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Bonds and yields

Yields up, prices down — and by how much. Price a bond from its coupon and yield, read premium or discount, and see which bond falls more.

  • Money
  • Untimed
  • Endless levels

How it plays

A bond is a list of payments: a coupon every year and the face value at the end. Its price is what those payments are worth today at the market's yield. A coupon above the yield sells above face; below, it sells at a discount.

Levels go from the direction alone, to current yield (coupon ÷ price), to pricing a coupon bond, to zero-coupon bonds, to which of two bonds loses more when yields rise a point — the longer one, and the one whose money comes later.

Every prompt states the convention: yearly coupons at year end, yearly compounding. Prices are asked to the nearest hundredth and a typed answer within 0.50 counts. The wrong options are the real slips: the face value forgotten, coupons left undiscounted, simple discounting where it compounds.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This makes reading and pricing simple bonds from their coupon and yield quicker and surer. The bonds are illustrative; it is not investment advice and claims nothing wider.

Learn

How to get better at bonds and yields: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Compare the coupon with the yield first. Coupon above yield: the bond trades above face. Equal: at face. Below: under face. That one comparison answers the first kind of question and bounds every price you work out.
  2. List the payments: the coupon at the end of each year, and the face value with the last coupon. Every bond here pays yearly, and every yield compounds yearly.
  3. Value each payment today by dividing by (1 + yield) once for each year it waits, and add them up. A zero-coupon bond has only the one payment.
  4. Before you choose, check the price sits on the right side of face, and not further from face than the coupon gap times the years.

Techniques and tricks

  • Coupon against yield

    If the coupon pays more than the market asks, buyers pay extra for the bond; if less, it sells cheap enough to make up the difference.

    e.g. An 8% coupon at a 6% yield trades at a premium; a 5% coupon at 7% at a discount.

  • Current yield is coupon over price

    Divide the coupon in money by the price paid, not by the face value. Below face it comes out above the coupon rate; above face, below it.

    e.g. ₹80 coupon on a bond at ₹1,060: 80 ÷ 1,060 = 7.55%.

  • Face plus the gap, discounted

    Price = face + (coupon − yield × face) × the annuity factor, where the annuity factor is (1 − 1/(1 + y)ⁿ) ÷ y. You only discount the gap, not every payment.

    e.g. 5 years, 8% coupon, 6% yield, face 1,000: gap 80 − 60 = 20, factor 4.2124, price 1,000 + 84.25 = 1,084.25.

  • The price lies between face and the straight-line guess

    Face ± years × the coupon gap overstates the premium or discount, because later gaps are discounted. The true price is always between that guess and face.

    e.g. For the bond above, the guess is 1,000 + 5 × 20 = 1,100; the price is 1,084.25.

  • Rule of 72 for zero-coupon bonds

    At y%, money doubles in about 72 ÷ y years, so a zero repaying 1,000 after that long is worth about 500 today; after twice as long, about 250.

    e.g. 9 years at 8%: 1,000 ÷ 1.08⁹ = 500.25.

  • Powers by squaring

    Build (1 + y)ⁿ by squaring instead of multiplying n times: square for 2 years, square again for 4, again for 8.

    e.g. 1.05² = 1.1025, 1.05⁴ = 1.2155, 1.05⁸ = 1.4775.

  • Later money moves more

    When yields rise, a payment far in the future loses more of its value than a near one. So the longer bond falls more, and of two bonds the same length, the zero-coupon one falls more.

As the levels climb

  • The kinds unlock one at a time: premium, par or discount at level 0; current yield at level 1; price from yield at level 3; zero-coupon bonds at level 5; which falls more at level 7.
  • Between unlocks, and for ever after, the bonds get longer: up to 10 years for a priced coupon bond and up to 30 for a zero. Longer bonds mean more payments to discount, so the face-plus-gap method saves more time as they grow.
  • Each notch of size also gives a little more time, 8% more per notch for the first twelve. Use it for the bound check, not for a second full calculation.
  • In “Which falls more?”, the bonds compared get longer as you climb and their lengths can differ by more. The rule does not change: the bond whose money arrives later falls more.

Traps to avoid

  • The face value forgotten: only the coupons discounted. The price comes out far below face, which a bond with a coupon near its yield cannot be.
  • Coupons left undiscounted: years × coupon plus the discounted face. It overvalues the bond because it treats later coupons as worth as much as cash today.
  • The straight-line guess, face ± years × the coupon gap, offered as the price. The true price is always nearer face than that.
  • Coupon and yield swapped: a premium price for a discount bond. Re-check the first comparison before you choose.
  • For a zero, simple discounting, 1,000 ÷ (1 + n × y), or a year too few. At 8% over 9 years simple discounting gives 581.40, against the true 500.25.

How to practise

  • Use “Kinds of question” to drill current yield and zero-coupon bonds on their own; they are one division and one power, and build the habits the coupon-bond prices need.
  • Switch Answer by to “Typing the answer” once choosing is easy: a typed price within 0.50 counts, and it makes you finish the discounting.
  • Pick your Currency; the arithmetic is the same in any of them, but it reads better in your own.
  • When you miss a price, read the working’s split between coupons and face: it shows which half of the sum went wrong.

Levels

It is untimed unless you turn the clock on. There is no last level: at least one dial turns without end. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

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