Rates are the price of time

M3 ended in statistical physics. M4 changes register completely — less mathematics, more institutions, and the context that explains why the moves in earlier modules happen when they do.

But it starts with a debt. M0.5 said bonds are "priced in yield, not just price" and moved on. The course has never said what a yield is, and every remaining lesson in this module depends on it.

Yield is a summary, not a rate you are paid

A bond is a fixed schedule of future cashflows. It has a price today. The yield to maturity is the single discount rate that reconciles the two:

                  C          C                C + F
      P  =  ───────── + ─────────── + … + ───────────
             (1 + y)     (1 + y)²           (1 + y)ⁿ

      P = price today    C = coupon    F = face value    n = years
      y = YIELD — the one rate that makes this equation true

Three consequences worth stating explicitly:

Price and yield move inversely, always. They are two views of one quantity. "Bonds sold off" and "yields rose" are the same sentence. Beginners lose a surprising amount of time to this.

Yield is a summary statistic, not a promise. It is the return you earn if you hold to maturity and reinvest every coupon at the same rate y. That reinvestment assumption is rarely true, which is why yield is a comparison device rather than a forecast.

It makes bonds comparable. M0.5's point lands here: one issuer floats many bonds with different coupons and maturities, so raw prices are meaningless across them. Yield puts them on one axis, which is why the market quotes it.

Duration — how much you lose when rates rise

The question that matters is sensitivity. Differentiate the pricing formula and you get a weighted average of the times at which cashflows arrive:

   Macaulay duration  D   =  Σ t·PV(cashflow_t) / P        (in years)

   Modified duration  MD  =  D / (1 + y)

                     ΔP/P  ≈  −MD × Δy

For a 10-year bond with a 4% annual coupon, priced at par:

   price 100.00     Macaulay duration 8.44 years
                    modified duration 8.11
                    convexity 80.8

So +100bp of yield costs about 8.1% of price. Duration is the single number that tells you what a rate move does to a bond, and it is why "long duration" and "rate-sensitive" mean the same thing.

Two things follow that are worth having as reflexes: duration rises with maturity (cashflows are further away) and falls with coupon (a high coupon returns your money sooner, so the average cashflow arrives earlier). A zero-coupon bond has duration exactly equal to its maturity, because there is only one cashflow.

Convexity — and why it is a gift

Duration is a first derivative, so it is a straight line approximating a curve. Push the yield far and the approximation drifts:

        Δy    actual price   actual %   duration says   + convexity
   ─────────────────────────────────────────────────────────────────
    −200bp        117.97      +17.97%       +16.22%        +17.84%
    −100bp        108.53       +8.53%        +8.11%         +8.51%
     −50bp        104.16       +4.16%        +4.06%         +4.16%
     +50bp         96.04       −3.96%        −4.06%         −3.95%
    +100bp         92.28       −7.72%        −8.11%         −7.71%
    +200bp         85.28      −14.72%       −16.22%        −14.61%
    +300bp         78.93      −21.07%       −24.33%        −20.70%

Look at the ±200bp rows together. Duration predicts a symmetric ±16.22%. The truth is −14.72% down, +17.97% up — you lose less than predicted and gain more than predicted, in both directions.

That asymmetry is convexity, the second derivative, and for an ordinary bond it is positive:

   ΔP/P  ≈  −MD·Δy  +  ½·C·(Δy)²
                        ────────────
                        always positive — it ADDS in both directions

Adding that term (the last column) tracks reality closely out to ±300bp. Positive convexity is genuinely valuable — it is optionality in a bond, and it is why convexity is quoted and paid for. Instruments with negative convexity, most notably mortgage-backed securities where borrowers refinance when rates fall, have exactly the opposite and much nastier profile.

2022, so this doesn't stay theoretical

Bonds are widely described as the safe asset. Here is what "safe" did in a single year:

   10-year Treasury yield,  2022-01-03  →  2022-12-30

              1.63%   →   3.88%          (+225 basis points)

   a 10-year bond over that move:   −17%

A −17% year in the asset held precisely because it is not supposed to do that. No default, no credit event, nothing exotic — just duration meeting a 225bp move. And bonds fell in the same year equities did, which is not supposed to happen either; that part is M4.6.

The mechanism is now completely unmysterious to you, which is the point of this lesson. Duration times the rate move. Everything else in M4 is about why the rate moved.

Source: any fixed-income primer covers this; Bruce Tuckman, Fixed Income Securities, ch.4–6 is the standard if you want it properly. The numbers above are from FRED series DGS10 and a page of arithmetic — worth reproducing, since duration is one of those quantities that only becomes intuitive once you have watched it mis-predict a large move.