Stochastic asset models for actuarial use Part II Dr Douglas - PDF document

Lecture 4 Lecture 5 Lecture 6 Stochastic asset models for actuarial use – Part II Dr Douglas Wright, Cass Business School Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Table of contents Lecture 4 1 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns Lecture 5 2 Dynamic ALM using the Wilkie model ARCH and regime-switching models for asset returns Economic-based models Lecture 6 3 Using SAMs to price investment guarantees Using SAMs to determine strategic asset allocation Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II

Lecture 4 Modelling asset returns Lecture 5 Wilkie model for equity returns Lecture 6 Wilkie model for bond returns Modelling asset returns (1) We now consider modelling asset returns , and will focus only on the two main asset classes for most institutional investors: domestic equities, and domestic long-dated fixed-interest government bonds. Because one of the main uses of stochastic asset models is to assist in financial planning, we are likely to want to project the cash flows arising from the assets (rather than just the total return achieved). Thus, we must consider income and capital gain components separately. Then, the total (gross) return over the year is given by: total return = asset price at end of year + income received over year − 1 asset price at start of year For fixed-interest bonds , the income component is fixed (and volatility in return comes from changes in capital value) – i.e. a change in GRY. Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Modelling asset returns Lecture 5 Wilkie model for equity returns Lecture 6 Wilkie model for bond returns Modelling asset returns (2) We define the following key asset variables: Y ( t ) is the equity dividend yield at time t , K ( t ) is the force of equity dividend growth in year ( t − 1 , t ), and C ( t ) is the (running) yield on undated fixed interest bonds at time t we will also refer to this is the “long-term interest rate” Note that, for simplicity, the Wilkie model assumes that fixed-interest bonds are undated (rather than redeemed at some specified future date). Then, level of dividend income on equities at time t , D ( t ), is given by: D ( t ) = D ( t − 1) × exp [ K ( t )] Without loss of generality, we assume that D (0) = 1 . 0 – as D ( t ) is an index , the absolute value at time 0 is not important. Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II

Lecture 4 Modelling asset returns Lecture 5 Wilkie model for equity returns Lecture 6 Wilkie model for bond returns Modelling asset returns (3) Define P ( t ) as the level of the share price index at time t . Then, by definition, we have: Y ( t ) = D ( t ) P ( t ) ⇒ P ( t ) = D ( t ) Y ( t ) Assuming equity dividends are payable in arrears and ignoring taxation of dividend income, the gross return on equities over the year ( t − 1 , t ) is: PR ( t ) = P ( t ) + D ( t ) − 1 L P ( t − 1) Define PR ( t ) to be the accumulation at time t of an initial investment of 1 made at time 0 in equities (assuming re-investment of dividend income). Then, we have: PR ( t )] = PR ( t − 1) × P ( t ) + D ( t ) PR ( t ) = PR ( t − 1) × [1 + L P ( t − 1) Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Modelling asset returns Lecture 5 Wilkie model for equity returns Lecture 6 Wilkie model for bond returns Modelling asset returns (4) Without loss of generality, assume that the annual income from an undated fixed-interest bond is 1.0. Then, the (running) yield on this investment at time t is given by: C ( t ) = income at time t 1 ⇒ price at time t = price at time t C ( t ) Assuming bond income is payable in arrears and ignoring taxation of bond income, the gross return on fixed-interest bonds over the year ( t − 1 , t ) is: 1 C ( t ) + 1 [ ] 1 CR ( t ) = − 1 = C ( t − 1) × C ( t ) + 1 − 1 L 1 C ( t − 1) Define CR ( t ) to be the accumulation at time t of an initial investment of 1 made at time 0 in fixed-interest bonds. Then, we have: [ 1 ] CR ( t ) = CR ( t − 1) × [1 + L CR ( t )] = CR ( t ) × C ( t − 1) × C ( t ) + 1 Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II

Lecture 4 Modelling asset returns Lecture 5 Wilkie model for equity returns Lecture 6 Wilkie model for bond returns Wilkie model for equity returns (1) Wilkie model has a hierarchical (or cascade ) structure. force of price inflation � � � � � equity dividend yield � � � � � � � � � � force of equity dividend growth yield on long-dated FIBs Figure 25: Cascade structure for Wilkie model Then, price inflation is the primary process driving the others. Also, for reasons discussed below, equity dividend yield influences both force of equity dividend growth and long-term interest rate . Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Modelling asset returns Lecture 5 Wilkie model for equity returns Lecture 6 Wilkie model for bond returns Wilkie model for equity returns (2) ��� ��� ������������������������������ ��������������������������� ��� ��� �� �� �� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ��� ���� Figure 26: Annual force of price inflation, I ( t ) and equity dividend yield, Y ( t ), in UK, 1923-2009 Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II

Lecture 4 Modelling asset returns Lecture 5 Wilkie model for equity returns Lecture 6 Wilkie model for bond returns Wilkie model for equity returns (3) Thus, equity dividend yield is much more stable than price inflation. There is some evidence of a positive correlation between the two variables (indeed, the correlation coefficient is c = 0 . 60). However, is is crucial to note that, in particular, it is a LARGE increase in price inflation (e.g. 1940 and 1974) that seems to lead to LARGE contemporaneous increases in the equity dividend yield. How can this relationship be explained? This illustrates one the key problems with time series modelling: there does not appear to be a simple linear relationship between these variables ideally, we want a system which allows for a positive correlation between the variables if there is a large change in I ( t ), but with little or no relationship otherwise but, such relationships cannot easily be modelled or parameterised Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Modelling asset returns Lecture 5 Wilkie model for equity returns Lecture 6 Wilkie model for bond returns Wilkie model for equity returns (4) We require that Y ( t ) is positive ⇒ use log transform. Then, the model for the equity dividend yield at time t , Y ( t ), is: ln [ Y ( t )] = YW × I ( t ) + ln YMU + YN ( t ) where Y ( t ) is the equity dividend yield at time t I ( t ) is the force of price inflation over the year ( t − 1 , t ) YN ( t ) = YA × YN ( t − 1) + YE ( t ) is an AR(1) process independent of price inflation YE ( t ) = YSD × YZ ( t ) is the random component of the equity dividend yield process at time t YSD is the (constant) standard deviation parameter of the equity dividend yield process YZ ( t ) is a standard N (0 , 1) random variable Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II

Lecture 4 Modelling asset returns Lecture 5 Wilkie model for equity returns Lecture 6 Wilkie model for bond returns Wilkie model for equity returns (5) Then, the (log) equity dividend yield is modelled as: a stationary AR(1) process (around a mean of ln YMU ) plus a direct influence from the current level of price inflation Then, fitting to 1923-2009 data gives parameter estimates as follows: YW = 1 . 55 YMU = 0 . 0375 YA = 0 . 63 YSD = 0 . 155 As expected, we have YW > 0 – implying a positive correlation between the current levels of price inflation and equity dividend yield. Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Modelling asset returns Lecture 5 Wilkie model for equity returns Lecture 6 Wilkie model for bond returns Wilkie model for equity returns (6) ��� ��� ��� �� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ������������������������������ ������������������������������������� ���� ���� Figure 27: Annual force of price inflation, I ( t ) and force of equity growth, K ( t ), in UK, 1923-2009 Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II