Replication and Relative Value

Previously we have discussed valuation of bonds and stocks: primarily how they can be modelled through decomposing instruments into sums over individual cashflows. Whilst analysing individual instruments is useful, more importantly this allows us to apply the same approach over sets of instruments; e.g. to rank instruments to research profitable investment opportunities or as part of managing risk exposure. This post looks at how one can replicate underlying cashflows in general and uses examples to illustrate the approach and explain why this problem requires both data aggregation and fast analytics capabilities.

Why Replicate?

The simplest way to obtain a stock price is to look it up on a financial news application. But, the information is limited: it tells you the price someone else has paid for it, which may not be representative of the stock’s underlying value. Comparing two stocks from the price alone is not meaningful; without the number of shares, the price per share does not even tell you the fraction of company ownership it buys.

Bond comparisons are subject to similar considerations and, as such, have similar solutions. Calculations like yield-to-maturity (ytm) or the tax adjusted yield-to-maturity allow sensible comparisons of bonds with different maturities. Similar metrics are used for this type of comparison with Stocks; e.g. price-to-earnings, debt-to-equity or earnings-per-share. Indeed, as stocks are more complicated financial instruments than bonds, the metrics used to analyse them are more numerous and considerably more varied than for bonds. Ultimately, each fails to capture some aspect of the relationship between a company and its stock.

In the same way that the price fails to allow comparison between stocks, simple metrics do not allow comparison between stocks and bonds; there is no price-to-earnings measure for a bond, though the dividend yield of a stock is somewhat analogous to a bond’s yield. Separately, none of these metrics are derived from the wider economic environment; whilst it seems evident that a change in bank lending rates should impact on a company with high debt-to-equity this does not immediately impact on metrics based on reported performance.

As financial engineering evolved, it was recognised that heuristic analytics often lead to “apples and oranges” analysis and a method providing sytematic comparison was needed. Recognising that all financial instruments grant rights to the ownership of cash or assets in some form and that they can be decomposed in terms of such underlying “atoms”, a method based on replication of the total value of a financial contract in terms of the sum of its underlying parts was developed¹.

Replication

Replication² provides a framework for valuing an instrument as the sum of prices of its individual constituents.

For example, the purchase of a stock traded on NYSE at price $X for a UK SIPP is comprised of

1. The purchase of $X at price £Y, followed by
2. The purchase of 1 unit of stock on NYSE for $X.

i.e. a currency exchange at spot rate of X/Y to convert GBP to USD is required in addition to the purchase of the stock.

Breaking an instrument down into constituents makes the assumptions used in valuation easier to surface and standardise: e.g. the same rate can (and should) be used to discount cashflows from dividends (stocks) and coupons (bonds) back to today, thereby making them comparable with other cash amounts received on different dates.

The accuracy of replication does not need to be perfect. For instance, a fund manager might construct a small set of stocks that track a target index (e.g. S&P500) to within some error tolerance to reduce the cost of rebalancing that portfolio and thereby be able to offer lower fees or reduce trading costs.

Replication detail also varies. A company might be analysed purely at the level of its dividends, as if it were a perpetual bond. Alternatively, a more detailed model suitable for analysis of a company investing in a new product area might use company information to include issued corporate bonds, their repayment schedules and how those match with expected growth in company revenue to see whether the investments are paying off, subject to some assumptions.

Assumptions: Uncertainty and Implied Probabilities.

If we lookup prices for stocks and use those to value our portfolio, we make the assumption that we can sell our stocks at those prices. If we only check the price daily, we assume that the movements between any two consecutive days will be small. Assumptions drive our investment choices and hence making good ones can significantly improve our returns.

Suppose one buys government bonds, should one worry about the government actually making a coupon payment on the dates the bond states? Probably not; when a government can

• for all intents and puroposes - print money, it is vanishingly unlikely not to make a payment. However, an investor in government debt needs to consider how much is being issued; too much and there is a risk that the value of the payments will decrease over time through one or both of currency devaluation and inflation.

Conversely, corporates have different risks: a company cannot print money and can default on its bonds if it does not have sufficient cash. For stocks, a company might choose to suspend a dividend, or pay dividends at a different frequency or unexpected amounts.

The dependency on how likely events are to occur naturally affects the value of cashflows. In the same way that (with all else equal) a dollar today is preferable to one tomorrow, a riskier payment is less desirable than a certain one.

In replicating instruments these risks should be accounted for. The key difficulty is how to estimate the probabilities to use. The usual two answers apply: historical estimates or implied observable values. In the same way that discount factors, implied from bond or futures, relate a future cashflow back to today prices can be used to infer probabilities of events.

For example, the price of a call option is its expectation. From a set of option prices, the underlying probability and cumulative density functions can be inferred³. The future distribution of variables such as the USDGBP FX rate can then be calculated and used in comparing (say) a US vs a UK bond for investment returns that are to be received in GBP.

$$E[V(k)] = \int \max(x-k,0)f(x)dx\;\;\;\text{\small (option)}$$ $$F_X(x)=P(X\le x)\;\;\;\;\;\text{\small (cdf)}$$ $$f(x)=\frac{dF(x)}{dx}\;\;\;\;\;\;\;\;\;\text{\small(pdf)}$$

Replication also enables a principled approach to risk analysis as the perturbation of the inputs to a model; its sensitivity to model assumptions and parameters.

Limitations

Whilst extremely useful, replication has limitations;

• Parameter estimation
  • if data is unavailable they are often set to an acceptable default, which may be unrepresentative of real-world behaviour.
• Assets are not always fungible (interchangeable) at low cost.
  • Where cash can be moved easily between (most) currencies, commodities require physical storage on delivery.
  • An oil future is a financial instrument but on settlement a physical delivery contract means actually receiving and storing (many) barrels of oil.
• Intangible Value
  • e.g. the value of brand value to a company is subjective and varies according to the individual
  • e.g. capitalisation of investment in software; there is not a 1:1 relationship to return.
• Transaction costs
  • buying and selling stocks is effectively instantaneous and costs a few basis points, where buying/selling a house may take a couple of months and cost a couple of percent of its value in fees to lawyers, estate agents and land registry.

An Example: Purchasing vs Renting Property.

To aid in gaining an intuitive grasp of replication, we will apply it to a common adult decision: is it economically preferable to purchase or rent?

How might you model prices for these flats independently of an estate-agent's?

Assumptions and Model

Firstly, we will use replication rather than the route of heuristic analytics such as price per square metre. We will model the values of purchasing a property and renting it in terms of the cashflows we would need to pursue either choice. The starting position is assumed to be the same; e.g. in making the choice to rent, the deposit the property purchase would require is invested instead.

Secondly, we are choosing to ignore valuation of items such as distance from schools, transport links and neighbourhood amenities as these are arguably intangibles.

Thirdly, to value the underlying cashflows we need the following assumptions:

• The mortgage period (time to maturity) is years.
• The target house price is £.
• % of the house price is placed as deposit, equivalently £12,500.
• Investments earn % per year and are reinvested.
• The mortgage rate is % on £237,500 principal, implying (£1,603.03) per month.
• Equivalent rental cost (yield) for a £250000 property is %, (£1,000) per month.
• The implied rental-mortgage payment gap of £603.03 per month is invested by the renter.
• House price growth over 20 years of makes the house value £500,000 at maturity.

These assumptions feed into our model for the costs in the two cases.

Rental: Costs over 20 years

Item	Frequency	Credit or (Debit)	Periods	Item Value
Rent	Monthly	(£1,000)	240	(£240,000)
Investment Interest	Monthly	£50.93 to £138.15	240	£20,666.22
Investment Principal	-	£12,500		£12,500
Investment of Mtg/Rent Gap	Monthly	£603.03	240	£246,314.18
TOTAL				£39,480.41

Purchase: Costs over 20 years

Item	Frequency	Credit or (Debit)	Periods	Item Value
Mortgage Payment	Monthly	(£1,603.03)	240	(£384,728.31)
Asset Value	-	£500,000		£500,000
TOTAL				£115,271.69

Suburbs/Centre Rental Costs

To account for the suburbs/centre difference additional assumptions are required:

• Cost of underground per year: £.
• Cost of rail/external travel per year: £.
• Rental discount percentage vs city: %.

The value of travel time, whether seen as a positive benefit that allows (say) reading, or a negative where time is lost standing is highly subjective and is ignored as intangible.

Item	Frequency	Credit or (Debit)	Periods	Item Value
Travel Difference	Annual	£4,000	20	£80,000
Rental Difference	Monthly	(£330)	240	(£79,200)
TOTAL				£800

Analysing Our Model

What can we conclude from our examples?

1. Comparing with previous examples on bonds, the monthly mortgage, rent and investment items behave exactly like the coupon and principal for bond valuation; they are calculated the same way. This is replication; reduction to common components.

2. For the rental vs mortgage question the default assumptions suggest that buying is preferable to renting. However, that conclusion is sensitive to a change in the expected house price growth. If growth was flat (x1) then this conclusion reverses. Conversely, we can see that the choice of where to rent probably does not matter much as savings on rent are lost in travel cost: this is roughly what we would expect in an efficient market. The examples illustrate how sensitivity to changes in model inputs can be used to assess stability of conclusions drawn from the model.

3. Even with a simple model there are four variables for expected returns or costs; mortgage, investment, rental and house price growth rates. This assumes they are constant over time and ignores the huge variability in investment returns depending on choice of asset. The input rates can be obtained from bank lending values on a given or returns to a proxy index such as the FTSE or S&P.

4. Careful observers will have realised that replication ignored discounting of future cash. Whilst technically wrong, this effect would be be dominated by inflation which is also not modelled. We also ignore the effect of fixing rates for only part of the term and then remortgaging in a different environment. Extending a model to handle these differences makes it considerably more complicated.

5. Whilst the example taken was mortgages and rent, it could as easily have been valuation of a solar farm, wind infrastructure or home renovation project. The difficulty in doing so is making obtaining the data with standardised units easy.

6. The limitations on writing more complex models are in managing the rapid increase in data requirements and the manipulation and addressing of that data by the analytics. This is very hard to achieve reliably in a simple script (e.g. the ~100 lines of JS used here) or spreadsheet. For instance, there is no validation on the input ranges in the logic above, nor checking of units in calculations (which makes adding up incompatible amounts easy).

To summarise, we have taken the abstract concepts of valuing stocks and bonds through replication and applied the same principles to a familiar example from daily life to illustrate the benefit of a systematic, principled approach to making investment decisions and to describe the difficulties one encounters along the way.