Pension funds are continuously seeking to match the duration of their liabilities to the duration of their assets in order to immunise themselves against the effects of changes in interest rates. This has become even more important due to restrictions imposed by governments with regard to interest rate risk and minimum funding ratios. For this reason, many pension funds are looking for long-dated bonds to extend the total duration of their portfolio. At the same time pension funds are also investing in equities, real estate and other assets to enhance returns. Unfortunately, these investments cannot be used to match liabilities as equities and real estate do not have any commonly accepted duration determination method.

The duration of assets other than bonds is difficult to measure because the value of these assets is not solely dependent on interest rates. Therefore, some have argued that it might even be pointless to investigate the matter. Nevertheless, there have been many attempts to devise some sort of

duration measure for equities and to a lesser extent property, with outcomes ranging from low durations of two to three years to more than 50 years.

In this article we are concerned with the duration of real estate. This asset class is interesting

from a duration perspective because property investments produce fairly stable cash flow

patterns (annual/monthly rents), albeit with some variation by sector and region. In particular we

are interested in the duration of closed-end real estate funds, as closed-end funds have a fixed

lifetime.

he cash flow scheme of real estate is rather similar to that of a bond. When invested, the asset will generate a cash flow during the lifetime of the fund and in the end a value is returned. The difference, however, is that rents (the periodical cash flows) are generally increased by inflation and sometimes adjusted for market rent movements. Also, the capital value, and therefore end value, is changing during the lifetime.

The final payout is therefore nominal invested capital adjusted for capital appreciation. This deviates from the general government bond cash flow pattern which yields constant payments and has a fixed final repayment (the principal). It deviates less, however, from a diversified set of credits or CMBS structures that can go into default and have an uncertain loss due to default probabilities and therefore an uncertain repayment. Interestingly, it is not uncommon to use duration measures for these assets.

he uncertainty surrounding the cash flows can be modelled by using a simulator that copies the working of a fund. As a first step towards this we have set up a simulation model which aims to mimic as closely as possible the evolution of a closed-end property fund.

Within this model, a number of features are assumed stochastic, ie: the parameters are liable to insecurity. These factors are size of properties in the portfolio, initial yield, inflation, lease expiries, changes in market rent and vacancy rate with many more features available if required. The number of properties in the fund and the lifetime can be set as required.

The actual input values will depend on the

country and/or sector that the fund covers, setting, for example, larger standard deviations for

office property than for retail property on many variables.

To keep things manageable, we have limited ourselves for the moment to the Dutch property market, for which a large amount of data is available.

Every run or trial during the simulation will result in a cash flow pattern determined by the interaction of our key variables and calibrated by historical patterns of individual property and market returns. For instance, rental income is dependent on inflation correction, market rental value changes and vacancy rates. Each of these variables follows

a certain statistical distribution. These are estimated by looking at the statistics of

historical property time series and cross-sectional data.

By the latter we mean that single properties do not necessarily follow the market average, while a large portfolio of properties will. Therefore, we have had to make some assumptions about the

possible bandwidth of the value of each variable

on an annual basis. Data on the Dutch markets is supplied to us by the ROZ/IPD benchmarking organisation, giving us the average value and cross-sectional standard deviation as well as

different interval levels, such as the 5% and 10% highest and lowest values.

We can see each run of the model as a possible realisation of the property environment we have created. Having simulated a stream of cash flows for the entire life of the fund, we can use the formal calculation method for Macauley Duration:

With t: year

n: lifetime of fund

C: annual payment (rental income)

TV: repayment/final value

i: discount factor

he discount rate is the most difficult factor to determine. For bonds, yield-to-maturity is used, typically the interest rate which applies at the moment. Therefore, the denominator in the

equation is the actual value of the bond, as it describes the formula to calculate bond value.

For real estate funds, several methods can be considered. One option is to use the expected IRR of the fund. Discounting with the expected IRR will bring us on average back to the initial investment value.

The second option would be to use the risk-free bond rate corresponding to the life of the fund and add a risk premium to that rate. The risk premium will typically be dependent on sector, region, property quality, and so on. This would allow for a high degree of subjectivity, which might not be preferable. However, this approach would provide a clear theoretical explanation for why value is not only determined by interest rates, as it is possible that a rise in interest rate will be offset by an unrelated decline in risk premium. Discounting with this factor could, however, lead to present values of the investment that are different from the actual investment value.

Whichever discount rate is used, each

simulation run yields a value for the Macauley Duration calculation. This is only one outcome out of many different possibilities.

The strength of the simulation process lies in the fact that it will produce a whole set of duration outcomes, enabling us to make inferences about the likely value of a fund's duration and to determine upper and lower bounds. At the very least these boundaries tell us that that duration is not zero. A somewhat bolder remark would be that duration will be close to the average simulated value.

here are many points of criticism that can be made towards this approach. The most fundamental concerns whether a duration measure can be applied to real estate. For bonds the duration gives a close approximation of the sensitivity of the price of a bond to interest rate changes. To calculate this effect, we need a slightly adjusted duration measure, the Modified Duration, which is defined as Macauley Duration divided by (1+i). For small changes in interest rate, the following equation holds:

With MD: Modified duration

B: Bond price

i: Yield-to-maturity

The equation basically states that the Modified Duration of a bond multiplied by the interest

rate change gives an approximation of the

price change of the bond due to the interest rate change.

This equality does not hold for real estate as the price function of a real estate fund (if it could be determined) is not simply a discounted cash flow formula, but seems to be dependent on many factors. We do not exactly know what the influence is of the interest rate in that real estate price function. Therefore, the modified duration measure gives no close approximation of the real estate price change when interest rates change. This means that real estate duration is less fit to match liability duration for interest rate immunization purposes.

hile the practical applications remain open for discussion, it is interesting to see what the Macauley Duration of a typical Dutch closed-end fund might be. We have simulated two virtual closed-end funds consisting of 10 properties of varying size. One is a core residential fund and the second is a core office fund. These alternatives have been chosen because they have historically shown very different performance characteristics in the Dutch market.

Office funds generally show a large spread in size/value of different properties, have a relatively high yield and a large spread in yields over different properties. Furthermore, office properties are much more dynamic when it comes to vacancy rates than properties in other sectors.

Residential properties have a low spread in property size/value, have low yields and low vacancy as well as low rental growth volatility both cross-sectional and over time.

Average yields, rental value growth, vacancy rate, and so on have all been set according to present market conditions. Variation over time

has been determined using historical series, while variation between properties has been set

using the cross-sectional data from ROZ/IPD. Both funds will be held and remain closed for two time periods; for 10 years and for 20 years. We have chosen to use the risk premium discount factor approach.

The discount rate has been set to 6.5%, assuming a risk premium of approximately 2.5%. Arguably, the discount rate could be different for different sectors, but for simplicity we have decided to stick with one factor.

For each different setup we have performed 10,000 simulations, calculating the average

duration and the bottom and top 5% duration.

The outcomes of these simulations are shown in the table in the next column.

The differences between the office and residential fund seem plausible. We would expect the duration of the residential fund to be somewhat higher because it has a lower yield than the office fund and therefore lower cash flows in the first years, shifting the duration balance somewhat to the right. Moreover, we would expect the bandwidth of durations to be larger for the office fund because it exhibits more volatility.

This approach can be used for any market that has sufficient data to estimate the statistical parameters of the model. It also allows for testing the effect of many different fund characteristics, for instance the effect of leverage.

uration matching has become an important topic in today's risk-controlled pension market. Whereas the emphasis has always been put on bond duration, many people argue that other assets must also have a duration. The closed-end real estate funds discussed in this article suffer from uncertainty in their cash flow patterns.

However, this feature has not stopped analysts from determining a duration for commercial mortgage-backed securitisation structures and credits. One way to handle the uncertainty in cash flows is to use a fund simulator. Generating thousands of possible cash flow scenarios, this simulator calculates lower and upper bounds for real estate fund duration.

While the applicability of the real estate duration measure is something for further discussion, this duration seems far from zero, which is in fact the value appointed to these investments by some pension fund regulators.

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