School Launch

The Bond Seesaw

A bond’s payments are fixed the day it is issued, so the only thing that can move is its price. When market yields rise, the price falls; when they fall, it rises. Set a bond, move the market, and watch the four yields the Series 65 asks about line up in order, while duration tells you how hard the seesaw swings.

Series 65 Lessons 4.4 and 4.5 · also Lessons 2.5 and 4.2 · about 25 minutes

The curve is the bond’s price at every market yield. Click or drag on the chart to set the market yield. With the chart focused, the arrow keys move the yield 0.05 point (Page Up and Page Down move 0.5).

Market price
Nominal yield (coupon rate)
Current yield
Yield to maturity
Yield to call
Duration (Macaulay)

The yields, ranked

Duration: how hard the seesaw swings

Macaulay duration
Modified duration
If yields move
Estimate: −modified × change
Actual, repriced
Gap (convexity)

The same thing in a spreadsheet

Settlement on a coupon date (here 15 January 2026) makes the spreadsheet’s answers match the page exactly. Prices are per $100 of par.

Where the price comes from

Each payment discounted back to today at the market yield. The prices add up to the bond’s price; the last column, divided by the price, adds up to the Macaulay duration.

PaymentYearCashWorth todayYear × worth

The bond

$
%
Coupons paid

The market

Type either one; the other follows.

$
%

The call

$

If market yields move

bp

100 basis points (bp) = 1 percentage point.

Try an example

From the Series 65 lessons; maturities the lessons leave open are filled in.

Four yields from one bond

The nominal yield is the coupon rate, printed on the bond and fixed for life: a 6% bond pays $60 a year on each $1,000 of par. The current yield asks what that coupon earns at today’s price:

current yield = annual interest ÷ market price

The yield to maturity (YTM) is the one rate that discounts every coupon and the par value at maturity back to exactly the price you pay. It counts the coupons and the gain (discount bond) or loss (premium bond) as the price is pulled to par. The yield to call (YTC) is the same calculation, stopped at the call date and the call price.

The ordering the exam loves

Premium bond (price above par): nominal > current > YTM > YTC. Every yield that counts the price falls below the coupon, and the call squeezes the loss down to par into fewer years, so YTC is lowest. A discount bond reverses the list: nominal < current < YTM < YTC. At par, nominal, current and YTM all equal the coupon rate. Brokers quote a premium callable bond at the lower of YTM and YTC: the worst case for the buyer.

The textbook order assumes the call price is par or close to it. A large call premium can lift YTC above YTM, and the page will tell you when that happens.

Price from yield

With f coupons a year, coupon C per payment, N payments and market yield y,

P = Σ C / (1 + y/f)k + par / (1 + y/f)N

Raise y and every term shrinks: that is the seesaw. There is no formula that runs the other way, so YTM and YTC are found by trial: the page (like a spreadsheet’s =YIELD) keeps halving the gap between a yield that is too low and one that is too high until the price matches to a fraction of a cent.

Duration

Macaulay duration is the average time you wait for your money, each payment weighted by what it is worth today. A zero-coupon bond pays everything at the end, so its duration equals its maturity. Coupons arrive earlier, so a coupon bond’s duration is shorter than its maturity; the lower the coupon and the longer the maturity, the longer the duration.

Modified duration = Macaulay ÷ (1 + y/f) turns that into a price sensitivity:

% change in price ≈ −modified duration × change in yield

Duration draws a straight line touching the curve. The real curve bends upward (convexity), so the true price is always a little higher than the straight-line estimate: losses are a bit smaller and gains a bit larger than duration predicts.

Check yourself

  1. A 6% bond trades at $1,200. Find the current yield and say where YTM falls.
    Answer

    $60 ÷ $1,200 = 5%. It is a premium bond, so YTM is lower still: with 10 years left it is about 3.60%. Nominal 6% > current 5% > YTM 3.60%. =YIELD(DATE(2026,1,15), DATE(2036,1,15), 6%, 120, 100, 2).

  2. A 7% bond trades at $940 with 8 years to run. Order its yields.
    Answer

    Nominal 7% < current 7.45% ($70 ÷ $940) < YTM about 8.03%. The $60 rise to par over eight years is the part current yield leaves out.

  3. An 8% bond maturing in 15 years trades at $1,080 and can be called at par in 4 years. Which yield should the customer be quoted?
    Answer

    Current yield 7.41%, YTM about 7.12%, YTC about 5.73%. The YTC is lowest because the $80 premium is lost in 4 years instead of 15, so it is the honest (worst-case) quote.

  4. Price a 15-year zero to yield 6% (yearly compounding), then at 7%. What did duration predict?
    Answer

    $1,000 ÷ 1.0615 = $417.27; at 7%, $362.45, a loss of 13.14%. Macaulay duration is 15, modified 15 ÷ 1.06 = 14.15, so duration predicted −14.15%. The actual loss is smaller: convexity.

  5. Which has the longer duration: a 10-year 8% bond or a 10-year zero? Check it with the page.
    Answer

    The zero: its duration is exactly 10 years. At an 8% yield, paid twice a year, the 8% bond’s duration is about 7.07 years, because its coupons return money early.

A bit of history: British consols paid their coupon forever and never matured; when the market demanded 6% instead of 4%, a £3 consol fell from £75 to £50 though no payment ever changed. The last of them were redeemed in 2015.

“The wicked borroweth, and payeth not again: but the righteous sheweth mercy, and giveth.” — Psalm 37:21