NISM Series IV Formula Sheet
Ten formulas, what each one is for, and a worked example.
Interest rate derivatives are priced off bond maths, so Series IV mixes yield and duration formulas with the settlement rules for futures, FRAs and swaps.
These ten formulas cover both, with a worked FRA settlement.
The 10 Formulas
1
Formula
Bond price
Expression
P = Σ C ÷ (1 + y)ᵗ + F ÷ (1 + y)ⁿ
Use it for
Price of the underlying bond at a given yield.
2
Formula
Modified duration
Expression
MD = Macaulay duration ÷ (1 + y)
Use it for
Price sensitivity: ΔP% ≈ −MD × Δy.
3
Formula
PVBP (DV01)
Expression
PVBP = MD × Price × 0.0001
Use it for
Rupee change in price for a one basis point move in yield.
4
Formula
Invoice price (bond futures)
Expression
Futures settlement price × Conversion factor + Accrued interest
Use it for
Amount paid on delivery of a bond against an IRF.
5
Formula
Cheapest to deliver
Expression
Choose the bond with the lowest Clean price − Futures price × CF
Use it for
The bond a seller will deliver.
6
Formula
Hedge ratio (duration-based)
Expression
Contracts = (PVBP of portfolio ÷ PVBP of futures) × Hedge share
Use it for
Number of IRF contracts to hedge a bond portfolio.
7
Formula
Implied forward rate
Expression
(1 + s₂)² = (1 + s₁) × (1 + f₁,₂)
Use it for
Forward rate implied by spot rates; the basis of FRA pricing.
8
Formula
FRA settlement
Expression
N × (L − K) × t ÷ (1 + L × t)
Use it for
Cash paid at the start of the FRA period, discounted from its end.
9
Formula
Swap net payment
Expression
N × (Fixed rate − Floating rate) × t
Use it for
Periodic net cash flow in an interest rate swap.
10
Formula
Futures on a T-bill (price)
Expression
Price = 100 − Annualised yield
Use it for
Short-term interest rate futures quote convention.
| # | Formula | Expression | Use it for |
|---|---|---|---|
| 1 | Bond price | P = Σ C ÷ (1 + y)ᵗ + F ÷ (1 + y)ⁿ | Price of the underlying bond at a given yield. |
| 2 | Modified duration | MD = Macaulay duration ÷ (1 + y) | Price sensitivity: ΔP% ≈ −MD × Δy. |
| 3 | PVBP (DV01) | PVBP = MD × Price × 0.0001 | Rupee change in price for a one basis point move in yield. |
| 4 | Invoice price (bond futures) | Futures settlement price × Conversion factor + Accrued interest | Amount paid on delivery of a bond against an IRF. |
| 5 | Cheapest to deliver | Choose the bond with the lowest Clean price − Futures price × CF | The bond a seller will deliver. |
| 6 | Hedge ratio (duration-based) | Contracts = (PVBP of portfolio ÷ PVBP of futures) × Hedge share | Number of IRF contracts to hedge a bond portfolio. |
| 7 | Implied forward rate | (1 + s₂)² = (1 + s₁) × (1 + f₁,₂) | Forward rate implied by spot rates; the basis of FRA pricing. |
| 8 | FRA settlement | N × (L − K) × t ÷ (1 + L × t) | Cash paid at the start of the FRA period, discounted from its end. |
| 9 | Swap net payment | N × (Fixed rate − Floating rate) × t | Periodic net cash flow in an interest rate swap. |
| 10 | Futures on a T-bill (price) | Price = 100 − Annualised yield | Short-term interest rate futures quote convention. |
What the Symbols Mean
- CF
- Conversion factor: the exchange's adjustment that makes each deliverable bond comparable.
- N, K, L
- Notional amount, the FRA contract rate and the reference rate on the fixing date.
- t
- Length of the FRA or swap period in years.
Free account, this exam preselected.
Worked Example: FRA Settlement
A ₹10 crore FRA has a contract rate of 6.5%. The reference rate fixes at 7% for a 91-day period.
- 1
Interest difference
10,00,00,000 × (0.07 − 0.065) × 91 ÷ 365 ≈ ₹1,24,658.
- 2
Discount to the settlement date
Divide by 1 + 0.07 × 91 ÷ 365 = 1.01745.
- 3
Settlement amount
≈ ₹1,22,519, paid by the FRA seller to the buyer, because rates rose above the contract rate.
Who Gains When Rates Rise?
The FRA buyer, who pays fixed, gains when rates rise. The buyer of a bond future gains when rates fall. Fix the direction first. The arithmetic only sets the size.
FAQs
Why is the FRA payment discounted?expand_more
FRAs settle at the start of the period, but the interest difference relates to its end, so it's discounted back at the reference rate.
What does a conversion factor do?expand_more
It adjusts the futures price for the coupon and maturity of the bond actually delivered, so different bonds can settle one contract.
Is PVBP the same as DV01?expand_more
Yes. Both are the price change for a one basis point change in yield.
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