Investing in Fixed Income Securities: formula sheet

Key formulas

Annual coupon

Annual coupon (₹) = Coupon rate × Face value

Always on face value, never on the market price or purchase price.

Periodic coupon

Coupon per period = Annual coupon ÷ Number of payments per year

For half-yearly payment, divide by 2.

Current yield

Current yield = Annual coupon ÷ Current market price

A simple measure. It ignores gain or loss at maturity.

Price-rate relationship

Market interest rates ↑ ⇒ Bond prices ↓ (and the reverse)

Holds for plain fixed coupon bonds.

Price vs face value

Coupon rate > market yield ⇒ price > face value (premium); coupon rate < market yield ⇒ price < face value (discount)

If equal, the bond trades at par.

Yield on price (annualised) for discount instruments

Yield = (Face value − Price) ÷ Price × (365 ÷ Days to maturity)

Used for the yield on price of T-bills, CP and CDs. Price is below face value. Return comes only from the discount. A discount rate is computed on face value instead: (Face value − Price) ÷ Face value × (365 ÷ Days).

Zero coupon bond price

Price = Face value ÷ (1 + r)^n

r is the annual yield and n is years. There is no coupon in between.

T-bill tenors

91 days, 182 days, 364 days

Issued at a discount, no coupon, redeemed at face value.

Money market limit

Original maturity ≤ 1 year

Longer than one year is a capital market (bond) instrument.

Bond price (annual coupons)

P = C ÷ (1+y) + C ÷ (1+y)² + … + (C + F) ÷ (1+y)ⁿ

C = annual coupon, F = face value, y = required yield, n = years to maturity.

Bond price (semi-annual coupons)

P = Σ [ (C÷2) ÷ (1 + y÷2)ᵗ ] + F ÷ (1 + y÷2)²ⁿ

Halve the coupon and yield, double the number of periods.

Coupon

C = Coupon rate × Face value

Coupon is computed on face value, not on market price.

Current yield

Current yield = Annual coupon ÷ Current market price × 100

Ignores capital gain or loss and time value of money.

YTM approximation

YTM ≈ [C + (F − P) ÷ n] ÷ [(F + P) ÷ 2]

Only an estimate. The exact YTM comes from trial and error or a calculator.

Price–yield and par/premium/discount rules

Coupon rate = YTM → Price = Face value; Coupon rate > YTM → Price > Face value; Coupon rate < YTM → Price < Face value

Relationship between price and yield is inverse.

Yield to call

Price = Σ coupons discounted + Call price ÷ (1+y)ᵗ

t = years to the call date. Solve for y.

Macaulay duration

D = Σ [t × PV(CFt)] ÷ Price

t is the time of each cash flow in years. PV(CFt) is the present value of that cash flow at the bond's yield. Price is the sum of all PVs.

Modified duration

Modified duration = Macaulay duration ÷ (1 + y ÷ n)

y is the annual yield and n is the number of coupon payments per year. For annual coupons, divide by (1 + y).

Approximate price change using duration

% change in price ≈ − Modified duration × Δy

Δy is the change in yield in percentage points. The sign is negative because price and yield move in opposite directions.

Price change with convexity adjustment

% change in price ≈ − Modified duration × Δy + ½ × Convexity × (Δy)²

Use Δy as a decimal here (0.01 for 1%). The convexity term is always positive for a plain bond, so it adds to the estimate.

Price value of a basis point (PVBP)

PVBP ≈ Modified duration × Price × 0.0001

Approximate rupee change in price for a 1 basis point (0.01%) change in yield.

Duration rules of thumb

Zero-coupon bond: D = maturity. Coupon bond: D < maturity.

Duration rises with longer maturity. It falls with a higher coupon and with a higher yield, other things equal.

Bond price using spot rates

P = C₁ ÷ (1 + s₁)¹ + C₂ ÷ (1 + s₂)² + … + (Cₙ + F) ÷ (1 + sₙ)ⁿ

Each cash flow is discounted at the spot rate for its own year. s is the spot rate, C the coupon, F the face value.

Forward rate from spot rates (annual compounding)

(1 + s₂)² = (1 + s₁) × (1 + f)

f is the one-year rate one year from now. Rearranged: f = (1 + s₂)² ÷ (1 + s₁) − 1.

Pure expectations idea

Long-term rate ≈ average of expected future short-term rates

Under this theory forward rates are unbiased predictors of future spot rates.

Liquidity preference idea

Long-term yield = expected average short rate + liquidity premium

The premium is positive and tends to rise with maturity.

Curve shape summary

Normal: long > short; Inverted: long < short; Flat: long ≈ short

Compare yields at the short and long ends of the curve.

Credit spread

Credit spread = Yield on corporate bond − Yield on government security of similar maturity

Wider spread means the market sees higher credit risk. Compare bonds of similar maturity only.

Investment grade boundary

Investment grade: BBB− and above. Speculative grade: below BBB−

Rating scale from top: AAA, AA, A, BBB, BB, B, C, D. D means default.

Rating change effect

Downgrade → required yield ↑ → price ↓. Upgrade → required yield ↓ → price ↑

Price and yield move in opposite directions.

Risk direction rules

Reinvestment risk: falling rates hurt. Call risk: falling rates raise it. Liquidity risk: thin trading raises it

Match each risk to its trigger to answer scenario questions.

Equal ladder amount per rung

Amount per rung = Total investment ÷ Number of rungs

A ladder normally splits money equally across maturities.

Ladder roll-over

When the shortest rung matures, reinvest in the longest maturity of the ladder

This keeps the ladder structure unchanged.

Barbell vs bullet

Barbell = short + long maturities, no middle. Bullet = maturities clustered at one point

Compare them at the same duration. The barbell has higher convexity.

Interest income tax

Tax = Interest × Slab rate

Interest on bonds is taxed at the investor's slab rate. Cess is added on top.

Listed bond capital gains (current rules)

Held more than 12 months: long-term, 12.5% with no indexation. Held 12 months or less: short-term, slab rate

The ₹1,25,000 exemption applies to listed equity gains, not to bonds.

Unlisted bonds and debentures (current rules)

Gain is taxed as short-term at slab rate, whatever the holding period

Check the workbook for the date from which this applies.

Quick revision

• Bond price is the present value of future coupons and principal, discounted at the required yield.
• Yield up means price down; yield down means price up.
• A bond trades at a discount when the coupon is below the required yield, and at a premium when it is above.
• Current yield = annual coupon ÷ current market price.
• Yield to maturity is the single discount rate that equates the present value of cash flows to the price.
• Longer maturity and lower coupon generally mean higher interest rate risk.
• Duration measures price sensitivity to yield changes; convexity corrects the duration estimate for larger moves.
• Zero coupon bonds have the highest duration for a given maturity, equal to their maturity.
• A normal yield curve slopes upward; an inverted curve slopes downward.
• Government securities carry negligible default risk but still carry interest rate risk.
• Reinvestment risk is higher for high-coupon bonds when rates fall.
• Credit ratings reflect default risk; a lower rating usually means a higher yield.

Common mistakes

• Calculating coupon on the market price instead of face value. Fix: Coupon = coupon rate × face value. Price affects yield, not the coupon in rupees.
• Calling all bonds risk free. Fix: Only sovereign securities are treated as free of default risk. All bonds carry interest rate risk, and non-government bonds carry credit risk.
• Saying T-bills pay a coupon. Fix: T-bills are discount instruments. You buy below face value and get face value at maturity.
• Treating SDLs as having the same credit backing as G-secs in every sense. Fix: SDLs are issued by state governments, not the Centre, and are not backed by the Central Government. They usually yield slightly more than G-secs of the same tenor.
• Calculating coupon on market price instead of face value. Fix: Coupon = coupon rate × face value always. Use market price only as the denominator in current yield.
• Treating current yield as YTM. Fix: Current yield ignores redemption gain or loss and time value. They are equal only when the bond sells at par.
• Using Macaulay duration directly to estimate price change. Fix: Price sensitivity uses modified duration. Divide Macaulay by (1 + y ÷ n) first.
• Getting the sign wrong and saying the price rises when yields rise. Fix: Write the minus sign in the formula every time. Yield up means price down.
• Calling a curve inverted when long yields are higher. Fix: Inverted means short-term yields exceed long-term yields. The curve slopes down.
• Discounting all cash flows at the bond's single yield when asked to use spot rates. Fix: When spot rates are given, discount each cash flow at the spot rate for its own maturity.

Exam tips

• Expect direct questions on definitions: face value, coupon, maturity and what a zero coupon bond is.
• In calculation questions, check the payment frequency before choosing an option.
• Watch for trap options that calculate coupon on market price.
• Questions on portfolio role usually test income, lower volatility and diversification, along with the caution that bonds carry interest rate and credit risk.
• Read the issuer carefully. Government issuer points to no default risk. Corporate issuer points to credit risk.
• Remember tenors: T-bills are 91, 182 and 364 days. Nothing above one year is a money market instrument.
• Link each issuer to its instrument: Government to G-sec and T-bill, state to SDL, bank to CD, company to CP.
• For yield questions, check the sign. A discount instrument price must be below face value.