Pricing Conventions, Discounting, and Interest Rates

A trader sees a 5-year Treasury bond quoted at 98.50 and a 5-year strip of zero-coupon Treasuries that, if assembled into the same cash-flow pattern, would cost 99.10. The same dollars on the same dates trade at two different prices. Either there's a free 0.60 of arbitrage, or one of the quoted prices is wrong. The whole structure of fixed-income pricing is built to make sure the answer is always "the prices are wrong."

Every fixed-income valuation reduces to a single primitive: the discount factor. $d(t)$ is the price today of a riskless dollar to be received at time t. A coupon bond paying $c$ at times $t_1, t_2, \ldots, t_n$ and principal $F$ at $t_n$ has price:

$$P = \sum_{i=1}^{n} c \cdot d(t_i) + F \cdot d(t_n)$$

Compounding convention sets the relationship between d(t) and the underlying rate:

$$d(t) = \frac{1}{(1+r)^t} \quad \text{(annual)}, \quad d(t) = \frac{1}{(1+r/2)^{2t}} \quad \text{(semiannual)}, \quad d(t) = e^{-rt} \quad \text{(continuous)}$$

The same dollar amount has the same discount factor across conventions; the rate that produces it is what differs. Continuous compounding is convenient analytically; semiannual is the U.S.

Common mistakes

• Using clean price for cash settlement. The buyer pays the dirty price. Clean is the quoted convention; dirty includes accrued. Trap: a question gives clean price and asks for the cash payment 90 days into a coupon period; the right answer adds accrued. Choice C reports clean and catches the candidate who skipped the convention.
• Bootstrapping with wrong cash flow timing. A 2-year bond has cash flows at 0.5, 1, 1.5, and 2 years (semiannual coupons). Treating it as having only annual cash flows omits the half-year and one-and-a-half-year payment dates.
• Forward rate sign confusion. The forward rate $f(t_1, t_2)$ covers borrowing FROM t1 TO t2. $f(1, 2)$ is the 1-year-out 1-year forward; $f(2, 1)$ is meaningless (negative duration). Market shorthand separates start and tenor: 1y2y starts in one year and lasts two years, so it is $f(1,3)$. Do not confuse it with the one-year rate $f(1,2)$.

Bottom line

• Discount factor $d(t)$ is the present value today of $1 received at time t; bond price equals the sum of cash flows times their discount factors.
• Law of one price: identical cash flows must carry identical prices, and any violation creates a riskless arbitrage opportunity.
• Spot rate $s(t)$ is a zero-coupon yield, forward rate $f(t_1,t_2)$ is today's rate for borrowing between t1 and t2, and par rate is the coupon making a bond trade at par; all three represent the same term structure.
• Forward rates are pinned by no-arbitrage: investing to t2 directly must equal investing to t1 and rolling forward over t1 to t2.

Exam shortcut

When a question gives par yields and asks for spot rates, you must bootstrap. Par yields are NOT spot rates. The shortest-maturity bond is already a zero (its yield is its spot); each subsequent step solves for one new unknown. When a question gives spot rates and asks for forwards, use the no-arbitrage identity directly.

The full lesson (about 5,044 words, 34 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

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