A riskless zero-coupon bond at 100 has a six-month forward of 102.531512, a premium. Give the same bond an 8% coupon and the forward drops to 98.511194, a discount, because the sign of the premium is the sign of the rate minus the coupon and nothing else. Quoting the forward at spot when the coupon is rich hands the other side a riskless 1.518802 per hundred, and the discrete-coupon version shows the answer also turns on whether a payment date falls before delivery.
A top-rated issuer picks the coupon that prices its ten-year bond at exactly 100 off its own flat 5 percent curve, and the same cash flows discounted off a swap curve 25 basis points lower come to 101.954087. Two facts do the work: a present value is strictly decreasing in every rate it is discounted at, and for a top-rated name the swap curve sits below its own bond curve because a swap risks no principal and is margined daily. A modified duration of 7.7217 times the spread accounts for 1.9304 of the lift, and a convexity of 74.9977 supplies the last two cents.
Sketch a one-year call struck at 100 with a five percent rate. Deep in the money the curve straightens into a line of slope one, and that line crosses at 95.122942 rather than at 100, so drawing it through the strike is out by 4.877058 for ever. That gap is the interest saved on the strike, and it is also why an American call on a share paying no dividends is never exercised early. Plot the same option against the futures price and the crossing returns to 100 while the slope drops to 0.951229.
A bond paying 100 in ten years costs 67.5564 at a four percent yield. The first two points of yield cost 11.7169 and the next two only 9.5201, so the curve bends. The usual explanation blames duration falling as yields rise, and this bond refutes it: with a single cash flow its Macaulay duration is exactly ten at every yield. The slope is minus duration times price over one plus the yield, and the general statement needs no duration at all, only that every discount factor is convex.
Heads pays $7 in eighteen months, tails costs $2 today, and the curve gives 12% for one year and 18% for two. Averaging the amounts gives $2.50, which is 38.68% too high, because expectation and discounting only commute when every cash flow lands on the same date. The answer is about $1.80, and four defensible compounding conventions spread it from 1.7862 to 1.8381, so one decimal is honest and two are not.
Because the horizons are ten and five, both sides of the no-arbitrage equation are fifth powers and the root disappears, leaving 1 + f = 1.15 squared over 1.10 = 529/440, so f = 89/440 exactly. Reflecting 10 percent around 15 to get 20 is low by exactly (b - a) squared over (1 + a), a square over a positive number, which is why the reflection can never overshoot for any pair of rates. Under continuous compounding the same problem is linear and 20 percent is exactly right, so the instinct is correct machinery pointed at the wrong convention.
Multiplying by the conjugate turns the difference into x over the sum of the same two terms, exactly, with no series and no error term, and dividing through by x reads off the limit one half. Completing the square gives a constant excess of a quarter, so the gap climbs toward a half strictly from below and never reaches it.