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Use Jensen's inequality to estimate the convexity effect.

FRM® Part II · Market Risk Measurement and Management · Expectations, Risk Premium, Convexity, and the Shape of the Term Structure · FRM2.MR.13.c

The idea

Jensen's inequality says that for a convex function f, the average of f is at least f of the average. In symbols, E[f(X)] ≥ f(E[X]), and the gap grows with the dispersion of X. A bond price is convex in the interest rate, so the expected price across uncertain future rates exceeds the price at the expected rate. This is the convexity effect, and it is why the yield implied by a zero-coupon price is lower than the expected average rate. Let C be convexity, σ the rate volatility and P the bond price. The gain in expected price is then about Gain = ½ × C × σ² × P. The effect rises with variance and roughly with the square of maturity. The convexity of a zero is roughly C = T × (T + 1) ÷ (1 + y)². With continuous compounding and a parallel random shift, the yield falls by Fall = ½ × σ² × T. Mistakes: reversing the inequality, omitting the one half, and using volatility where variance belongs.

Formulas

Convexity gain in a bond's expected price

Gain = ½ × C × σ² × P

C
the bond's convexity
σ
the volatility of the rate, as a decimal
P
the bond price

From Jensen's inequality: the expected price exceeds the price at the expected rate by this amount. Use variance σ², not σ, and keep the one half.

Approximate convexity of a zero-coupon bond

C = T × (T + 1) ÷ (1 + y)²

T
the bond's maturity, in years
y
the annually compounded yield

For annual compounding. Convexity grows roughly with the square of maturity, which is why the convexity effect is much larger for long zeros.

Fall in yield from convexity under a random parallel shift

Fall = ½ × σ² × T

σ
the standard deviation of the parallel rate shift, per year
T
the bond's maturity in years

Continuous compounding with a normal parallel shift held for the bond's life: the yield equals the expected rate less this amount. It grows linearly with maturity.

A practice question

From the question bank, with the reason each wrong option is wrong.

A one-year zero-coupon bond has a face value of 100. Next year's discount rate is equally likely to be 2% or 8%. The expected price of the bond is closest to:

  1. 95.24Why not: Prices the bond at the expected rate of 5%, 100/1.05, which misses the Jensen gap.
  2. 95.32 - correct
  3. 95.16Why not: Reflects the gap around 95.24 to the other side, treating the convexity effect as a loss.
  4. 98.04Why not: Uses only the low-rate scenario of 2% instead of averaging the two outcomes.

E[P] = 0.5 x 100/1.02 + 0.5 x 100/1.08 = 0.5 x 98.039 + 0.5 x 92.593 = 95.316. The price at the expected rate is 95.238, so Jensen's gap is +0.08.

A harder one

A five-year zero-coupon bond yields 4.00% (annual compounding) and has a price of 82.19 per 100 face. Convexity is approximately T(T+1)/(1+y)². If the yield level is uncertain with a standard deviation of 1.00% and an unchanged mean, the convexity effect on the expected price is closest to:

  1. 0.23Why not: Omits the one half in the formula 0.5 x convexity x variance x price, doubling the effect.
  2. 0.09Why not: Uses T²/(1+y)² = 23.1 as convexity instead of T(T+1)/(1+y)² = 27.7.
  3. 0.11 - correct
  4. 0.14Why not: Applies the effect to the face value of 100 instead of the bond's price of 82.19.

Convexity = 5 x 6 / 1.04² = 27.74. Effect = 0.5 x 27.74 x (0.01)² x 82.19 = 0.114.

The hardest tier

A model uses continuously compounded rates. The average expected rate over the next ten years is 4.00%. All rates will shift in parallel by a random amount that is normally distributed with mean zero and standard deviation 2.00% per year, and the shift is constant over the bond's life. The yield on a ten-year zero-coupon bond, using the approximation yield = expected rate - 0.5 x sigma² x T, is closest to:

  1. 2.00%Why not: Uses T² instead of T in the yield adjustment, which applies to the log of the price rather than the yield.
  2. 3.80% - correct
  3. 3.60%Why not: Omits the one half, subtracting sigma² x T.
  4. 4.20%Why not: Adds the convexity effect to the expected rate instead of subtracting it.

ln E[P] = -rT + 0.5 x sigma² x T², so the yield is r - 0.5 x sigma² x T = 4.00% - 0.5 x 0.0004 x 10 = 3.80%. Convexity pulls the yield below the expected rate.

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