Bond valuation and the yield curve

A bond's price is just the present value of the cash it will pay, and once you can compute that you can see why a rate rise hurts long bonds more than short ones, why a Debt Long-Short strategy might short the 10-year and buy the 3-year, and what a "steepening" trade means. The exam asks for the price given a yield, and for the yield curve shapes and theories.

Price as present value

Discount every coupon and the final principal at the required yield and add them up. The yield is the market's required return for that issuer and maturity; the coupon is fixed when the bond is issued. When they differ, the price adjusts.

Three rules fall out of the table. When yield is above the coupon the bond trades at a discount; below the coupon, at a premium; equal to the coupon, at par. The relation between price and yield is inverse and convex: a 1% fall in yield (8% to 7%) raised the price by ₹2.58, while a further 1% fall (7% to 6%) raised it by ₹2.67. Prices rise faster than they fall for the same yield change, which Chapter 11 measures as convexity.

Indian G-Secs pay coupons every six months, so in practice you halve the coupon and the yield and double the number of periods: a 7.26% 10-year G-Sec at 7% yield is valued as 20 payments of ₹3.63 discounted at 3.5% per period plus ₹100 at the end. The exam usually keeps to annual coupons for arithmetic, but may ask which adjustment semi-annual compounding needs.

Finding the yield from the price

YTM cannot be solved directly; it is found by trial or a financial calculator. If the price is below par, try a yield above the coupon; above par, try below. A quick approximation for the exam:

For the 3-year 7% bond at ₹97.42: [7 + (100 − 97.42) ÷ 3] ÷ [(100 + 97.42) ÷ 2] = 7.86 ÷ 98.71 = 7.96%, close to the true 8%. The approximation is good enough to pick the right option in a multiple-choice question.

Clean price, dirty price and accrued interest

Bonds are quoted at the clean price, but the buyer pays the dirty price: clean price plus the interest accrued since the last coupon, which belongs to the seller. For Indian G-Secs the day count is 30/360 (each month counted as 30 days, year as 360); for T-bills and money market instruments it is actual/365.

A 7.26% G-Sec, semi-annual coupon of ₹3.63, 60 days after the last coupon date under 30/360: accrued = 7.26 × 60 ÷ 360 = ₹1.21 per ₹100. If the clean price is ₹99.50, the settlement price is ₹100.71. On the coupon date accrued interest drops to zero and the dirty price falls by the coupon, which is not a loss to anyone; it simply means the coupon has been paid.

The yield curve

Plot the YTM of government securities against maturity and you get the yield curve (or term structure of interest rates). Four shapes matter:

• Normal (upward sloping): longer maturities yield more, the usual state, reflecting expected growth and the premium for tying up money longer.
• Inverted: short rates above long rates, often when the central bank has tightened hard and the market expects cuts; historically a recession signal.
• Flat: little difference across maturities, typical at turning points.
• Humped: medium maturities yield most.

Three theories explain the shape. The expectations theory says long rates are the average of expected future short rates, so an upward slope means rates are expected to rise. The liquidity preference theory adds a premium for longer maturities because investors dislike price risk, so the curve slopes up even if rates are expected to stay flat. The market segmentation theory says each maturity bucket has its own buyers (banks at the short end, insurers and pension funds at the long end) and supply and demand in each sets its yield.

From the curve you can extract spot rates (the yield on a zero coupon bond of each maturity) and forward rates (the rate implied for a future period). If the 1-year spot is 6.5% and the 2-year spot is 7.0%, the implied 1-year rate one year from now is (1.07² ÷ 1.065) − 1 = 7.50%. Forward rates are the fair prices for interest rate futures and FRAs, which is why the yield curve is the pricing engine for Module 3.

What curve moves mean for a portfolio

A parallel shift moves every maturity by the same amount; duration (next chapter) captures it. A steepening raises long yields more than short ones; a flattening does the reverse. A Debt Long-Short strategy that expects steepening would buy short-dated bonds and short long-dated futures; a mutual fund cannot take the short leg. A credit spread is the gap between a corporate yield and the G-Sec of the same maturity; it widens in stress and narrows in calm, independent of the risk-free curve.