Term structure and spot curves

If yields are the weather report, the term structure is the climate map — it tells you how the market expects interest rates to behave across time, not just right now.

You already know how to price bond cash flows and the yield measures that come with them, and you remember day count conventions like old friends who sometimes ghost you on weekends. Now we level up. This piece builds on those ideas and on the statistical thinking from risk, return, and probability: how to extract a full timeline of discount rates from market prices, why that timeline must obey no-arbitrage logic, and how forward rates sneak out of the cracks.

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What is the term structure and what is a spot curve

• Term structure of interest rates is the relationship between interest rates and their maturities. It is the market's schedule of rates for lending from today to various future dates.
• A spot curve (a.k.a. zero curve) is the term structure expressed as spot rates: each point is the yield on a zero-coupon instrument that delivers one unit of currency at a single future date.

Think of the spot curve as the price list you get when you ask: how much does the market demand for me to wait until year t to get one dollar? Unlike yield to maturity, which smashes all cash flows of a coupon bond into one single number, the spot curve gives you a separate discount rate for each cash flow date. That is why we bootstrap it from coupon bond prices.

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Why the spot curve matters (practical reasons)

• Accurate pricing of any bond or structured product: present value = sum of cash flows discounted by spot rates for their specific dates.
• Implied forward rates: expectations and hedging of future short rates come from spot geometry.
• No-arbitrage valuation: mispriced spot curves create arbitrage opportunities across maturities.

If you only use YTM or crude average yields, you are pricing with a hammer and thinking every problem is a nail. The spot curve is the surgical tool.

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How to get a spot curve: bootstrapping in plain English

Bootstrapping is the stepwise recovery of spot rates from market prices of coupon-bearing bonds. Imagine you have a set of market-traded bonds with different maturities. Use the shortest maturity first since it gives a direct spot. Then use that result to solve for the next, and so on.

Example (numbers you can actually calculate at a party)

Given per 1 unit of face value:

• 1-year zero price P1 = 0.96. That gives spot rate s1 where P1 = 1/(1 + s1)

  s1 = 1 / 0.96 - 1 = 0.0416667 -> 4.1667%
  

• 2-year annual coupon bond, coupon 5% (so cash flows 0.05 at year 1 and 1.05 at year 2), market price = 0.99.

  Price equation:

  0.99 = 0.05/(1 + s1) + 1.05/(1 + s2)^2
  

  Plug s1 = 0.0416667, solve for s2:

  0.05/1.0416667 = 0.0480
  Remaining = 0.99 - 0.0480 = 0.9420
  1.05/(1 + s2)^2 = 0.9420 -> (1 + s2)^2 = 1.05 / 0.9420 = 1.1150
  s2 = sqrt(1.1150) - 1 = 0.055 -> 5.5%
  

• 3-year annual coupon 6%, price 1.01. Repeat solving for s3 using s1 and s2.

  The idea is the same: use known discount factors to peel off early coupon PVs, leaving a single unknown for the final payment. Step forward until you have a discount factor (or spot rate) for each maturity.

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Mathematical relationships you should memorize (but not neuroses over)

• Discount factor to time t:

  P(0,t) = 1 / (1 + s_t)^t
  

  where s_t is the t-period spot rate with the chosen compounding convention.

• Present value of a coupon bond using spot rates:

  Price = sum_{i=1}^T CF_i * P(0,i)
  

• Forward rates from spot rates (discrete compounding):

  (1 + s_n)^n = (1 + s_m)^m * (1 + f_{m,n-m})^{n-m}
  

  which for adjacent periods simplifies to:

  f_{t-1,1} = (1 + s_t)^t / (1 + s_{t-1})^{t-1} - 1
  

Forward rates are the market-implied rates for lending over a future short interval, implied by today's spot curve. They are not perfect predictions of future short rates, but they are the equilibrium arbitrage-free prices.

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Par curves, yield curves, and how they differ from spot curves

Curve type	What it quotes	Use case
Spot (zero) curve	Zero-coupon rates for each maturity	PV of any cash flow schedule, bootstrapping
Par curve	Coupon rates that make bond price = par	Quoting market coupon yields, useful for swaps
Yield curve (YTM)	Single yield that equates price to cash flows	Quick comparatives, but masks cash flow timing

Spot is the most fundamental; par and YTM are convenient aggregations.

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Common mistakes and traps (and how to avoid them)

• Mixing compounding conventions. Annual, semiannual, and continuous compounding change numbers dramatically. Always match the market convention to your formulas.
• Forgetting day count and settlement conventions. Your earlier work on day counts matters here; misapplied day counts give wrong discount factors.
• Using YTM to discount each coupon. That double-counts risk timing and will misprice the bond unless the curve is flat.
• Blind interpolation. If market maturities are sparse, interpolating spot rates is necessary, but choose a method that preserves no-arbitrage (e.g., interpolate discount factors or use spline methods built for curves).

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Quick intuition: why no-arbitrage forces the spot curve to behave

If the implied discount factors from bonds do not multiply consistently, arbitrageurs can create zero-cost portfolios that pay a guaranteed profit. Bootstrapping enforces that the combination of short-dated instruments price longer-dated ones consistently. This is the same logic you use in risk and return when ensuring expected payoffs are matched; here it's deterministic and sharper.

If prices don’t line up, there is a pizza-eating-arbitrage sitting on the table waiting for someone with the guts to take it. Markets generally close that pizza down quickly.

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Takeaways and next moves

• The spot curve is the backbone of fixed income valuation. It gives the discount rate for each cash flow date and is derived from coupon bond prices via bootstrapping.
• From the spot curve you can get forward rates, value swaps, price tricky structured products, and test no-arbitrage conditions.
• Practical work: build a bootstrapping spreadsheet, mind compounding and day counts, and practice with market data to see how par, spot, and forward curves relate dynamically.

If you want, I can provide a downloadable bootstrapping spreadsheet example, or walk through constructing a curve from semiannual coupon bonds with day-count conventions included. Warning: once you start seeing curves, you may never look at a single YTM the same way again.