
- 2
Amount to sell
4,13,928 / 6,467.50 = 64.00, so sell about ₹64 crore face of Bond C.
- 3
Check
Selling ₹64 crore of C gives PV01 of minus ₹4,13,920, which offsets the book within ₹8 per basis point.
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Why the Hedge Is Not Perfect
PV01 matching protects against a parallel shift: every yield moving by the same amount. If short yields and long yields move differently, a book hedged with one bond can still gain or lose. Banks therefore set PV01 limits by tenor bucket, not only in total.
Worked Example 3: VaR on the Bond Book
Illustrative. Daily yield volatility is 6 basis points. Confidence 99%, so z = 2.33.
- 1
1-day VaR
4,13,928 x 6 x 2.33 = ₹57,86,713, about ₹57.9 lakh.
- 2
Read it
On 99 days out of 100, the book should not lose more than about ₹57.9 lakh in a day. On the remaining day, it can lose more.
- 3
10-day VaR
57,86,713 x square root of 10 (3.1623) = ₹1,82,99,195, about ₹1.83 crore.
Worked Example 4: VaR on a Forex Position
Illustrative: a ₹10 crore long dollar position, daily volatility of USD/INR 0.40%, confidence 95% (z = 1.65).
1-day VaR
Working
10,00,00,000 x 0.0040 x 1.65
Result
₹6,60,000
10-day VaR
Working
6,60,000 x 3.1623
Result
₹20,87,103
| Step | Working | Result |
|---|---|---|
| 1-day VaR | 10,00,00,000 x 0.0040 x 1.65 | ₹6,60,000 |
| 10-day VaR | 6,60,000 x 3.1623 | ₹20,87,103 |
PV01 vs VaR
Question it answers
PV01
How much for a 1 bp move?
VaR
How much could I lose, at what probability?
Inputs
PV01
Duration, price, position
VaR
Position, volatility, confidence, holding period
Covers
PV01
Interest rate risk only
VaR
Any market risk: rates, forex, equity
Weak spot
PV01
Assumes parallel shifts; ignores convexity
VaR
Says nothing about losses beyond the confidence level; relies on past volatility
| PV01 | VaR | |
|---|---|---|
| Question it answers | How much for a 1 bp move? | How much could I lose, at what probability? |
| Inputs | Duration, price, position | Position, volatility, confidence, holding period |
| Covers | Interest rate risk only | Any market risk: rates, forex, equity |
| Weak spot | Assumes parallel shifts; ignores convexity | Says nothing about losses beyond the confidence level; relies on past volatility |
How the CTP Exam Tests This
- check_circlePV01 from modified duration and price, then scaled to a position size.
- check_circleWhich bond has more PV01: the one with higher duration and value, not simply the larger face value.
- check_circleHedge ratio: PV01 of the position divided by PV01 of the hedge.
- check_circleVaR concepts: confidence level, holding period, square-root-of-time scaling.
- check_circleLimits: who sets them (the Board, within RBI's rules) and where PV01 and VaR limits replace gap limits.
FAQs
What is the difference between PV01 and DV01?expand_more
None in practice. Both mean the change in value for a 1 basis point change in yield. PV01 is the term used in Indian bank treasuries and in the CTP syllabus.
How do I calculate PV01 of a bond portfolio?expand_more
Work out PV01 for each holding (face value x price / 100 x modified duration x 0.0001) and add them, counting short positions as negative.
Why is 10-day VaR the 1-day VaR times the square root of 10?expand_more
If daily changes are independent, variance grows in line with time, so volatility grows with the square root of time. It is a simplifying assumption, not a law.
What z-value is used for 95% and 99% VaR?expand_more
1.65 for 95% and 2.33 for 99%, the one-tailed values of the normal distribution.
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