Correction to: Yield curve shapes and the asymptotic short rate distribution in affine onefactor models
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Correction to: Finance Stoch. (2008) 12: 149–172 https://doi.org/10.1007/s007800070059z
I should like to thank Ralf Korn for alerting me to an error in the original paper [2]. The error concerns the threshold at which the yield curve in an affine short rate model changes from normal (strictly increasing) to humped (endowed with a single maximum). In particular, it is not true that this threshold is the same for the forward curve and for the yield curve, as claimed in [2]. Below, the correct mathematical expression for the threshold is given, supplemented with a selfcontained and corrected proof.
1 Setting

normal if it is a strictly increasing function of \(x\),

inverse if it is a strictly decreasing function of \(x\),

humped if it has exactly one local maximum and no local minimum in \((0,\infty )\).
Condition 1.1
2 Corrections to results
Theorem 3.1 in [2] should be replaced by the following corrected version.
Theorem 2.1
 (1)
The yield curve\(Y(\cdot ,r_{t})\)can only be normal, inverse or humped.
 (2)DefineThe yield curve is normal if\(r_{t} \le b_{\mathrm{y}\text{}\mathrm{norm}}\), humped if\(b_{\mathrm{y}\text{}\mathrm{norm}}< r_{t} < b_{\mathrm{inv}}\), and inverse if\(r_{t} \ge b_{\mathrm{inv}}\).$$\begin{aligned} b_{\mathrm{y}\text{}\mathrm{norm}} &:= \frac{1}{c}\int _{c}^{0} \frac{F(u)  F(c)}{R(u)  1} du, \qquad \\ b_{\mathrm{inv}} &:= \textstyle\begin{cases} \frac{F'(0)}{R'(0)}, &\quad \textit{if}\ R'(0) < 0, \\ +\infty , &\quad \textit{if}\ R'(0) \ge 0. \end{cases}\displaystyle \end{aligned}$$
Remark 2.2
The correction only concerns the expression for \(b_{\mathrm{y} \text{norm}}\), which was called \(b_{\mathrm{norm}}\) in [2] and erroneously given as \(b_{\mathrm{norm}}= F'(c) / R'(c)\). All other parts of the theorem are the same as in [2, Theorem 3.1].
Corollary 3.11 in [2] should be replaced by the following result.
Theorem 2.3
 (1)
The forward curve\(f(\cdot ,r_{t})\)can only be normal, inverse or humped.
 (2)
The forward curve is normal if\(r_{t} \le b_{\mathrm{fw}\text{}\mathrm{norm}}\), humped if\(b_{\mathrm{fw}\text{}\mathrm{norm}} < r_{t} < b_{\mathrm{inv}}\), and inverse if\(r_{t} \ge b_{\mathrm{inv}}\).
Remark 2.4
We have intentionally renamed the result from corollary to theorem, since the correction changes the logical structure of the proof. Note that the above result is equivalent to [2, Corollary 3.11] up to the notational change from \(b_{\mathrm{norm}}\) to \(b_{\mathrm{fw}\text{}\mathrm{norm}}\). Note that \(b_{\mathrm{y}\text{}\mathrm{norm}}\neq b_{\mathrm{fw}\text{}\mathrm{norm}}\) in general, while in [2] it was erroneously claimed that \(b_{\mathrm{y}\text{}\mathrm{norm}}= b_{\mathrm{fw} \text{}\mathrm{norm}}\).
Corollary 3.12 in [2] should be replaced by the following result.
Corollary 2.5
The error also affects [2, Fig. 1], where the expression for \(b_{\mathrm{norm}}\) should be replaced by the correct value of \(b_{\mathrm{y}\text{norm}}\). It also affects the application section [2, Sect. 4], where the values of \(b_{\mathrm{norm}}\) and \(b_{\mathrm{inv}}\) are calculated in different models. The corrections to [2, Sect. 4] are as follows.
Since the resulting expressions are quite involved, we omit the calculations for the extended CIR model [2, Eq. (4.7)].
3 Corrected proofs
 (1)
\(F\) is either strictly convex or linear; the same holds for \(R\). Both functions are continuously differentiable on the interior of their effective domains.
 (2)
The function \(B\) is strictly decreasing with limit \(\lim _{x \to \infty } B(x) = c\).
 (3)
\(F(0) = R(0) = 0\) and \(R'(c) < 0\). In addition, \(F'(0) >0\) if \(D = {[0,\infty )}\).
 (4)Either
 a)
\(D = {[0,\infty )}\), or
 b)
\(D = \mathbb{R}\) and \(R(u) = u/c\) with \(c < 0\).
 a)
 (1)
At least one of \(F\) and \(R\) is strictly convex.
In addition, we introduce the following terminology. Let \(f: (0, \infty ) \to \mathbb{R}\) be a continuous function. The zero set of \(f\) is \(Z := \left \{ x \in (0,\infty ): f(x) = 0\right \} \). The sign sequence of \(Z\) is the sequence of signs \(\left \{ +,\right \} \) that \(f\) takes on the complement of \(Z\), ordered by the natural order on ℝ. For example, the function \(x^{2}  1\) on \((0,\infty )\) has the finite sign sequence \((+)\); the function \(\sin x\) has the infinite sign sequence \((++\cdots )\). An obvious, but important property is the following: Let \(g: (0,\infty ) \to (0,\infty )\) be a positive continuous function. Then \(fg\) has the same zero set and the same sign sequence as \(f\).
Proof of Theorem 2.3
(a) Assume that \(r_{t} \in D = {[0,\infty )}\). By (P2), \(B(x)\) is strictly decreasing, and by (P1′), either \(F'\) or \(R'\) is strictly increasing. Thus if \(r_{t} > 0\), it follows that \(k(x)\) is a strictly decreasing function. If \(r_{t} = 0\), then \(k\) is either strictly decreasing (if \(F'\) is strictly convex) or \(k\) is constant (if \(F\) is linear). By (P1), these are the only possibilities. In addition, the case \(F = 0\) is ruled out by the assumptions.
(b) Assume that \(r_{t} \in D = \mathbb{R}\). In this case, \(R(u) = u/c\), and hence \(R'(u) = 1/c\) is constant and \(F'\) is strictly increasing, by (P1′). We conclude that \(k\) is strictly decreasing.
Proof of Theorem 2.1
Proof of Corollary 2.5
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