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Developing a yield curve model

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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]

				
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