Case study: modified duration and elasticity

The exact slope of a bond price

A bond pays $c_t$ at each time $t$ (in years). At the yield $y$, that payment is worth $c_t(1 + y)^{-t}$ today.

As a function of $y$, this is a power of the inner function $1 + y$, whose slope is 1. The power rule (lesson 2.3.1) with the chain rule (lesson 2.3.3) gives

The sum rule (lesson 2.3.1) adds these over all payments, so $P'(y)$ is exact, with no bump. Each term is $-t$ times the payment's present value $PV_t = c_t(1 + y)^{-t}$, times one extra factor $\frac{1}{1 + y}$.

Duration is a balance point

Picture each payment's present value as a weight placed on a timeline at its time. The Macaulay duration is the point where the timeline balances.

For the 5-year 4% bond at 5%, the coupons weigh 3.81, 3.63, 3.46 and 3.29. The last payment weighs 81.49, which is 85% of the price.

Dividing by $1.05$ turns 4.6203 into the modified duration 4.4003, and 4.3570 into 4.1496, the 8% bond's duration in case study 2.2.4. A zero-coupon bond has one weight, at $T$, so $D_{\text{Mac}} = T$ and $D = \frac{T}{1 + y}$: the clean case of case study 2.2.4, now from the formula.

On the desk: conventions and edge inputs

Case study 2.2.4 covered signs, units and DV01. The exact formula adds a convention to get right and some inputs to guard against.

The compounding convention

The factor $\frac{1}{1 + y}$ came from differentiating $(1 + y)^{-t}$, so it depends on how the yield compounds. Many government bonds pay coupons twice a year, with the yield compounded twice a year. Then a payment $c$ at time $t$ is worth $c(1 + \frac{y}{2})^{-2t}$, and the chain rule, with inner slope $\frac{1}{2}$, gives

So the factor becomes $\frac{1}{1 + y/2}$. Paid as 2 every half year, the 5-year 4% bond at 5% has $D_{\text{Mac}} = 4.5695$ and $D = \frac{4.5695}{1.025} = 4.4581$, which a bump confirms. Dividing by 1.05 instead gives 4.3519, off by 0.1.

Edge inputs

• Negative yields have happened on some government bonds. The formula still works while $1 + y > 0$: a 5-year 1% bond at $-0.5\%$ is priced at 107.61, with $D = 4.9308$.
• A yield of $-100\%$ or below makes $1 + y$ zero or negative. The discount factors $(1 + y)^{-t}$ then divide by zero or mean nothing, so a function should refuse such a yield.
• Payments that move. The formula, like the bump, assumes the payments $c_t$ do not depend on $y$ (case study 2.2.4). For bonds the issuer may repay early, they do.

Duration is still a slope. For moves of 1% or more, the bend of the price curve matters: that is convexity, in case study 2.4.5.

With continuous discounting, duration is an average time

Discount continuously instead (case study 1.4.6): each payment is worth $c_t e^{-yt}$. Lesson 2.3.4 gives $\frac{d}{dy}e^{-yt} = -te^{-yt}$, so

There is no $1 + y$ to divide by: the duration is the present-value-weighted average time. It is also $-\frac{d}{dy}\ln P$, by the log-of-a-function rule of lesson 2.3.4: duration is the relative rate at which the price falls.

Elasticity

How much does demand $Q$ respond to the price $p$? The slope $\frac{dQ}{dp}$ depends on units: quote prices in cents instead of dollars and it shrinks a hundredfold. The elasticity compares percentage changes instead, so it has no units.

Constant elasticity. For $Q = 1000\,p^{-1.5}$, the power rule gives $\frac{dQ}{dp} = -1500\,p^{-2.5}$. Then $E = -1.5$ at every price: any $Q = kp^{a}$ has elasticity $a$.

Straight-line demand. For $Q = 500 - 20p$, $E = \frac{-20p}{500 - 20p}$ changes along the line. It is 0 at $p = 0$, $-0.67$ at $p = 10$, $-1$ at $p = 12.5$ and $-4$ at $p = 20$. At $p = 25$ demand is 0 and $E$ is undefined; just below it, $E$ falls without limit ($-24$ at $p = 24$).

The same demand with prices in cents is $Q = 500 - 0.2c$. Its slope is $-0.2$ per cent, not $-20$ per dollar, but at $c = 1000$ cents the elasticity is still $-0.67$.

Revenue and the product rule

Revenue is price times quantity, $R = pQ$. The product rule (lesson 2.3.2) gives its slope, and the elasticity appears inside it.

For $Q = 500 - 20p$, the slope of revenue is $500 - 40p$. At $p = 10$, $Q = 300$ and $E = -\frac{2}{3}$, so the slope is $300 \times (1 - \frac{2}{3}) = 100$. At $p = 20$ it is $-300$.

It is 0 at $p = 12.5$, where revenue peaks at 3125.

Elasticity is a slope

Like duration, elasticity describes small changes. For $Q = 1000\,p^{-1.5}$, a 2% price rise cuts the quantity by about $1.5 \times 2\% = 3\%$; the curve gives 2.93%.

For a 20% rise the estimate is 30%, but the curve gives only 23.93%. For big moves, use the demand curve itself.

def _price(c, F, T, y, freq=1):
    return sum((c / freq) / (1 + y / freq) ** k for k in range(1, T * freq + 1)) + F / (1 + y / freq) ** (T * freq)


def _bump(c, F, T, y, freq=1, h=1e-6):
    return -(_price(c, F, T, y + h, freq) - _price(c, F, T, y - h, freq)) / (2 * h) / _price(c, F, T, y, freq)


def test_five_year_bond():
    """The 5-year 4% bond at 5% has duration about 4.40031"""
    expect_close(exact_duration(4, 100, 5, 0.05), 4.400306998058356, what="exact_duration(4, 100, 5, 0.05)", rel=1e-9)


def test_matches_bump():
    """It agrees with a bump estimate for a 10-year 6% bond at 3%"""
    expect_close(exact_duration(6, 100, 10, 0.03), _bump(6, 100, 10, 0.03), what="exact vs bump", rel=1e-6)


def test_zero_coupon():
    """A 7-year zero-coupon bond at 4% has duration 7 / 1.04"""
    expect_close(exact_duration(0, 100, 7, 0.04), 7 / 1.04, what="exact_duration(0, 100, 7, 0.04)")


def test_one_year():
    """A 1-year bond has duration 1 / (1 + y)"""
    expect_close(exact_duration(5, 100, 1, 0.05), 1 / 1.05, what="exact_duration(5, 100, 1, 0.05)")


def test_semiannual():
    """With freq=2, the 5-year 4% bond at 5% has duration about 4.45806, not 4.35191"""
    expect_close(exact_duration(4, 100, 5, 0.05, freq=2), 4.458056324352842, what="exact_duration(4, 100, 5, 0.05, freq=2)", rel=1e-9)


def test_semiannual_matches_bump():
    """With freq=2, it agrees with a bump estimate for a 30-year 5% bond at 4.5%"""
    expect_close(exact_duration(5, 100, 30, 0.045, freq=2), _bump(5, 100, 30, 0.045, freq=2), what="exact vs bump, freq=2", rel=1e-6)


def test_negative_yield():
    """A negative yield is allowed: a 5-year 1% bond at -0.5% has duration about 4.93079"""
    expect_close(exact_duration(1, 100, 5, -0.005), 4.9307916502902485, what="exact_duration(1, 100, 5, -0.005)", rel=1e-9)


def test_rejects_impossible_yield():
    """A yield of -100% or below has no discount factors: raise ValueError"""
    for y in (-1, -1.5):
        try:
            exact_duration(4, 100, 5, y)
        except ValueError:
            continue
        raise AssertionError(f"expected ValueError for y = {y}")