NISM Certifications · NISM-Series-XXI-A: Portfolio Management Services (PMS) Distributors

Investing in Fixed Income Securities (NISM XXI-A): formula sheet

Full chapter guide

Key formulas

Annual coupon amount: Coupon amount = Coupon rate × Face value; Coupon is always applied to face value, not to market price. Check how often it is paid (annual or half-yearly).
Current yield: Current yield = Annual coupon ÷ Market price × 100; Uses market price. It differs from the coupon rate whenever price differs from face value.
Price versus face value: Premium: price > face value. Par: price = face value. Discount: price < face value.; If market yield is above the coupon rate, the bond usually trades at a discount. If below, at a premium.
Zero-coupon bond: Return comes only from (Face value − Issue price); No periodic interest is paid. Treasury Bills work this way.
Discount-priced instrument price: Price = Face value ÷ (1 + Yield × Days ÷ 365); Used for T-bills, CP and CDs when yield is quoted as an annualised yield. Use the day count given in the question.
Yield on a discount instrument: Yield = (Face value − Price) ÷ Price × 365 ÷ Days; Divide by the price paid, not by the face value.
Absolute return over holding period: Return = (Face value − Price) ÷ Price; Annualise by multiplying by 365 ÷ Days.
Repo interest: Interest = Cash lent × Repo rate × Days ÷ 365; Repurchase price = Sale price + Interest.
Bond price: P = C ÷ (1+y) + C ÷ (1+y)² + … + (C + F) ÷ (1+y)ⁿ; C = coupon per period, F = face value, y = yield per period, n = number of periods.
Annuity form of price: P = C × [1 − (1+y)⁻ⁿ] ÷ y + F ÷ (1+y)ⁿ; Useful when there are many coupons.
Zero coupon bond price: P = F ÷ (1+y)ⁿ; No coupons. Only one cash flow at maturity.
Current yield: Current yield = Annual coupon ÷ Market price × 100; Uses market price, not face value.
Coupon rate: Coupon rate = Annual coupon ÷ Face value × 100; Fixed at issue. Do not confuse with current yield.
Par, premium, discount rule: Yield = coupon rate → par; yield > coupon rate → discount; yield < coupon rate → premium; Price and yield move in opposite directions.
Approximate YTM: YTM ≈ [C + (F − P) ÷ n] ÷ [(F + P) ÷ 2]; A quick estimate only. The exact YTM needs trial and error or a calculator.
Semi-annual adjustment: Periods = years × 2; yield per period = annual yield ÷ 2; Apply when coupons are paid twice a year.
Modified duration: Modified Duration = Macaulay Duration ÷ (1 + y/n); y is the yield to maturity as a decimal; n is the number of coupon payments per year. For annual coupons, divide by (1 + y).
Price change using duration: ΔP/P ≈ −Modified Duration × Δy; The minus sign shows the inverse relationship. Use Δy as a decimal, or in % to get the answer in %.
Convexity adjustment: ΔP/P ≈ −MD × Δy + ½ × Convexity × (Δy)²; The convexity term is always positive for a plain bond, so it adds to the price change in both directions.
Duration of a zero-coupon bond: Macaulay Duration = Time to maturity; For a coupon-paying bond, Macaulay duration is less than maturity.
Direction rule: Yield ↑ → Price ↓; Yield ↓ → Price ↑; Holds for plain fixed-rate bonds.
Forward rate from spot rates: (1 + S₂)² = (1 + S₁) × (1 + f₁,₂); S₁ and S₂ are 1-year and 2-year spot rates. f₁,₂ is the 1-year rate starting one year from now. Use annual compounding unless told otherwise.
General forward rate: (1 + Sₙ)ⁿ = (1 + Sₘ)ᵐ × (1 + f)ⁿ⁻ᵐ; f is the annual forward rate from year m to year n.
Curve shapes: Normal: long yield > short yield | Inverted: long < short | Flat: equal; A humped curve peaks at medium maturities.
Spread (slope): Slope = Long-term yield − Short-term yield; Positive for normal, negative for inverted, near zero for flat.
Credit spread: Credit spread = Yield on the corporate bond − Yield on a risk-free government security of similar maturity; Compare bonds of similar maturity. A wider spread means the market sees higher credit risk.
Spread direction rule: Downgrade or weaker outlook → spread widens → price falls; Upgrade does the reverse: spread narrows and price rises.
Rating bands: AAA to BBB = investment grade; BB and below = speculative grade; D = default; Higher letters mean lower default risk and usually lower yield.
Expected loss idea: Expected loss = Probability of default × Loss given default; Loss given default is the share of the investment lost if default happens. Recovery reduces it.
Immunisation condition: Portfolio duration = investment horizon (time to liability); Price risk and reinvestment risk offset. Needs periodic rebalancing, and works best for small parallel yield shifts.
Ladder allocation: Equal amount in each maturity bucket; reinvest maturing proceeds at the long end; Gives regular cash flow and averages reinvestment rates.
Barbell vs bullet: Barbell: short + long maturities. Bullet: one maturity cluster; For similar duration, a barbell generally has higher convexity than a bullet.
Riding the yield curve: Buy maturity longer than horizon; sell at horizon; gain from roll-down on an upward-sloping curve; Active strategy. Loses if yields rise more than the roll-down gain.
Passive vs active: Passive: no rate view (buy and hold, ladder, immunisation, indexing). Active: rate or curve view; Classify first when a question asks which type a strategy is.

Quick revision

Bond price and yield move in opposite directions.
A bond's price is the present value of its coupons and principal, discounted at the required yield.
A bond with a coupon equal to its yield trades at par; a lower coupon means a discount, a higher coupon a premium.
Longer maturity and lower coupon generally mean higher duration and higher interest rate risk.
Duration estimates the percentage price change for a small change in yield.
Convexity captures the curvature that duration misses, and it matters more for large yield moves.
Zero coupon bonds have no reinvestment risk on coupons but carry high price sensitivity.
Money market instruments are short-term debt, with maturity up to one year.
An upward sloping yield curve means longer maturity bonds yield more than shorter ones.
A lower credit rating means higher credit risk and a higher yield demanded.
Credit rating is an opinion on the ability to repay, not a guarantee.
Passive strategies such as buy and hold differ from active strategies that take views on rates or credit.

Common mistakes

Applying the coupon rate to market price instead of face value. Fix: Coupon amount = coupon rate × face value. Market price is only used for current yield.
Saying Treasury Bills pay a coupon. Fix: T-Bills are issued at a discount and redeemed at face value. They pay no coupon.
Saying commercial paper is issued by banks. Fix: Link CP to companies and CD to banks. Write this pair in your notes.
Thinking T-bills pay periodic interest. Fix: T-bills are zero-coupon discount instruments. Return comes only from redemption at face value.
Dividing the coupon by face value and calling it current yield. Fix: Current yield always uses the market price in the denominator.
Using the annual yield with semi-annual coupons. Fix: Halve the yield and double the number of periods, and use half the annual coupon.
Using Macaulay duration directly as the percentage price change. Fix: Use modified duration for price change. Divide Macaulay duration by (1 + y/n) first.
Forgetting the negative sign. Fix: Write the sign before calculating. Yield up means price down.
Averaging spot rates to get the forward rate. Fix: Always use the compounding equation. The forward rate is the rate that links the two spot rates, not their average.
Calling an inverted curve one where yields rise with maturity. Fix: Inverted means short-term yields are higher than long-term yields, so the curve slopes down.

Exam tips

Expect direct questions matching an instrument to its issuer. Learn the issuer list cold.
Read numerical options carefully. Wrong options often use market price in place of face value.
Remember that discount instruments carry no coupon. This is a frequent trap.
For G-Sec versus corporate bond questions, answer with credit risk and yield first.
Wrong answers cost marks in this exam, so skip a question only if you cannot eliminate any option.
Always tie the instrument to its issuer first. Most trap options swap CP and CD.
Remember that money market means original maturity up to one year. Reject options with longer maturity.
For price questions, check that your answer is below face value. This catches sign errors quickly.