Why the Yield Convention Matters
Convexity measures the curvature of a price-yield relationship, but its numerical value depends on the yield variable used to take the second derivative. A price differentiated twice with respect to an annual effective yield does not generally have the same convexity as that price differentiated twice with respect to a nominal rate convertible \(m\) times per year. The time unit used to describe cash-flow dates must also match the compounding convention.
The calculations below use a fixed set of promised cash flows, with no change in default, call, or prepayment assumptions as the yield changes. Convexity is the second derivative of price with respect to the stated yield, divided by price. Yields are entered as decimals: a change of 0.01 means one percentage point. The resulting convexity is therefore tied to the yield variable and its units.
Annual Effective Yield and Annual-Time Units
Suppose cash flow \(C_t\) is paid at time \(t\) years, and a single annual effective yield \(i\) discounts every cash flow. The price is \(P(i)=\sum_t C_t(1+i)^{-t}\). The times \(t\) are measured in years; they need not be whole numbers. Differentiating twice with respect to \(i\) gives the annual-effective-yield convexity:
The last expression shows that this convexity is a price-weighted average of the factors \(t(t+1)/(1+i)^2\). The time \(t\) is in years because the yield variable is the annual effective yield. Using the number of semiannual or quarterly payment periods in place of \(t\) would change the units and give the wrong result.
Worked Example: Annual Effective Yield for a Three-Year Zero-Coupon Bond
A hypothetical bond pays $1,000 at the end of 3 years and nothing earlier. Its annual effective yield is \(i=5\%\). Find its annual-effective-yield convexity. State the method and check the result.
Method. Use the annual-effective-yield formula for a single cash flow, with \(t=3\) years. The price is \(P=1000(1.05)^{-3}=1000/1.157625=863.837598\ldots\) dollars.
For this one cash flow, the price cancels from \(P''/P\), leaving \(t(t+1)/(1+i)^2\). Substitution gives \(3(4)/(1.05)^2=12/1.1025=10.88435374\ldots\). Thus the convexity is 10.8844 to four decimal places.
Check and mistake avoided. The annual-time input is 3, not a number of coupon periods. Direct differentiation gives \(P''=1000(3)(4)(1.05)^{-5}\); dividing by \(P=1000(1.05)^{-3}\) gives \(12(1.05)^{-2}\), as above. Using the payment count would be inappropriate for this annual-effective yield.
Nominal Yield Convertible \(m\) Times per Year
Now let \(j=i^{(m)}\) be a nominal annual yield convertible \(m\) times per year. A cash flow at time \(t\) years is discounted by \((1+j/m)^{-mt}\). If payments occur exactly on conversion dates, write \(k=mt\) for the number of payment periods. Differentiating price with respect to the nominal annual yield \(j\) gives:
The two forms are equivalent because \(mt(mt+1)/m^2=t(t+1/m)\). In the first form, the period count \(mt\) is explicit. In the second, time is in years. Both differentiate with respect to \(j\), the nominal annual yield, not the yield per payment period.
To see why a factor of \(m^2\) appears, set \(q=j/m\), the yield per conversion period. Since \(q=j/m\), the chain rule gives \(d/dj=(1/m)d/dq\), and therefore \(d^2/dj^2=(1/m^2)d^2/dq^2\). Thus, if period convexity means \(P''(q)/P\), nominal-annual-yield convexity is period convexity divided by \(m^2\). This is a conversion of the derivative variable, not a change in the underlying cash flows or price.
Worked Example: Quarterly Nominal Yield for a One-Year Payment
A hypothetical zero-coupon bond pays $1,000 in one year. Its nominal annual yield convertible quarterly is \(j=8\%\). Find its convexity with respect to \(j\), and verify the conversion from payment-period units.
Method. Here \(m=4\), \(t=1\) year, and \(q=j/m=0.08/4=0.02\). The price is \(P=1000(1.02)^{-4}=1000/1.08243216=923.845426\ldots\) dollars.
Using annual-time units, the nominal-yield convexity is \(t(t+1/m)/(1+j/m)^2 =1(1+1/4)/(1.02)^2 =1.25/1.0404 =1.20146098\ldots\). The answer is 1.2015 to four decimal places.
As a check, the payment-period count is \(k=mt=4\). Convexity with respect to the quarterly yield \(q\) is \(k(k+1)/(1+q)^2=4(5)/(1.02)^2=19.2233756\ldots\). Dividing by \(m^2=16\) gives \(19.2233756\ldots/16=1.20146098\ldots\), agreeing with the annual-time calculation.
Mistake avoided. Reporting \(19.2234\) as the convexity with respect to the nominal annual yield would omit the chain-rule factor \(1/16\). The period-yield measure and the nominal-annual-yield measure use different derivative variables.
Equivalent Yields Do Not Make Convexities Equal
Two yield conventions can produce the same price at a particular point without producing the same second derivative. For example, a nominal yield \(j=i^{(m)}\) and annual effective yield \(i\) are equivalent when \(1+i=(1+j/m)^m\). This makes the discount factors agree, but the relationship between \(i\) and \(j\) is nonlinear. A second derivative must account for both the first and second derivatives of that conversion.
If \(i=(1+j/m)^m-1\), then \(di/dj=(1+j/m)^{m-1}\) and \(d^2i/dj^2=((m-1)/m)(1+j/m)^{m-2}\). For a price expressed as \(P(i)\), the chain rule gives:
The second term is essential. It arises because a small change in \(j\) does not translate into a constant-sized change in \(i\). Consequently, multiplying one convexity by a squared conversion factor alone is generally insufficient.
Worked Example: Comparing Equivalent Annual and Semiannual Yields
A hypothetical zero-coupon bond pays $1,000 at the end of 2 years. Its nominal annual yield convertible semiannually is \(j=10\%\). Compare convexity with respect to \(j\) with convexity with respect to the equivalent annual effective yield.
Method. With \(m=2\), the semiannual yield is \(j/m=0.05\), so the price is \(P=1000(1.05)^{-4}=1000/1.21550625=822.702474\ldots\) dollars. The equivalent annual effective yield is \(i=(1.05)^2-1=0.1025\), or \(10.25\%\). This produces the same price because \((1+i)^2=(1.05)^4\).
For the nominal annual yield, \(t=2\) and \(t+1/m=2.5\). Therefore, \(P''(j)/P=2(2.5)/(1.05)^2=5/1.1025=4.53514739\ldots\), or 4.5351 to four decimal places.
For the annual effective yield, \(P''(i)/P=2(3)/(1.1025)^2=6/1.21550625=4.93621485\ldots\), or 4.9362 to four decimal places. The yields give the same price, but these convexities differ because they are derivatives with respect to different variables.
The chain rule verifies the nominal-yield result. At this point, \(di/dj=1.05\), \(d^2i/dj^2=0.5\), and \(P'(i)/P=-2/1.1025\). Hence \(4.93621485(1.05)^2+(-2/1.1025)(0.5)=5.44217687-0.90702948=4.53514739\ldots\).
Mistake avoided. Treating equivalent yields as interchangeable derivative variables would incorrectly report the same convexity under both conventions. The chain rule, including its second-derivative term, resolves the difference.
Common Unit Errors
A reliable calculation begins by writing down the yield variable and the time unit attached to it. In an annual-effective-yield calculation, use years and factors \(t(t+1)/(1+i)^2\). In a nominal-yield calculation, use years and \(t(t+1/m)/(1+j/m)^2\), or use period counts and then divide period convexity by \(m^2\). Keep the discounting convention and differentiation variable consistent throughout.
Decide whether the derivative is with respect to annual effective \(i\), nominal annual \(j=i^{(m)}\), or a period yield \(q=j/m\).
Use \(t\) in years with annual-time formulas. Use \(k=mt\) for payment-period counts when payments align with conversion dates.
For period convexity converted to nominal annual yield, divide by \(m^2\). For equivalent annual and nominal yields, use the full chain rule.
Confirm that the discount factors used to price the cash flows match the yield convention used in the derivatives.
A common wrong calculation is to take quarterly period convexity \(19.2234\) from the preceding one-year example and report it directly as nominal annual-yield convexity. The correction is \(19.2234/4^2=1.2015\). Another wrong approach is to apply \(t(t+1)/(1+i)^2\) to a nominal yield without converting the derivative variable. Use \(t(t+1/m)/(1+j/m)^2\) for that nominal yield instead.
Check Your Understanding
For each problem, identify the yield variable before calculating. Round numerical convexities to four decimal places.
- A hypothetical zero-coupon bond pays $1,000 in 2.5 years and has an annual effective yield of \(4\%\). Find its annual-effective-yield convexity. (Answer: \(8.0899\))
- A hypothetical zero-coupon bond pays $1,000 in 3 years. Its nominal annual yield is \(6\%\) convertible semiannually. Find convexity with respect to that nominal annual yield. (Answer: \(10.5/(1.03)^2=9.8973\))
- A hypothetical zero-coupon bond pays $1,000 in 2 years and has nominal annual yield \(8\%\) convertible quarterly. Find its nominal-yield convexity and its convexity with respect to the equivalent annual effective yield. (Answer: nominal \(4.3253\); annual effective \(5.1209\))
- A calculation gives period convexity \(24.5\) for a nominal yield convertible quarterly. Convert it to convexity with respect to the nominal annual yield, and state the conversion factor. (Answer: \(24.5/16=1.5313\); divide by \(m^2=16\))