Modelling long term interest rates for pension funds

Modelling long term interest rates for pension funds Michel Vellekoop Netspar and the University of Amsterdam Actuarial and Risk Measures Workshop on Pension Plans and Related Topics University of Piraeus, October 2014 Joint work with Jan de Kort Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 1 / 22

Overview Provisions for funded pension system Inter- and extrapolation problems for long term discounting Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 2 / 22

Overview Provisions for funded pension system Inter- and extrapolation problems for long term discounting Extrapolation using an ultimate forward rate assumption Alternative formulations Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 2 / 22

Overview Provisions for funded pension system Inter- and extrapolation problems for long term discounting Extrapolation using an ultimate forward rate assumption Alternative formulations Conclusions & Future Research Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 2 / 22

Motivation In collective funded pension schemes which provide annuities at retirement, the participants share Interest rate Risk, since price of funding long-term liabilities depends on current term structure in market-consistent approach Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 3 / 22

Motivation In collective funded pension schemes which provide annuities at retirement, the participants share Interest rate Risk, since price of funding long-term liabilities depends on current term structure in market-consistent approach Equity Risk, when proceeds are partially invested in stocks in an attempt to compensate for inflation in pension payments Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 3 / 22

Motivation In collective funded pension schemes which provide annuities at retirement, the participants share Interest rate Risk, since price of funding long-term liabilities depends on current term structure in market-consistent approach Equity Risk, when proceeds are partially invested in stocks in an attempt to compensate for inflation in pension payments Longevity Risk, since expected remaining lifetime at pension age is currently increasing over time. Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 3 / 22

Motivation Risk sharing over different generations makes sense for the first two risks, if we believe that economic cycles may generate lucky and unlucky generations in investments: Pension of older generations is not reduced immediately in bad economic times (dampening of effects of underfunding) Buffers above what is needed for indexation of existing pensions are kept for younger generations (dampening of effects of overfunding) Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 4 / 22

Motivation Risk sharing over different generations makes sense for the first two risks, if we believe that economic cycles may generate lucky and unlucky generations in investments: Pension of older generations is not reduced immediately in bad economic times (dampening of effects of underfunding) Buffers above what is needed for indexation of existing pensions are kept for younger generations (dampening of effects of overfunding) Longevity risk is currently unidirectional and highly correlated across ages so diversifying risk over generations seems less effective. Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 4 / 22

Motivation Pension fund may thus provide a collective risk-sharing contract over generations which cannot be found in the marketplace. Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 5 / 22

Motivation Pension fund may thus provide a collective risk-sharing contract over generations which cannot be found in the marketplace. Agreement on valuation principles essential for fairness of collective schemes: Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 5 / 22

Motivation Pension fund may thus provide a collective risk-sharing contract over generations which cannot be found in the marketplace. Agreement on valuation principles essential for fairness of collective schemes: Being overly optimistic in valuation is beneficial for older participants (and for pension fund managers?) Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 5 / 22

Motivation Pension fund may thus provide a collective risk-sharing contract over generations which cannot be found in the marketplace. Agreement on valuation principles essential for fairness of collective schemes: Being overly optimistic in valuation is beneficial for older participants (and for pension fund managers?) Being overly pessimistic in valuation is beneficial for younger participants (and for regulators?) Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 5 / 22

Motivation Incorporating market information whenever possible is useful in the search for objective criteria. But bond prices and swap rates are not available beyond a certain maximal maturity. Even before that maturity, illiquidity in long-term fixed income products may make informatio on higher maturities considerably less reliable. In times of severe market distress, even shorter maturities may not give consistent information. Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 6 / 22

Motivation Incorporating market information whenever possible is useful in the search for objective criteria. But bond prices and swap rates are not available beyond a certain maximal maturity. Even before that maturity, illiquidity in long-term fixed income products may make informatio on higher maturities considerably less reliable. In times of severe market distress, even shorter maturities may not give consistent information. Concrete subproblem in this talk: how can we use market prices for fixed income products to generate discount curves that extrapolate beyond maturities for which reliable information is available? Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 6 / 22

Approach of European Insurers To generate official discount curves European insurance regulator EIOPA uses information from coupon bonds or swap quotes for maturities up until 20 yrs a given constant asymptotic value (UFR) for forward rates after 60 yrs interpolation (up until maturity 20 yrs) and extrapolation (from 20 to 60 yrs) The UFR (ultimate forward rate) is assumed to be constant although the evidence for this is limited. We propose methods to estimate asymptotic forward rates which are consistent with the methodology proposed by EIOPA but without the assumption that the UFR is constant. Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 7 / 22

Approach of European Insurers To generate official discount curves European insurance regulator EIOPA uses information from coupon bonds or swap quotes for maturities up until 20 yrs a given constant asymptotic value (UFR) for forward rates after 60 yrs interpolation (up until maturity 20 yrs) and extrapolation (from 20 to 60 yrs) The UFR (ultimate forward rate) is assumed to be constant although the evidence for this is limited. We propose methods to estimate asymptotic forward rates which are consistent with the methodology proposed by EIOPA but without the assumption that the UFR is constant. This allows us to check that assumption using unsmoothed market information of liquid tradeable assets without making additional assumptions on model structure. Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 7 / 22

Preliminaries Denote the amount to be paid at a time t 0 to receive a certain single euro at time T t, the zero-coupon bond price, by p(t, T ). Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 8 / 22

Preliminaries Denote the amount to be paid at a time t 0 to receive a certain single euro at time T t, the zero-coupon bond price, by p(t, T ). The continuous-time yield y(t, T ) and forward rate f (t, T ) are then implictly defined by T p(t, T ) = exp( (T t)y(t, T ) ) = exp( f (t, u)du). t Assume given fixed income instruments indexed by i I which pay cashflows c ij at times u j (j J ) and have a current price m i. An interpolating curve p(0, t) must thus satisfy m i = j J c ij p(0, u j ) Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 8 / 22

Schweikert functions as solution to interpolation problem Interpolating curve is chosen by EIOPA to have the form p(0, t) = (1 + g(t))e f t, g(t) = j J η j W (t, u j ) Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 9 / 22

Schweikert functions as solution to interpolation problem Interpolating curve is chosen by EIOPA to have the form p(0, t) = (1 + g(t))e f t, g(t) = j J η j W (t, u j ) with f the UFR, (η j ) j J appropriately chosen weights and W the exponential tension spline base functions (Schweikert, 1994) which are also called Smith-Wilson functions (Smith & Wilson, 2001): W (t, u) = α min(t, u) 1 2 e α t u + 1 2 e α(t+u) Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 9 / 22

Schweikert functions as solution to interpolation problem Interpolating curve is chosen by EIOPA to have the form p(0, t) = (1 + g(t))e f t, g(t) = j J η j W (t, u j ) with f the UFR, (η j ) j J appropriately chosen weights and W the exponential tension spline base functions (Schweikert, 1994) which are also called Smith-Wilson functions (Smith & Wilson, 2001): W (t, u) = α min(t, u) 1 2 e α t u + 1 2 e α(t+u) Lemma For given u > 0 the function W (t, u) is the only twice continuously differentiable solution to 2 t W (t, u) = α 2 W (t, u) α 3 min(t, u) (1) which is zero for t = 0 and has a finite limit for t. Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 9 / 22

Covariance structure and Existence of Inverse Functions W are, according to EIOPA, related to covariance function of integrated Ornstein-Uhlenbeck processes but they do not match exactly (Andersson & Lindholm, 2013). In fact Proposition Let Z t be a standard Brownian Motion and L t be an Ornstein Uhlenbeck process L t = t 0 e α(t s) dv s with V a Brownian Motion independent of Z and α > 0 a given constant. Then W (t, u) = α cov(z t + L t, Z u L u ). Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 10 / 22

Covariance structure and Existence of Inverse Functions W are, according to EIOPA, related to covariance function of integrated Ornstein-Uhlenbeck processes but they do not match exactly (Andersson & Lindholm, 2013). In fact Proposition Let Z t be a standard Brownian Motion and L t be an Ornstein Uhlenbeck process L t = t 0 e α(t s) dv s with V a Brownian Motion independent of Z and α > 0 a given constant. Then W (t, u) = α cov(z t + L t, Z u L u ). This characterizes W as a covariance between two different processes. Also equals (auto)covariance process of a single Gaussian process on any finite interval [0, T ]. Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 10 / 22

Covariance structure and Existence of Inverse Proposition For a given T > 0 and α > 0 let (z n ) n N be the countably infinite number of solutions to the equation and define z 3 tan(zαt ) + (1 + z 2 ) 3 2 tanh((1 + z 2 ) 3 2 αt ) = 1 sin(αtz n ) ψ n (t) = z n cos(αtz n ) + sinh(αt 1 + z 1 + zn 2 n 2 ) cosh(αt 1 + zn 2 ) X t = 1 α n=0 ψ n (t)ɛ n ψ n 2 z n 1 + z 2 n with (ɛ n ) n N iid standard Gaussian. Then X has covariance function W on domain [0, T ] i.e. E(X t X u ) = W (t, u). Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 11 / 22

Covariance structure and Existence of Inverse Proposition For a given T > 0 and α > 0 let (z n ) n N be the countably infinite number of solutions to the equation and define z 3 tan(zαt ) + (1 + z 2 ) 3 2 tanh((1 + z 2 ) 3 2 αt ) = 1 sin(αtz n ) ψ n (t) = z n cos(αtz n ) + sinh(αt 1 + z 1 + zn 2 n 2 ) cosh(αt 1 + zn 2 ) X t = 1 α n=0 ψ n (t)ɛ n ψ n 2 z n 1 + z 2 n with (ɛ n ) n N iid standard Gaussian. Then X has covariance function W on domain [0, T ] i.e. E(X t X u ) = W (t, u). Corollary: matrix with elements w ij = W (u i, u j ) is invertible. Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 11 / 22

Constrained variational problem Idea Smith & Wilson: interpolating discount curves p(0, t) = (1 + g(t))e f t, g(t) = j J η j W (t, u j ) should be required to be sufficiently smooth. Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 12 / 22

Constrained variational problem Idea Smith & Wilson: interpolating discount curves p(0, t) = (1 + g(t))e f t, g(t) = j J η j W (t, u j ) should be required to be sufficiently smooth. Functions W are solutions to variational problem min L[g], L[g] := g F 0 on the Sobolev space 0 [g (s) 2 + α 2 g (s) 2 ] ds F a = {g C 2 (R + ) : g(0) = a, g E, g E} E = {g L 2 (R + ) : lim g(t) = 0}. t and η = C T ( CW C T ) 1 m, with C ij = c ij e f u j and m i = m i j J C ij. Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 12 / 22

Unconstrained variational problem Our proposal: remove asymptotic constraint on forward rate but keep the optimization criterion as before. This picks the ultimate forward rate which creates minimal tension in discount curves. Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 13 / 22

Unconstrained variational problem Our proposal: remove asymptotic constraint on forward rate but keep the optimization criterion as before. This picks the ultimate forward rate which creates minimal tension in discount curves. We thus solve min f min g H f L[g] on space H f = { g C 2 (R + ) g (0) = 0, j J C ij g(u j ) = m i, for all i I } Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 13 / 22

Unconstrained variational problem Theorem The optimized ultimate forward rate f = f solves ( ) (m CD f e) T (CD f WD f C T ) 1 CD f U e + WD f C T (CD f WD f C T ) 1 (m CD f e) = 0 with W ij = W (u i, u j ), D f ij = e f u j 1 {i=j}, U ij = u j 1 {i=j}, e i = 1. If the cashflow matrix C is invertible this simplifies to (u j π j e f u j ) W 1 jk (π k e f u k 1) = 0 with π = C 1 m. j J k J A = @(D) (C*D*W*D*C ); goal = @(D) ((m-c*d*e) )*(A(D)\(C*D*diag(u)*(e+(W*D*C )*(A(D)\(m-C*D*e))))); ufr = fzero(@(f) goal(diag(exp(-f*u))), 0.02), Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 14 / 22

Example: Euro Swap Rates, 2 Jan 2001 0.06 02.01.01 0.07 0.055 0.06 Yield 0.05 0.05 Forward Rates 0.045 0 10 20 30 40 50 60 0.04 Maturity Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 15 / 22

Example: Dutch regulator curve, 31 March 2013 0.05 31.03.13 0.05 0.04 0.04 Yield 0.03 0.02 0.03 0.02 Forward Rates 0.01 0.01 0 0 10 20 30 40 50 60 0 Maturity Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 16 / 22

Alternative formulation on different function space Last example shows that smoothest convergence discount curve does not translate into smoothest convergence forward rate. Given UFR philosophy that would be more natural criterion. Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 17 / 22

Alternative formulation on different function space Last example shows that smoothest convergence discount curve does not translate into smoothest convergence forward rate. Given UFR philosophy that would be more natural criterion. Our alternative formulation is therefore in terms of forward rates g min L[g] g H on function space { H = g C 2 (R + ) g(0) = a, g (0) = lim t g (t) = 0, j J c ij e u j 0 g(s) ds = m i, for i = 1,..., J }. Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 17 / 22

Alternative formulation on different function space Last example shows that smoothest convergence discount curve does not translate into smoothest convergence forward rate. Given UFR philosophy that would be more natural criterion. Our alternative formulation is therefore in terms of forward rates g min L[g] g H on function space { H = g C 2 (R + ) g(0) = a, g (0) = lim t g (t) = 0, j J c ij e u j 0 g(s) ds = m i, for i = 1,..., J }. Notice that we assume that short rate g(0) is observed. It can be estimated during the optimization as well. Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 17 / 22

Solution Theorem A solution of this problem must take the form g(t) = g(0) + ζ i c ij π j H(t, u j ), i I j J ( αt cosh(αu) 1 H(t, u) = 1 e 1 + 1 {t u} 2 α2 u 2 with the (ζ i ) i I and (π j ) j J solving the equations m i = j J c ij π j, ln π k = g(0)u k + i I cosh(α(u t)) 1 1 2 α2 (u t) 2 1 2 α2 u 2 ) uk ζ i π j c ij H(s, u j )ds Functions H start at H(0, u) = 0 and converge to lim t H(t, u) = 1 with 1 2 H(0, u) = 0. They are smoother than W. j J 0 Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 18 / 22

Example: Dutch regulator curve, 31 March 2013 0.05 31.03.13 0.05 0.05 31.03.13 0.05 0.04 0.04 0.04 0.04 Yield 0.03 0.02 0.03 0.02 Forward Rates Yield 0.03 0.02 0.03 0.02 Forward Rates 0.01 0.01 0.01 0.01 0 0 10 20 30 40 50 60 0 Maturity 0 0 10 20 30 40 50 60 0 Maturity Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 19 / 22

Solution The UFR follows directly from the optimization. Denote by y(u k ) = ln p(0, u k )/u k the yield for maturity u k, and let y(u 0 ) := y(0) be the short rate. Theorem If the cashflow matrix is invertible then f = The coefficients (v k ) equal n v k y(u k ) k=0 v k = n j=1 G 1 jk, v 0 = 1 n v k, k=1 G kj = 1 uk H(s, u j )ds. u k 0 Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 20 / 22

Example: Euro Swap Rates 2001-2007 Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 21 / 22

References Anderssson, H. and Lindholm, M. (2013). On the relation between the Smith-Wilson method and integrated Ornstein-Uhlenbeck processes. Research Report 2013-01 Mathematical Statistics, Stockholm University. EIOPA (2013). Technical Findings on the Long-Term Guarantees Assessment. Published 14 June 2013. Smith, A. and Wilson, T. (2000) Fitting Yield Curves with Long Term Constraints. Bacon & Woodrow Research Notes. Schweikert, D.G. (1994) An interpolation curve using a spline in tension. J. Math. and Physics 45, 312-317. Michel Vellekoop (University of Amsterdam) Long term rates for Pension Funds Piraeus, October 2014 22 / 22