Case study: convexity and risk aversion

The price curve bends up

A bond pays $c_t$ at each time $t$, in years. At the yield $y$ (a decimal: 5% is 0.05), its price is $P(y) = \sum_t c_t(1 + y)^{-t}$.

Case study 2.2.4 found the tangent to this curve always below the true price. That is a concave-up curve: $P''(y) > 0$.

Differentiate one payment twice, by the power and chain rules (lessons 2.3.1 and 2.3.3). The first derivative is $-t\,c_t(1 + y)^{-t-1}$, as in case study 2.3.7. The second is $t(t + 1)\,c_t(1 + y)^{-t-2}$, positive for every payment, so $P'' > 0$ for any bond with positive payments.

Duration plus convexity

Duration alone is the tangent: a straight-line estimate of the price change. Adding a term for the bend gives a parabola that follows the curve much further. Here $\Delta y$ is the change in yield.

For a 2% rise, $-D\,\Delta y = -8.8006\%$ and $\frac{1}{2}C(\Delta y)^2 = +0.4894\%$. The estimate is a fall of 8.3112%, to 87.7192; the true fall is 8.3318%, to 87.6994. For a 2% fall the same correction is added again: a rise of 9.2900% against a true 9.3124%.

Duration alone is off by 0.45 for a 2% rise; with convexity, by 0.02. Doubling the move roughly quadruples the duration error, as for any tangent. The convexity estimate's error grows about eight times, from 0.0025 to 0.0198, and it changes sign: it overshoots slightly when yields rise.

Convexity on the desk

Convexity grows quickly with maturity. A zero-coupon bond paying only at $T$ has $C = \frac{T(T + 1)}{(1 + y)^2}$: for 5 years at 5%, $\frac{30}{1.05^2} = 27.21$. At the same 5% yield:

Convexity grows much faster than duration. From 5 to 10 years, the 4% bond's duration grows 1.81 times and its convexity 3.2 times. For long bonds the correction matters even for moderate moves.

Risk aversion: a concave utility

Economists describe how much a person values wealth $w$ with a utility function $u(w)$. A common choice is $u(w) = \ln w$.

Its slope $u'(w) = \frac{1}{w}$ is positive: more is better. Its second derivative $u''(w) = -\frac{1}{w^2}$ is negative: each extra dollar adds less than the one before (lessons 2.3.4 and 2.4.2).

Offer someone with $100,000 a fair coin flip: win or lose $20,000. Their expected wealth, the average of the two equally likely outcomes, is still $100,000. But their expected utility is $\frac{1}{2}\ln 120{,}000 + \frac{1}{2}\ln 80{,}000 = 11.4925$, below $\ln 100{,}000 = 11.5129$.

The reason is the shape. The chord between the two outcomes lies below a concave curve, so the average of the two utilities sits below the utility of the average. The loss hurts more than the gain helps.

Here $\text{CE} = e^{11.4925} \approx$ $97,979.59, so the risk premium is $2,020.41. The figure shows a larger flip, $50,000 either way, where the shapes are easier to see.

How big is the premium?

How strongly a person dislikes risk is the curvature relative to the slope: the Arrow–Pratt coefficient $A(w) = -\frac{u''(w)}{u'(w)}$. For $\ln w$ it is $\frac{1}{w}$. For a fair flip of $\pm s$, the premium is about $\frac{1}{2}A(w)\,s^2$: half the curvature times the square of the move, like the convexity term.

Doubling the stake roughly quadruples the premium. The estimate is good for small stakes and too low for large ones. A richer person minds less: with $1,000,000, the same $20,000 flip has a premium of only $200.02.

The sign of $u''$ decides

For the $20,000 flip on $100,000:

A concave utility is risk averse: the premium is positive. A straight line is risk neutral: only the expected wealth matters. A convex utility is risk seeking: the person would pay to take the gamble.

A log-utility investor treats ruin as infinitely bad: as wealth falls towards 0, $\ln w$ falls without limit. So a model must check that every outcome leaves positive wealth before taking logs.

def test_convexity():
    """The 5-year 4% bond at 5% has convexity about 24.47"""
    expect_close(convexity(4, 100, 5, 0.05), 24.472405046329648, what="convexity(4, 100, 5, 0.05)", rel=1e-4)


def test_zero_coupon_convexity():
    """A 5-year zero at 5% has convexity 5 × 6 / 1.05²"""
    expect_close(convexity(0, 100, 5, 0.05), 30 / 1.05**2, what="convexity(0, 100, 5, 0.05)", rel=1e-4)


def test_convexity_positive():
    """Bond convexity is positive"""
    c = convexity(7, 100, 20, 0.06)
    assert c > 0, f"expected positive convexity, got {c}"


def test_certainty_equivalent():
    """$100,000 facing a fair ±$20,000 flip: certainty equivalent about $97,980"""
    expect_close(certainty_equivalent(100_000, 20_000, 20_000), 97979.58971132721, what="certainty_equivalent", rel=1e-9)


def test_risk_aversion():
    """The certainty equivalent is below the expected wealth"""
    ce = certainty_equivalent(50_000, 10_000, 10_000)
    assert ce < 50_000, f"expected a certainty equivalent below 50,000, got {ce}"


def test_ruin():
    """A loss that would leave no wealth raises ValueError"""
    try:
        certainty_equivalent(20_000, 20_000, 20_000)
    except ValueError:
        return
    raise AssertionError("expected ValueError when the loss would leave no wealth")