Term structure — spot, forward, and par rates — CFA L2 FI

Term structure is the relationship between yield and maturity. CFA L2 tests bootstrapping spot rates, computing forward rates, and pricing bonds in arbitrage-free framework.

Foundation

**Spot rate (z_t)**: yield on a zero-coupon bond maturing at time t. PV factor = 1/(1+z_t)^t. **Forward rate (f_{m,n})**: rate agreed today for borrowing from time m to time n. (1+z_n)^n = (1+z_m)^m × (1+f_{m,n-m})^{n-m} For 1-period forwards from spots: (1+z_2)² = (1+z_1)(1+f_{1,1}) → f_{1,1} = (1+z_2)²/(1+z_1) – 1. **Par rate**: coupon rate that makes a coupon bond trade at par. Equivalent to yield-to-maturity for that bond. Bootstrapping: derive spot curve from observed coupon-bond prices iteratively.

Deep Dive

Arbitrage-free valuation: a coupon bond is a portfolio of zeros. Each cash flow discounted at corresponding spot rate. P = Σ CF_t / (1+z_t)^t If market price ≠ this, arbitrage exists (strip and reconstitute). Forward rates and expectations: - Pure expectations: f_{m,n} = E(z_{m,n}). Forward = expected future spot. - Liquidity preference: f_{m,n} > E(z_{m,n}) by liquidity premium. - Preferred habitat: shape driven by maturity-segmented demand/supply. Indian context: G-sec curve typically upward-sloping. Premium of long-dated over short = 100-200 bps. Forward-rate analysis informs RBI watch.

Advanced

L2 vignette traps: - Annualised vs periodic rates — be careful when squaring/multiplying. - Semi-annual compounding (US convention) vs annual (academic). - Continuously compounded forwards: f = ln(P_m/P_n) etc. Using forward curve to price a bond: each future cash flow discounted by chain of one-period forwards. Arbitrage-free. Negative forward rates: possible when spot curve is steeply inverted. Reflects expected easing.

Regulatory references

CFA Institute FI curriculum
RBI G-Sec Trading Manual

Common mistakes & pitfalls

Mixing simple and compound rates.
Using YTM as if it were spot rate (works for zero-coupon only).
Ignoring day-count conventions in forward calculations.

Frequently asked

What is the difference between YTM and spot rate?

YTM is single discount rate that prices bond. Spot rate is YTM of zero-coupon bond at that maturity. Bond price = sum of discounted CFs at spot rates.

When does forward rate equal expected spot?

Under pure expectations theory. Empirically, forwards include liquidity premium → biased upward.

Practice questions

Click each question to reveal the answer and explanation.

Q 1

Bootstrapping derives:

(a)Forward curve from coupon bonds
(b)Spot curve from coupon bonds
(c)YTM from spot
(d)Par from forward

Correct: (b) Spot curve from coupon bonds

Bootstrapping: extract zero-coupon (spot) rates from coupon-bond prices iteratively.

Q 2

If z_1 = 5%, z_2 = 6%, then f_{1,1} ≈

(a)5.5%
(b)6.0%
(c)7.0%
(d)5.0%

Correct: (c) 7.0%

f_{1,1} = (1.06²/1.05)-1 = 7.0%.

Q 3

Arbitrage-free coupon bond price:

(a)Single discount at YTM
(b)Sum of CFs at corresponding spot rates
(c)Always equal to par
(d)CF / r

Correct: (b) Sum of CFs at corresponding spot rates

Each CF discounted at its own spot rate. Otherwise arbitrage by stripping.

Q 4

Pure expectations theory states:

(a)Forward = expected future spot
(b)Forward > spot always
(c)Forward = par
(d)No relation

Correct: (a) Forward = expected future spot

Pure expectations: forwards are unbiased predictors of future spots.

Q 5

Negative forward rates suggest:

(a)Arbitrage
(b)Steeply inverted curve / expected easing
(c)Bond default
(d)Tax effect

Correct: (b) Steeply inverted curve / expected easing

Inverted curve → negative implied forwards → market expects falling rates.

Educational purposes only. The numbers, returns, and examples used in this lesson are illustrative. Past performance does not guarantee future results. Mutual fund and securities investments are subject to market risks. This lesson is not investment advice; for advice tailored to your circumstances, consult a SEBI-registered Investment Adviser. Read our full disclaimer.