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.
A 20-year 7 percent bond on a flat 10 percent curve prices at 744.5931, and a one-point rise costs exactly 63.1262. The tangent alone says 67.7028, and adding the second-order term of 4.8416 lands at 62.8612, inside 27 cents of the truth. Note that the correction and the error it corrects are two different numbers, which is why the estimate ends up on the wrong side of the answer.
An eight per cent thirty-year bond at par loses 27.49 dollars when its yield rises 25 basis points, and its yield moves because the principal is collateralised in United States Treasuries. The answer that circulates, about thirty-five dollars, needs a duration of fifteen, and a par bond at an eight per cent yield cannot have one: its modified duration is its own annuity factor, capped at 12.5 at any maturity whatsoever. The pass-through, the only soft number in the chain, is swept from an eighth to a half.
A share at 150, a call struck at 100, a year to run and no dividend: cashing out pays 50 while the option is worth 54.97. The floor S minus X e to the minus rT assumes no distribution at all, and it beats immediate exercise by exactly one year of interest on the strike, 4.8771. The article carries the cusp where the gap peaks at 10.4506, the shelf it settles onto far in the money, and the dividend condition that makes early exercise optimal after all.
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.