Institutional Investments & ALM: Case

Author: Hidde Hovenkamp (2541936)

May 27, 2015

1 Question 1

We are asked to look at cash flows and liabilities for a pension fund. There are cash flows running from maturity 1

until maturity 60. Using the zero rate for each maturity we can calculate the present value of the liabilities per year.

Summing this up we get a total present value of the liabilities of 14,518,902,923 . Figure 1 shows the distribution

of the present value of the expected pension payments over the years.

Figure 1: Present value of expected pension payments

Next, we calculate the modified duration for cash flow with maturity i as follows:

MDt =t

(1 + yo,t)

Uing this modified duration we can now calculate the BPV for each liability using the following formula:

BPVt =MD · PV

10, 000

Summing up the BPV’s of all liabilities equals 24,347,751. To hedge the interest rate off the balance sheet we have

to hedge the BPV of the assets. Under 110% funding ratio, this amounts to 26,782,526 .

Let us now look at 10 year, 15 year and 30 year swaps. Based on the zero rates we find the following swap rates:

k10 = 2.20%, k15 = 2.43% and k30 = 2.71%. We can use these swaps to hedge the interest rate off the balance sheet

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by matching the BPV of assets as previously calculated. We do so by hedging the interest rate risk coming from

liabilities from 1 to 10 years maturity with the 10 year swap, from 10 to 15 years maturity with the 15 years swap

and all other maturities (15-60) with the 30 years swap. Using this approach, we need a 10 year swap with notional

3,120,623,892, a 15 year swap with notional 2,579,438,972 and a 30 year swap with notional 10,420,614,064.

This hedge works very well for all buckets between 1 and 30 years maturity, but the liabilities with maturities

form 30 to 60 years cannot be hedged by these swaps. If we take the situation where all zero rates between 30 and

60 years drop with 0.3%, we see a value increase of the liabilities of 245,697,275. The value of the assets does not

change however because this drop of the zero rates does not afffect the swap rates for lower maturities. So we can

see that the hedge does not work very well for changes in the swap rates at the very long end of the curve. We

would need an extra swap with for example 60 years maturity.

2 Question 2

Now, let us perform a small ALM study for a pension fund. We do not have to worry about contributions, the

nominal funding ratio is 110% and we have a constant inflation rate of 1.5% per year. The indexation policy is as

follows: full indexation if the funding ratio is higher than 115% and otherwise no indexation.

For our portfolio we can choose from several asset classes: nominal matching portfolio, global equities and

cash. The nominal matching portfolio is constructed using zero coupon bonds that exactly match the the nominal

liabilities. Therefore, the 1 year expected return is equal to the 1-year zero rate (1.636%) and the 5-year expected

return to the 5-year zero rate (1.82%). For global equities we have used the return data on Global Equity Fund

index to find the returns and volatility and further we assume returns and volatility are lower on a longer horizon

(e.g. mean reverting). For cash we take the average deposit rate of the ECB over the last couple of years (about

0.2%). We also assume the lowest volatility of all three assets, as we think cash is the least risky asset. We assume

correlations with cash and other assets to be zero and correlation between global equities and the nominal matching

portfolio to be -0.4 (as suggested in the literature). All values for expected returns, volatility and correlations

between assets are show in table 1 and 2.

Asset class Return Volatility Corr. Global Equity Corr. Cash Corr. NMP

Global Equity 0.08 0.12 - 0 -0.4Cash 0.002 0.0006 0 - 0Nominal matching portfolio 0.016 0.08 -0.4 0 -

Table 1: Proposed parameters for 1-year returns, volatility and correlations

Asset class Return Volatility Corr. Global Equity Corr. Cash Corr. NMP

Global Equity 0.04 0.06 - 0 -0.4Cash 0.001 0.0004 0 - 0Nominal matching portfolio 0.0182 0.04 -0.4 0 -

Table 2: Proposed parameters for 5-year returns, volatility and correlations

With these proposed parameters we can now simulate 1,000 scenarios of the three asset classes. We use a

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multivariate normal distribution and simulate the values in MATLAB. With these scenarios we are interested in

three different investment strategies for the pension fund: 100% nominal matching portfolio, 50% nominal matching

portfolio and 50% global equities and 80% nominal matching portfolio, 40% global equities and -20% cash. Let us

now look at the average norminal funding ratio, funding ratio risk (defined as the probability of funding ratio below

100%) and indexation.

1-year horizon 5-year horizonStrat. 1 Strat. 2 Strat. 3 Strat. 1 Strat. 2 Strat. 3

Average nominal funding ratio 115% 118% 118% 115% 116% 115%Funding ratio risk (below 100%) 6.1% 0.1% 0.5% 0.2% 0.0% 0.0%Average indexation 0.7% 1.0% 0.9% 0.7% 0.9% 0.9%

Table 3: Results for three different investment strategies based on simulation of 1,000 scenarios.

Based on the results shown in table 3 I would advise strategy 2 to the Board of this pension fund. For the

1-year invesment horizon we see the lowest funding ratio risk, highest average nominal funding ratio and the highest

average indexation over all scenarios. For the 5-year investment horizon the average nonimal funding ratios are

slightly lower as well as the average indexation, but this comes with a lower risk of the funding ratio going below

100%. The results for the 5-year horizon support the conclusion to go for strategy 2. The reason strategy 2 works

well probably has to with the fact that it diversifies between equity and bonds, which are negatively correlated.

Therefore, it can create a higher return but partially offsets the risk through this diversification.

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