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Course 1, Semester 2, 2008 Assignment: Asset allocation of a pension fund Developing a yield curve model 1. The covariance matrix V of the monthly yield movements along the swap curve during the data period may be calculated as (see ‘Covariance’ tab of spreadsheet): 180 day 2 year 3 year 4 year 5 year 7 year 10 year 180 day 5.8% 5.9% 5.2% 4.8% 4.4% 3.8% 3.2% 2 year 5.9% 8.6% 8.4% 8.1% 7.7% 7.0% 6.5% 3 year 5.2% 8.4% 8.6% 8.5% 8.1% 7.6% 7.1% 4 year 4.8% 8.1% 8.5% 8.4% 8.1% 7.7% 7.3% 5 year 4.4% 7.7% 8.1% 8.1% 8.0% 7.6% 7.3% 7 year 3.8% 7.0% 7.6% 7.7% 7.6% 7.5% 7.4% 10 year 3.2% 6.5% 7.1% 7.3% 7.3% 7.4% 7.4% [1 mark] The process set in Appendix B of the assignment may be applied (see ‘Eigenvalue’ tab) to give the three largest eigenvalues as: 0.492 0.042 0.007 These have the following eigenvectors: [5 marks] It may be seen from the chart above that the eigenvector, associated with the largest eigenvalue (0.492), gives fairly constant weight to all terms along the yield curve, and this may be regarded as a ‘shift’ factor. Similarly the second largest eigenvalue (0.042) has an eigenvector of approximately linear shape, and this may be regarded as a ‘tilt’ factor. The third largest eigenvalue (0.007) has an eigenvector of a humped shape, and may be regarded as a ‘curvature’ factor. [1 mark] The sum of the largest three eigenvalues is 0.541. The total variance in yield movements along the yield curve (i.e. the sum of the diagonal elements in the matrix V is 0.542, which is equal to the sum of all eigenvalues). Thus the three factors referred to above explain over 99% of historical yield movements. [1 mark] By construction, the three factors are statistically independent, and hence each may be modelled as an independent normal distribution with a variance equal to its associated eigenvalue. Thus the 3×1 vector x for the three factors can be simulated. A model of yield movements r at all points on the yield curve may be obtained as r = Wx , where W is the 7 × 3 matrix of eigenvectors, set out under the chart above. [1 mark] The above model of yield curve movements may be converted into a model for the yield curve simply by summing yield movements over time, starting from the current yield curve. This assumes that the current yield curve is an unbiased predictor of future yields, which is reasonable. [The average historical monthly yield movement during the data period was not zero: 180 day 2 year 3 year 4 year 5 year 7 year 10 year Average 0.004 -0.008 -0.012 -0.015 -0.017 -0.019 -0.021 but this has no particular significance for modelling future yields.] [1 mark] 2. Investment in bank bills with a flxed/floating swap differs from investment in a government bond only in terms of credit risk. Hence swap yields may be expected generally to exceed bond yields by a credit margin. If this margin is constant, then yield movements on both swaps and bonds should be similar. [1 mark] This can be verified by looking at the relationship between the yield movements on 10 year swaps ys and bonds yb : ys = 0.0028 + 0.9811 yb . 47.42 0.51 (t-statistics underneath parameter estimate) with R 2 = 93% . On this basis swap and bond yield movements are hardly distinguishable. [1 mark] 3. Suppose P0 , P denote the values of a debt portfolio at the start and 1 end of a period respectively, and i0 ,i1 denote the yields on the portfolio at those times. Then the return on the portfolio is the sum of its income and capital gain components: P − P0 return = i0 + 1 P0 [1 mark] However if only i0 , i1 are known, along with the duration of the portfolio at start of the period, it is convenient to approximate the return by regarding the change in yield (and thus capital value) to take place at start of year, followed by an income return during the period of i1 . This gives the following approximation: return = (% change in capital value)× (1 + i1 ) − 1 (*) = [1 − (i1 − i0 ) ⋅ duration ](1 + i1 ) − 1 Strictly speaking, it is modified duration that should be used in the formula above. [1 mark] If convexity of the portfolio at start of year is also known, the approximation can be improved to: return = (% change in capital value)× (1 + i1 ) − 1 ⎡ = ⎢1 − (i1 − i0 ) ⋅ duration + (i1 − i0 )2 convexity⎤(1 + i ) − 1 ⎥ 1 ⎣ 2 ⎦ [1 mark] 4. Using the 10 year bond yields, it is possible to estimate monthly returns on a portfolio of any reasonably long duration using the formula in (*) in [3]. The duration can be chosen so as to minimise the squared differences between such estimated returns and those of the UBS Australia Composite Bond 10+ index. This minimisation can be accomplished using SOLVER (see the ‘swap rates’ tab), giving a duration of approximately 7.7 years and the following fit: [1 mark] An average duration on the index of 7.7 years is reasonable, given that coupons on a 10 year bond comprise a significant component of its cash flows. For example, the duration of a 10 year bond with coupon 6% at a yield of 6% is 7.8 years. [1 mark] An asset model 5. Using the eigenvectors in [1] to determine the shift, tilt and curvature factors, it is straightforward to compute the annualised historical covariances of the financial variables (see ‘model’ tab): AEQ Shift Tilt Curve CPI AEQ 1.341% -0.020% -0.004% -0.006% -0.013% Shift -0.020% 0.059% 0.000% 0.000% -0.003% Tilt -0.004% 0.000% 0.005% 0.000% 0.001% Curve -0.006% 0.000% 0.000% 0.001% 0.000% CPI -0.013% -0.003% 0.001% 0.000% 0.014% The annualised covariances are found as 12 times the monthly ones, assuming independence of returns between months. It is evident that there is zero covariance for the shift, tilt and curvature factors, confirming their construction. [1 mark] For CPI, which is given only quarterly, a constant CPI increase over a quarter is taken. The calculated monthly covariance is then multiplied by a factor of 3 to allow for the ‘smoothing’ over a quarter. [1 mark] It may be noticed that the estimated volatility for AEQ is only 11.5% pa. This reflects the shortness of the data period (from only 1994 onwards, omitting the 1987 crash). In practice a fuller investigation would be made, resulting in more realistic volatility of up to 20% pa. However the historical figure will be used for the remainder of this report. [1 mark] 6. Note for markers: stability can be approached in many different ways; several tests for stationarity or homoscedascity can be applied.] A plot of the errors (i.e. deviations from historical mean) in the variables is as follows: [1 mark] This suggests that the magnitude of the errors is likely to be stationary, except possibly for AEQ. A more formal test is that of Dickey-Fuller 1 , whereby the following regression is undertaken: Δε t = η + δε t −1 + z t This gives the following results: Dickey Fuller AEQ Shift Tilt Curve CPI δ -1.100 -0.963 -0.876 -1.001 -0.274 std error 0.078 0.079 0.074 0.081 0.055 t-statistic -14.153 -12.188 -11.827 -12.391 -4.986 from which it can be concluded that δ < 0 and that the individual errors are stationary. This test can be generalised to examine stationarity of covariances as follows: ε t = η + Dε t −1 + z t . The matrix D is estimated as: -0.098 -0.178 -3.009 4.174 -1.046 0.022 0.042 0.317 -1.249 -0.359 -0.005 0.043 0.121 0.076 0.000 -0.006 0.012 0.012 -0.047 0.033 -0.002 0.024 0.031 0.178 0.720 1 http://en.wikipedia.org/wiki/Dickey-Fuller_test whose eigenvalues are contained within [− 1,1] - this can be shown using the Power Method. Thus there is stationarity of the error structure as a whole. [1 mark] Nonetheless these tests of stationarity are specific to the data period. Inclusion of AEQ data for earlier periods is very likely to change the results. 7. Expectations of the variables may be assessed as follows: The current 10 year bond yield is 6.5% pa (compared to the current 10 year swap yield of 7.3% pa) and this provides the best estimate of a forward return expectation for AFI. The rest of the current bond yield curve is taken to have the same (inverted) shape as the current swap yield curve. [1 mark] The shift, tilt and curvature factors may be expected to have zero mean in the long term. Thus the long term yield curve is on average the same as the current one. This assumes that the current yield curve is the best predictor of future ones [1 mark] Current inflation levels exceed 4% pa, but it may be expected that it will moderate in future towards the RBA’s target. Thus 3.5% pa seems to be a reasonable assumption. [1 mark] An Equity Risk Premium of 5% pa, relative to debt, can be justified on current PE levels of about 14x at the present time, which happens to be the same as long term historical levels. This is relative to a long term historical ERP of 5-6%. Thus a long term expectation of AEQ returns of 11.5% pa is assumed. [1 mark] 8. There are various ways to derive a Cholesky matrix for the covariance matrix. Possibly the simplest is to use SOLVER to solve for a triangular Cholesky matrix. Alternatively the Cholesky- Banachiewicz 2 algorithm may be applied to derive a triangular matrix as follows (see ‘Cholesky’ tab): 2 http://www.answers.com/topic/cholesky-decomposition?cat=technology 11.579% 0.000% 0.000% 0.000% 0.000% -0.171% 2.424% 0.000% 0.000% 0.000% -0.030% -0.002% 0.713% 0.000% 0.000% -0.048% -0.003% -0.002% 0.282% 0.000% -0.114% -0.118% 0.115% 0.154% 1.173% : [1 mark] If z is a 5× 1 standard normal variable with unit variance matrix, then the covariance of Cz is: cov(Cz ) = E (Czz T CT ) = CE (zz T )CT = CICT = V Hence Cz simulates the variables that have a covariance matrix of V. [1 mark] Application of the asset model 9. The shift, tilt and curvature factors allow the yield curve to be simulated as in [1]. From these simulations: The cash rate can be taken as the lower end of the yield curve [1 mark] The AFI return can be taken as that on a debt portfolio of duration (and convexity) equal to that of the liabilities as in [3]. [1 mark] 10. At any time the liabilities are assessed as the present value of future claim payments, using a discount rate consistent with the duration of the liabilities and the yield curve applicable at that time. [1 mark] Allowance is made for future CPI increases at a fixed rate of 3.5%, consistent with current inflation expectations. This is reasonable as the volatility of CPI is limited.. [1 mark] The discount rate is determined as follows: An estimate is made of duration of liabilities, based on the discount rate applicable in the previous period; A yield on the current yield curve is determined based on this estimated duration; A revised estimated of duration is based on this current yield [2 marks] 11. The Trustee is concerned with a deficiency at any time in the future, as this may need to be funded by an external sponsor (or benefits reduced). It is the absolute amount of this deficiency that is relevant. The deficiency in relation to liabilities (i.e. as a solvency ratio) is not so relevant, as clearly the liabilities will reduce over time. A measure of downside risk may thus be taken as the maximum expected deficiency over the lifetime of the Fund, in present value terms. It would be reasonable to discount deficiencies in future years at the current bond yield, that is the downside risk measure could be taken as { t [ Downside = E min 1.065−t ( Assett − Liabilityt ) ]} [1 mark] The Trustee may to some extent be concerned about the upside of investment strategy. A future surplus may simply used regarded as a buffer against future experience. In addition surplus arising towards the end of the Fund’s lifetime may be applied to increase benefits (though this is not the primary concern of the Trustee). An upside measure could be taken as the discounted value of terminal surplus: { } Upside = E 1.065−ω ( Assetω − Liabilityω ) where ω = 35 years is the lifetime of the fund. [1 mark] [Note to markers: this is clearly a very subjective issue, and marks should be awarded for reasonable alternatives.] 12. For simplicity, only allocations to AEQ and AFI are considered. Though an allocation to cash is possible, it is clear that AFI dominates cash in the sense of matching liabilities. If a normal current yield curve is assumed (unlike the assumption in [7]), then AFI would dominate AFI completely. [1 mark for this or similar comment about cash] For various allocations to AEQ/AFI, 1000 simulations of the Downside and Upside measures in [11] were made to provide the following results: Downside Upside AEQ % ($000) ($000) 10% -114 -46 20% -116 17 30% -119 104 40% -126 196 50% -132 332 60% -142 474 70% -143 655 80% -153 850 90% -165 1,121 100% -189 1,471 [2 marks] The trade-off between Downside and Upside is approximately linear, as shown below: Thus every reduction of $1 in the Downside measure may be expected to be rewarded with $20 of Upside. [1 mark] The course of action in these circumstances should be to quantify precisely the Trustee’s risk/return preferences. It may, however, be presumed that these will be skewed heavily towards Downside 3 , so that an allocation to AEQ of less than 30% would likely be the final outcome. [1 mark] 13. Some shortcomings that may be observed in the asset model and its application are as follows: The investor objectives are concerned only with the maximum extent of deficiencies, and not with their frequency. The long term distribution of the yield curve is very much driven by its assumed starting level. It would be desirable to have a yield curve model that allows for its long term level to differ from its starting level, and for faster reversion to it (for example with serial correlation of yield movements). It would also be desirable to incorporate into the model other information about long term bond yields, e.g. futures prices. A yield was chosen for valuing liabilities in such a way that its position on the yield curve equals the duration of the liabilities (calculated at that yield). However the yield curve is itself a series of yields at which bonds for various maturities are 3 i.e. the Trustee’s indifference curve is very curvaceous traded: the duration of those bonds are not equal to their maturities. It would be more appropriate for a zero-coupon yield curve to be modelled for this purpose Inflationary expectations are a fundamental variable that impacts on how real liabilities are assessed. In the asset model such expectations are assumed fixed at 3.5% pa, which is unrealistic. It may be more appropriate to model real discount rates, rather than nominal ones. Fund deficiencies are particularly sensitive to non-stationary volatility of AEQ returns, as may be observed in historical data. A GARCH or other non-stationary process could be assumed for the AEQ error structure. [1 mark per point, up to 4 marks] Implementation 14. An immunised debt portfolio may be regarded as a passive means of investment, as many of the decisions to be made are dictated by the liabilities. However, there are still decisions to be made about the structure and parameters of the portfolio, and the following issues need to be addressed. Manager Selection Credit rating and liquidity. The manager must have expertise and systems for trading in all types of debt markets. This may require an international presence To the extent that hedging with interest rate and currency risks is required, experience in these areas is desirable. The manager must be able to respond quickly to instructions for changes in the portfolio structure. Are appropriate protocols in place? Are managers sufficiently different in style to warrant multiple appointments? Are the management fees competitive? [1 mark] Permitted investments It must be decided what securities are permitted in terms of: Credit rating and liquidity country and currency allocations Derivatives Other debt investment vehicles (e.g. CDOs) Security selection How manager discretions for security selection to be exercised: [1 mark] By use of credit analysis By reference to external bond benchmarks [1 mark] Rebalancing Does the manager have a process in place for: ensuring liquidity adjusting duration and other portfolio characteristics. [1 mark] Controls and monitoring The manager mandate should cover: Controls on portfolio duration, convexity Cash, country, credit, currency limits Limits on derivative exposure Frequency of reporting Consequences for breaches of mandate [1 mark] [Total 50 marks]