Asymmetric Prices For Equal Yield Shocks

Why Equal Yield Shocks Do Not Have Equal Price Effects

For a bond with positive convexity, a yield decrease of a given size generally produces a larger price gain than the price loss caused by a yield increase of the same size. The two shocks have equal magnitude, but the price-yield relationship is curved rather than linear. This tutorial compares the exact effects with duration-convexity estimates and shows how the curvature creates the asymmetry.

We use an annual-compounding model. The bond has fixed cash flows $C_t$ at times $t=1,\ldots,n$, and its yield $i$ is an effective annual rate. Assume $i>-1$, the cash flows do not change when yields move, and at least one cash flow is positive. In particular, the results below apply to fixed, nonnegative cash flows; they do not describe a security whose cash flows change in response to the yield shock.

Definition: Let $P(i)$ be the present value of a bond’s fixed cash flows at yield $i$. For a shock of size $h>0$, the price gain from a yield decrease is $G=P(i-h)-P(i)$, and the price loss from a yield increase is $L=P(i)-P(i+h)$. A positive $L$ means the bond’s price fell.

For an exact comparison, both shocked prices must be valued at their new yields. A first-order duration estimate alone predicts equal-sized percentage changes in the two directions. Positive convexity adds a positive adjustment to both directions: it increases the estimated gain when yields fall and reduces the estimated loss when yields rise.

Price Curvature and the Asymmetry

Under the stated annual-compounding assumptions, the price is the sum of the discounted cash flows. Differentiating this price function twice shows why its curvature matters.

P(i)=\sum_{t=1}^{n}\frac{C_t}{(1+i)^t}, \qquad P''(i)=\sum_{t=1}^{n}\frac{t(t+1)C_t}{(1+i)^{t+2}}.

For $i>-1$, each denominator in the second expression is positive. If the cash flows are nonnegative and at least one is positive at a time $t>0$, then $P''(i)>0$. Thus price is a strictly convex function of yield in this model. The convexity measures used in the local approximation are defined here using derivatives with respect to the effective annual yield:

Definition: The modified duration and convexity at yield $i$ are $D(i)=-P'(i)/P(i)$ and $C(i)=P''(i)/P(i)$. Under the fixed, nonnegative cash-flow assumptions above, $C(i)>0$.

The duration-convexity approximation for a small yield change $\Delta i$ is

\frac{\Delta P}{P(i)} \approx -D(i)\Delta i+\frac{1}{2}C(i)(\Delta i)^2.

For equal shocks, substitute $\Delta i=-h$ for the yield decrease and $\Delta i=+h$ for the increase. The linear duration terms have opposite signs; the quadratic convexity terms have the same positive sign. The estimated gain and loss are therefore

G\approx P(i)\left(D(i)h+\frac{1}{2}C(i)h^2\right), \qquad L\approx P(i)\left(D(i)h-\frac{1}{2}C(i)h^2\right).

Subtracting gives the approximate excess gain over loss:

G-L\approx P(i)C(i)h^2.

The approximation explains the direction and approximate size of the asymmetry: the duration effects cancel in the difference, leaving the convexity contribution. This is a local estimate, not an exact repricing formula. If the shock is not small, calculate both shocked prices directly.

Result: With fixed, nonnegative cash flows and positive convexity, an equal-sized yield decrease produces a larger price gain than the price loss from a yield increase. The exact comparison is $P(i-h)+P(i+h)>2P(i)$. The duration-convexity approximation estimates the difference as $P(i)C(i)h^2$.

Why the Exact Comparison Holds

The exact result follows from convexity without relying on a Taylor approximation. Define the symmetric price difference $g(h)=P(i-h)+P(i+h)-2P(i)$. At $h=0$, $g(0)=0$, and $g'(0)=0$. For shocks that keep both yields above $-1$,

$$ g''(h)=P''(i-h)+P''(i+h)>0. $$

Because $g''(h)>0$, the derivative $g'(h)$ increases from zero, so $g'(h)>0$ for $h>0$. Consequently $g(h)>g(0)=0$. Rewriting this inequality gives $P(i-h)-P(i)>P(i)-P(i+h)$, which is exactly $G>L$. In words, the two prices at yields equally far from $i$ have an average above the price at $i$.

Worked Examples

Worked Example: Exact Prices and a Convexity Estimate

A hypothetical bond pays \$100 at the end of year 1 and \$1,100 at the end of year 2. Its annual effective yield is 10%. Compare the exact price effects of a 200-basis-point yield decrease and increase. Then compare them with the duration-convexity estimates.

Method and starting price. Discount each cash flow at the stated yield. The initial present values are $100/1.10=\$90.9091$ and $1{,}100/(1.10)^2=\$909.0909$. Thus $P(0.10)=\$1{,}000.00$. The present-value weights are $0.0909091$ at time 1 and $0.9090909$ at time 2.

Exact repricing. The shocked yields are 8% and 12%. At 8%, $P(0.08)=100/1.08+1{,}100/(1.08)^2=\$92.5926+\$943.0727=\$1{,}035.6653$. The exact gain is $\$1{,}035.6653-\$1{,}000.00=\$35.6653$.

At 12%, $P(0.12)=100/1.12+1{,}100/(1.12)^2=\$89.2857+\$876.9133=\$966.1990$. The exact loss is $\$1{,}000.00-\$966.1990=\$33.8010$. The gain exceeds the loss by $\$35.6653-\$33.8010=\$1.8643$.

Duration-convexity estimate. For these annual cash flows, the derivative formulas give $D(i)=\sum w_t t/(1+i)$ and $C(i)=\sum w_t t(t+1)/(1+i)^2$, where $w_t$ is the time-$t$ present value divided by the total price. At $i=0.10$,

D(0.10)=\frac{0.0909091(1)+0.9090909(2)}{1.10} =1.73553719,

and

C(0.10)=\frac{0.0909091(1)(2)+0.9090909(2)(3)}{(1.10)^2} =\frac{5.6363636}{1.21} =4.65815177.

With $h=0.02$, the duration contribution is $D h=1.73553719(0.02)=0.03471074$, and the convexity contribution is $\tfrac12 C h^2=0.5(4.65815177)(0.0004)=0.00093163$. The estimated gain is $1{,}000(0.03471074+0.00093163)=\$35.6424$. The estimated loss is $1{,}000(0.03471074-0.00093163)=\$33.7791$, giving an estimated difference of $\$1.8633$. The exact and estimated differences are close, and both show a larger gain than loss.

Check and common mistake avoided. The exact prices give a gain of $\$35.67$ and a loss of $\$33.80$, confirming the sign and size of the estimated asymmetry. The weights must be based on present values, not on the undiscounted cash flows; using $100/1{,}000$ and $900/1{,}000$ as weights would give incorrect duration and convexity.

Worked Example: A Two-Year Zero-Coupon Bond

A hypothetical zero-coupon bond pays \$121 in two years. Its annual effective yield is 10%. Find the exact gain and loss under equal 10-percentage-point yield shocks, and compare their difference with the duration-convexity estimate.

Method and initial price. A zero-coupon bond has one cash flow, so $P(i)=121/(1+i)^2$. At 10%, $P(0.10)=121/(1.10)^2=121/1.21=\$100.00$.

Exact effects. A yield decrease gives a 0% yield, so $P(0)=121/1^2=\$121.00$, a gain of $\$21.00$. A yield increase gives a 20% yield, so $P(0.20)=121/(1.20)^2=121/1.44=\$84.0278$, a loss of $\$100.00-\$84.0278=\$15.9722$. The exact excess gain is $\$21.00-\$15.9722=\$5.0278$.

Approximate effects. For a single cash flow at time 2, $D=2/(1+i)$ and $C=2(3)/(1+i)^2$. At 10%, $D=2/1.10=1.8181818$ and $C=6/1.21=4.9586777$. With $h=0.10$, $Dh=0.1818182$ and $\tfrac12 Ch^2=0.5(4.9586777)(0.01)=0.0247934$. The estimated gain is $100(0.1818182+0.0247934)=\$20.6612$; the estimated loss is $100(0.1818182-0.0247934)=\$15.7025$. The estimated difference is $100(4.9586777)(0.01)=\$4.9587$.

Check and common mistake avoided. Exact repricing and the approximation both show that the gain exceeds the loss. The approximation is a local estimate; it need not equal the exact difference, particularly for a large shock. Do not treat the duration-only estimate as the exact change.

Worked Example: Converting the Asymmetry to Dollars

A hypothetical bond portfolio is worth \$250,000. At its current annual yield, its modified duration is 6.4 and its convexity is 52. Estimate the dollar price effect of equal yield shocks of 50 basis points in either direction.

Method. Fifty basis points is $h=0.005$, not $0.50$. The duration contribution is $Dh=6.4(0.005)=0.032$, or 3.2%. The convexity contribution is $\tfrac12 Ch^2=0.5(52)(0.005)^2=0.00065$, or 0.065%.

For the yield decrease, the estimated percentage gain is $0.032+0.00065=0.03265$, or 3.265%. In dollars, the gain is $\$250{,}000(0.03265)=\$8{,}162.50$. For the yield increase, the estimated percentage loss is $0.032-0.00065=0.03135$, or 3.135%. In dollars, the loss is $\$250{,}000(0.03135)=\$7{,}837.50$. The estimated gain exceeds the loss by $\$8{,}162.50-\$7{,}837.50=\$325.00$.

Check and common mistake avoided. The duration-only estimate would be a \$8,000 change in either direction. Convexity adds \$162.50 to the gain and subtracts \$162.50 from the loss, producing the \$325.00 difference. Converting basis points to a decimal yield change before squaring avoids a factor-of-10,000 error.

Interpreting the Result and Avoiding Traps

The comparison applies to percentage changes and dollar changes, provided both are measured from the same initial price. Multiplying the percentage estimates by $P(i)$ converts them to dollar effects. The same yield change should be used in both directions, and $h$ must be expressed as a decimal rate: 25 basis points is $0.0025$.

The condition that both shocked yields remain above $-1$ is necessary for the annual-compounding discount factors used here. Positive convexity follows from the fixed, nonnegative cash flows in the model. If a security’s cash flows can change with yields, or the price function does not have positive convexity, the conclusion should not be assumed without checking the actual price function.

Mixing up the comparison. Define gain and loss as positive quantities before subtracting. The excess gain is $G-L$, not the change in price from the high-yield scenario to the low-yield scenario.
Using duration alone. A first-order estimate gives equal magnitudes for equal shocks. It omits the curvature that creates the asymmetry.
Applying the convexity term with the wrong sign. The quadratic term is positive for both signed yield changes. It increases the gain and reduces the loss.
Using the wrong weights. When calculating duration or convexity from cash flows, normalize discounted present values, not undiscounted cash amounts.
Confusing percentage points and decimals. Convert the shock to a decimal before using it, and square that decimal in the convexity term.

Key takeaway: For fixed, nonnegative cash flows with positive convexity, equal-sized upward and downward yield shocks do not have equal price effects. Exact repricing gives a larger gain for the yield decrease; locally, the duration effects cancel in the comparison and convexity explains the difference.

Check Your Understanding

For each item, identify whether it asks for an exact comparison or a duration-convexity estimate, and keep the yield shock in decimal form.

A fixed-cash-flow bond has positive convexity. Its yield rises or falls by the same positive amount, with both resulting yields above $-1$. Which is larger: the price gain after the fall or the price loss after the rise? (Answer: The price gain after the yield fall is larger.)
A portfolio is worth \$2,000, has modified duration 3.5 and convexity 18, and faces equal yield shocks of 1%. Estimate the dollar gain for the decrease, the dollar loss for the increase, and their difference. (Answer: Gain $\$71.80$; loss $\$68.20$; difference $\$3.60$.)
A two-year zero-coupon bond pays \$1,100 and has a 5% annual effective yield. Find the exact price at the initial yield and at yields 4% and 6%, then compare the gain and loss. (Answer: Initial price $\$997.73$; price at 4% $\$1{,}017.01$, gain $\$19.28$; price at 6% $\$979.00$, loss $\$18.74$; the gain is larger.)
A bond’s prices at yields equally spaced around its current yield are \$1,034, \$1,000, and \$969, respectively, from lower yield to current yield to higher yield. Find the gain, loss, and excess gain. What does the price comparison indicate about convexity over this interval? (Answer: Gain $\$34$; loss $\$31$; excess gain $\$3$; the symmetric price difference is positive, consistent with positive convexity over this comparison.)
A portfolio worth \$50,000 has modified duration 5 and convexity 40. Estimate the gain and loss for equal yield shocks of 2%, and compare with the duration-only estimate. (Answer: Gain $\$5{,}400$; loss $\$4{,}600$; duration-only estimates are $\$5{,}000$ in either direction; the estimated gain exceeds the loss by $\$800$.)

What You'll Learn