Fair value of insurance liabilities A basic example of the assessment of MVM s and replicating portfolio. The following steps will need to be taken to determine the market value of the liabilities: 1. determine per assumption the best estimate (E) parameter 2. determine per assumption an appropriate MVM (except for interest) 3. calculate the liability cash flows under different interest rate scenarios 4. determine the replicating portfolio 5. determine the average NPV of liability cash flows 6. Determine the MVM for interest rate risk elow an example these 6 steps; for 1 and 2, only an example with regard to mortality rates is given. Step 1: determine per assumption the best estimate parameter (only mortality is considered in this example) est estimate assumptions can be determined in many different ways; the relatively simple approach described below is only one of them. First of all, mortality tables are determined for prior years based on the actually observed death rates in those years: one mortality table will be constructed for each of those years. These tables are smoothed within one age over consecutive years and within one year over consecutive ages to eliminate (most of) the volatility included in the observations (required for step 2). For example, at year-end 1999 we could construct mortality tables for the years 1999, 1998, 1997, etc., based on the occurred deaths in those years. Smoothening would take place over the years and between ages to get the tables that we need. Then, for a certain insurance portfolio (for instance the portfolio that exists at year-end 1999), the reserves present value of benefits, expenses and premiums- are calculated for each of these tables. Since they are based on the same set of insurance liabilities (the year-end 1999 portfolio) and on smoothed mortality rates, their change over time reflects the structural mortality development. The resulting graph forms the basis for the determination of the future best estimate mortality rates: Liabilities 97 rates 98 rates Year-end 99 liabilities at actual rates of prior years 99 rates Estimation of trend A A : best estimate trend development (translate into x % mortality improvement per year) asic example 1
The above graph shows what the impact of decreasing mortality rates could be on, for instance, annuity reserves: decreasing mortality rates lead to larger required reserves. ased on this development, the future improvement can be estimated in terms of expected change in reserves one year from now. This change needs to be translated into an expected change in mortality rates. Suppose that the trend would indicate that reserves would need to be 1 % higher at the end of the year 2000. Then, if we would assume, for argument s sake, that reserves would be a linear function of the mortality rates, this could be translated in a mortality improvement of 1 % for all ages for the coming year. This improvement should be taken into account in calculating the reserves (in practice the mortality improvement will be determined for the whole life of the business). The above example calculation is based on a relatively simple model that extrapolates previous years observed mortality rates; more sophisticated models (like, for example, developed in the USA, the UK and The Netherlands) might be appropriate. Step 2: determine per assumption an appropriate MVM (only mortality is considered in this example) The MVM is the price the market would charge for running the uncertainty risk: risk related to misestimating of parameters used in setting the best estimate assumptions. Therefore, the basis of the MVM is likely to be the standard deviation of the prior years reserves relative to the estimated trend. The MVM can then be defined as a multiple of this standard deviation. The multiple reflects the market price per unit of uncertainty. The larger the price per unit of uncertainty or the uncertainty itself, the larger the MVM and visa versa. The multiple also determines what part of the uncertainty will be included in the market value of the reserves. If for example a company would like to be able to withstand adverse developments with a 99,95 % chance over a 1-year time horizon, then (based on a normal distribution) 3,29 times the standard deviations should be held as a buffer above the best estimate reserves. A portion of this buffer is the MVM; the larger the MVM is, the smaller the part that will be held as capital. Suppose that the market would charge 1,3 1 times the standard deviation (90,32 % chance that the liabilities will be sufficient): liabilities A II I A : best estimate trend development (translate into x % mortality improvement per year) A I: 90,32 % confidence interval mortality improvement (translate into y % improvement per year; y > x) A II: 99,95 % confidence interval mortality improvement (translate into z % improvement per year; z > y) 1 Over time, the assessment of an appropriate level will be improved (guidance from IAA, IASC, IAIS, etc. and studies analysing the distribution functions). The 1,3 used here, is based on experience in Canada, where levels between 1 and 1,5 are typically used for these kind of calculations. asic example 2
Similar the translation of the expected change in reserves into a best estimate mortality development in step 1; mortality improvements can be determined suitable to quantify the MVM, as well as the amount of solvency capital that this company would hold based on a 99,95 % confidence interval for this particular risk. Example: the market value of the reserves will not be determined on the basis of the best estimate mortality improvement of 1 % (refer to step 1), but on a higher percentage (for instance 1,5 %) as a result of including the MVM. If this additional margin turns out to be unnecessary over time, it will be released as profit. For solvency purposes, the company could hold an additional amount of capital that would, together with the MVM, add up to the required confidence level as desired by the particular company or the regulators. Step 3: calculate the liability cash flows under different interest rate scenarios (all interest sensitive assumption must depend on the interest rate used). Suppose the interest generator has produced three scenarios for future rates (in practice, thousands of interest rate scenarios will be needed): Year Low rates ase case rates High rates 1999 4.00% 5.00% 6.00% 2000 4.10% 5.10% 6.10% 2001 4.20% 5.20% 6.20% 2002 4.20% 5.30% 6.10% 2003 4.10% 5.40% 6.00% 2004 4.00% 5.50% 6.50% 2005 4.00% 5.60% 6.50% The base case is to represent the current yield curve ased on these rates, the future liability cash flows are projected. (If profit sharing depends on specific asset returns, then the crediting rates must reflect this: the appropriate assets must be modelled simultaneously.) The cash flows that form the basis for the market value of the liabilities only contain the real money flows between policyholder and insurance company; below is an example of such cash flows. As mentioned, all assumptions are set at their best estimate value plus MVM; the interest rate scenario used is the base case as stated above: Year Premium Income Claims Expenses Commission Profit sharing (+) (-) (-) (-) (-) Liability Cash Flow 1999 5,924 4,771 175 217 39 722 2000 5,198 4,536 171 220 37 234 2001 4,956 4,985 166 204 34-433 2002 4,852 5,206 162 188 32-735 2003 4,775 5,288 158 172 29-872 2004 4,267 5,473 153 157 27-1,543 2005 2,414 68,285 122 72 0-66,064 asic example 3
For all three scenarios, this calculation is performed, resulting in the liability cash flows below: Liability cash flows for 3 different interest rate scenarios Year Low rates ase case rates High rates 1999 1,522 722-78 2000 1,034 234-566 2001 367-433 -1,233 2002 65-735 -1,535 2003-72 -872-1,672 2004-743 -1,543-2,343 2005-70,864-66,064-61,264 Positive numbers in this table reflect net cash inflow; while negative numbers are net cash outflows. Higher rates can make it more advantageous for policyholders to lapse their policies, leading to a higher net cash outflow. The opposite is true for lower rates; the minimum guaranteed interest rates provide a valuable asset to the client. Step 4: determine the replicating portfolio Per interest rate scenario, one set of liability cash flows will be generated (see above). ased on an optimisation rule (for instance, to minimise the sum of the annual asset minus liability cash flows over all scenarios), an asset portfolio can be determined that minimises the risk, given the liability and asset behaviour under the different interest rate scenarios (interest sensitivity). The optimisation rule chosen, will define what we mean by risk. The portfolio that minimises the risk, will be defined to be the replicating portfolio. Only assets that are actually available in the market and are consistent with regulatory constraints on investments can be included in the replicating portfolio. A possible additional constraint on the replicating portfolio could be, that it needs to be able to generate a minimum amount of profit sharing in order to remain in business. Given all these restrictions, the assets need to be as risk free as possible. The asset portfolio will, if available in the particular market, contain derivatives to hedge the embedded option. In practice, an almost unlimited number of potential replicating portfolios could be constructed. y using common sense and professional judgement, the number of different assets can be cut down and a limited number of asset portfolios can be designed. For instance, - portfolio I: government bonds up to 10 years, plus certain swaptions - portfolio II: government bonds up to 30 years, no swaptions - portfolio III: government bonds up to 30 years, plus certain swaptions - etc. Suppose that from all assets available we constructed (only) two sets of investment opportunities (A and ) from which we will select the assets that will form the replicating portfolio (in practice more sets will need to be defined). Furthermore, assume that we can construct from both sets of assets only one portfolio that has the potential to become our replicating portfolio (in practice more portfolios will need to be constructed from each set of assets). In determining the potential replicating portfolios, one must keep in mind that the market value of the liabilities that we will eventually determine, must be equal to the market value of the replicating asic example 4
assets. Since the assets in the replicating portfolio have a known (quoted) market value, it is relatively easy to determine their market value. If in step 6 the market value of the liabilities turns out to be lower than this value, the calculations must be repeated with a smaller number of assets. (In this example, we assume that we have already done this calibration.) Next, we determine for each of the two potential portfolios (A and ), the asset cash flow for each of the generated interest scenarios (3 in our case): Asset cash flow from for two possible initial asset portfolios based on above described interest rates A Year Low rates ase case rates High rates Low rates ase case rates High rates 1999 7 43 198 60 143-47 2000 64 71 592 135 34 661 2001 3 458 1,133 56 524 1,133 2002 33 717 1,515 26 717 1,439 2003 272 836 1,698 10 837 1,736 2004 650 1,624 2,499 796 1,602 2,598 2005 70,764 65,854 61,000 70,599 65,749 60,896 In order to find the portfolio that matches our liabilities best, the difference or mismatch between liability and asset cash flows is calculated for each of the two asset portfolios and for each of the three interest rate scenarios: Liability cash flow -/- asset A cash flow Liability cash flow -/- asset cash flow Year Low rates ase case rates High rates Low rates ase case rates High rates 1999 1529 765 120 1582 865-125 2000 1098 305 26 1169 268 95 2001 370 25-100 423 91-100 2002 98-18 -20 91-18 -96 2003 200-36 26-62 -35 64 2004-93 81 156 53 59 255 2005-100 -210-264 -265-315 -368 NPV* 3,207 1,271 532 3,345 1,444 821 * These are the NPV of the absolute difference between asset and liability cash flows The optimization rule defines how we determine the ALM-risk and thus which portfolio we will eventually chose as our replicating portfolio. Suppose this rule would be to: minimize the sum of the discounted absolute values of liability minus asset cash flow over the 7 years and the 3 interest rate scenarios or if one prefers formulae: liab asset s, t s, t A, s= scenario t= year + MIN CF Then asset portfolio A would be chosen as the replicating portfolio (sums of the NPV are 5,009 for portfolio A versus 5,610 for ). Step 5: determine the average NPV of liability cash flows t ( 1 ) disc s, t Next, the different liability cash flows are discounted at the return on the replicating asset portfolio, resulting in one NPV of future liability cash flows per interest rate scenario. The average of all scenarios is defined to be the average NPV liabilities. CF asic example 5
From the (known) market value of the replicating assets and the three asset cash flow patterns, three sets of discount rates can be determined: one for each of the liability cash flows. In case the replicating portfolio does not contain any derivatives and all asset cash flows are insensitive to changes in interest rates (for instance if all investments were done in normal bonds), then under all scenarios the same implicit yield c.q. discount rate would result. From the market value and the asset cash flow, different yield curves could be constructed. The most straightforward way to determine a unique yield curve is probably to deduct a fixed spread from each of the interest rate scenarios until the present value of the asset cash flows equals the known market value of the assets: Implicit asset yield per interest rate scenario Year Low rates ase case rates High rates 1999 5.47% 4.49% 4.52% 2000 5.57% 4.59% 4.62% 2001 5.67% 4.69% 4.72% 2002 5.67% 4.79% 4.62% 2003 5.57% 4.89% 4.52% 2004 5.47% 4.99% 5.02% 2005 5.47% 5.09% 5.02% spread 1.47% -0.51% -1.48% These discount rates reflect the fact that the asset cash flows are sensitive to changes in interest rates. If they were not sensitive, the three discount rate patterns would preferably- be identical (note however, that the above-described procedure, can lead to different asset yields, even if the asset cash flows are identical: the basis - the underlying interest rate scenarios- all have different shapes). If we use these discount rates to determine the NPV of the different liability cash flows, the result is: NPV liabilities per year Year Low rates ase case rates High rates 1999 1,443 691-74 2000 928 214-517 2001 311-377 -1,074 2002 52-610 -1,282 2003-55 -687-1,340 2004-540 -1,152-1,746 2005-48,797-46,678-43,482 TOTAL 46,657 48,598 49,515 average NPV liabilities = 48,257 Step 6: Determine the MVM for interest rate risk The mismatch between liabilities and replicating assets will partly be included in the MV liabilities by means of an MVM for interest uncertainty. The mismatch needs to be capitalised to such an extent that with 99.95 % chance, enough capital is available. This implies a capital of 3.3 times the standard deviation of the mismatch distribution (if normal distributed): 1.3 * σ will be included in the MV liabilities while the remaining 2 * σ will be included in the EC as ALM business risk. The mismatch between assets and liabilities has already been quantified in step 4, where the NPV was determined of the difference between the absolute asset and liability cash flows. asic example 6
In practice, more scenarios will need to be calculated to get a statistically acceptable result. Suppose that besides the above-described three interest scenario, there were 8 others with the following NPV of the absolute differences between asset and liability cash flows (in practice thousands of scenarios will be used): interest rate scenario NPV difference 1 3,207 2 1,271 3 532 4 1,963 5 130 6 1,049 7 41 8 1,065 9 1,008 10 2,162 stdev: 974 The outcomes of the first three scenarios equal those in step 4, asset portfolio A Graphically: NPV of ALM mismatch 4 4 3 3 2 2 NPV difference 1 1 0 500 1,000 1,500 2,000 2,500 3,000 3,500 The MVM for interest uncertainty will be equal to 1.3 * the standard deviation. In our example this is 1,266 Determine the market value of the liabilities & discount rates The market value of the liabilities equals the average NPV of liability cash flows plus MVM for interest uncertainty, or 48,257 + 1,266 = 49,523 (Note: as already mentioned the market value of the liabilities should equal the market value of the replicating portfolio; if it does not, ideally one should go back to step 4. As a shortcut, the asset portfolio can be scaled down.) Finally, the (implicit) discount rates can be derived from the market value of the liabilities and the base case liability cash flow projection. This base case is defined as the projection of the future liability cash flows with all assumptions at the E + MVM, while for interest rates the current yield curve is used. The MVM for interest that is added to the average NPV of liabilities, will thus be asic example 7
translated into a negative adjustment to the discount rates (the higher this MVM, the lower the implied discount rate: Year ase case liability cash flows base case interest rate correction on base case interest corrected disco s value liabilities 1998-722 5.00 % -0.796 % 4.20 % -693.2 1999-234 5.10 % -0.796 % 4.30 % -215.5 2000 433 5.20 % -0.796 % 4.40 % 380.4 2001 735 5.30 % -0.796 % 4.50 % 616.6 2002 872 5.40 % -0.796 % 4.60 % 696.3 2003 1,543 5.50 % -0.796 % 4.70 % 1171.0 2004 66,064 5.60 % -0.796 % 4.80 % 47567.3 49523.0 The derived rates can be interpreted as option-adjusted discount rates. (Note the implied yield on the replicating portfolio will generally be below the current yield curve because of derivatives.) asic example 8