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Investment strategies, lazy evaluation and memoization

This article will cover an interesting problem: given a set of possible investments, each with different tax rates, yearly rates and minimum time until withdrawal, what is the best investment strategy for the next 10, 20 or n years?

For instance, given the following investments:

  • i_1 = 9\% yearly rate, 25% taxes on profits upon withdrawal 1 year later
  • i_2 = 8\% yearly rate, 15% taxes on profits upon withdrawal 5 years later
  • i_3 = 7\% yearly rate, 0% taxes and withdrawal 3 years later

If we want to maximize earnings over 10 years, should we purchase i_1 ten times, i_2 twice, i_3 three times and one i_1, i_3 once and i_2 once and i_1 twice or some other combination?

Before we go into programming, let’s do some basic math/algorithms. This is all simple math, so don’t worry. You can also skip to programming if you prefer.

Theorem 1: The final value to withdraw with any investment i such as the ones exemplified can be written as product of the initial value and a factor defined by the investment i, e(i) = 1 + (1 - taxes) \cdot ((1 + yearlyRate)^{time} - 1)

Proof: Given some initial value v and composite yearly interest rates r, investment time t and taxes on profits taxes:
profitsBeforeTaxes = v \cdot (1 + r)^t - v = v \cdot ((1 + r)^t - 1)
profitsAfterTaxes = profitsBeforeTaxes \cdot (1 - taxes)
profitsAfterTaxes = v \cdot ((1 + r)^t - 1) \cdot (1 - taxes)

finalValue = v + profitsAfterTaxes = v + v \cdot ((1 + r)^t - 1) \cdot (1 - taxes)
finalValue = v \cdot (1 + ((1 + r)^t - 1) \cdot (1 - taxes))

The previous line proves the existence of the factor. Now, since finalValue=v \cdot e(i):

e(i) = finalValue / v = 1 + (1 - taxes) \cdot ((1 + r)^t - 1)

Theorem 2: Given a set of possible investments and a deadline n, the best investment strategy s_n=e(i_1) \cdot e(i_2) \cdot ... \cdot e(i_j), any sub-strategy contained in s_n is the best strategy for the sum of the times of the investments contained in it.

Proof: Without loss of generality, let us consider s_n=s_a \cdot s_b, with a+b=n and 0<a<n and assume the contrary: s_a is not the best strategy for time a, but s_n is the best strategy for time n. So there must be s'_a > s_a, and that would mean s'_n = s'_a \cdot s_b > s_n, which contradicts s_n being the best strategy. Therefore, there can be no s'_a > s_a and s_a is optimal.


Maybe you skipped the last part, but don’t worry. I’ll just roll out the recursive solution to the problem. How do we describe the list of investments to be made that maximizes earnings after some time n?

s_0 = 1
s_n = max \{s_1 \cdot s_{n-1}, s_2 \cdot s_{n-2}, ..., s_{n-1} \cdot s_1, i \}\text{, with } i =\text{ investment with largest }e(i)\text{ of all possible investments of } time=n

This basically means we test every possible combination, which is not very smart, of course. The advantage of finding a recursive solution is that we can compute and store calculations for use later on. This is what we call memoization.

Also, we only need to check up to s_{floor(n/2)} \cdot s_{n - floor(n/2)}, since every other check is redundant.

How do we implement this? In a procedural language we could use an array of size n and fill it up with all solutions from 0 to n. In Haskell we generally don’t want to use mutable data structures nor do we want to specify the order of evaluation of things, so we must find another tool in the toolbox to do this; it turns out that lazy evaluation is just that!

Lazy evaluation is, roughly speaking, a mechanism by which a value is only computed when required by some function. This means that we can define a data structure in terms of itself, and that it can even be infinite. Take the following example:

repeat :: a -> [a]
repeat x = x : repeat x

What happens here is that the function repeat takes an object of some type a and returns a possibly infinite list of a. The list will grow in size as more elements of it are demanded by evaluation. Let us take this idea to implement our bestStrategyFunctionBad:

— We use the “investment” below to make sure the algorithm always returns some strategy, even if it means leaving you money in the bank
investmentLeaveInTheBank :: Investment
investmentLeaveInTheBank = Investment { name = “Leave it in the bank account”, rate = 0, taxes = 0.0, time = 1 }
withMax :: Ord a => (b -> a) -> [b] -> Maybe b
withMax f xs = snd maybeRes
  where maybeRes = foldl’ (\acc el ->case acc of
    Just (maxVal, maxEl) -> let cmp = f el in if cmp > maxVal then Just (cmp, el) else acc
    Nothing -> Just (f el, el)) Nothing xs

withMax1 :: Ord a => (b -> a) -> b -> [b] -> b
withMax1 f firstEl xs = snd $ foldl’ (\acc@(maxVal, _) el -> let cmp = f el in if cmp > maxVal then (cmp, el) else acc) (f firstEl, firstEl) xs
bestStrategyBad :: Int -> [Investment] -> [Investment]
bestStrategyBad timeInYears invs’ = go !! timeInYears
  where invs = investmentLeaveInTheBank : invs’
              factorStrategyBad is = product $ fmap factorInvestment is
              bestStrat desiredTime = withMax1 factorStrategyBad (maybeToList (bestInvestmentWithTime desiredTime)) (allCombinations desiredTime)
              bestInvestmentWithTime desiredTime = withMax factorInvestment $ filter (\i -> time i == desiredTime) invs
              — For desiredTime=7 “allCombinations” returns strategies e1 ++ e6, e2 ++ e5 and e3 ++ e4
              allCombinations desiredTime = let halfTheTime = floor (fromIntegral desiredTime / 2)
                                                                       in fmap (\i -> go !! i ++ go !! (desiredTime – i)) [1..halfTheTime]
              go :: [[Investment]]
              go = [] : fmap bestStrat [1..]

There is nothing magical about the code above. When demanding \text{go !! 20}, for instance, the function bestStrat will be called with the value 20, which will demand all possible strategy investments (as defined by our equations). Demanding all combinations will once again require \text{go !! 19}, \text{go !! 18} and many others, which will repeat the process for a smaller n (the fact that they are smaller is crucial for our recursion to converge).

What is different from recursion in imperative languages is that go is not a function: it is a list whose values are lazily calculated. As values are demanded from it, they are calculated only once, so you don’t have to worry about what order to evaluate things in. In C# this is sort of like a List<Lazy<Investment[]>>.

This is nice! Still, there are two bad things about this solution:

1. go is a list, so accessing \text{go !! n} is O(n). If this were an array this would be better. We will not tackle this issue for now, but feel free to do so!
2. We are creating a large number of lists with the (++) function, not to mention that once we combine two strategies we have to go through every investment in the combined strategy to calculate its complete factor, when we could do better.
So now let’s go and solve issue number 2.

More lazy evaluation \text{and a little abstraction}

So, how can we solve issue number 2?
Combining two strategies leads to a strategy with a factor that is the product of the factors of each strategy. There is no need to concatenate lists to discover the best strategy of some given size. To avoid needless work, we need more lazy evaluation. Let’s add some functions to our code and create the bestStrategyGood function:

data StrategyCalc = StrategyCalc [Investment] Double

factorStrategyGood (StrategyCalc _ x) = x

combine :: StrategyCalc -> StrategyCalc -> StrategyCalc
combine (StrategyCalc s1 f1) (StrategyCalc s2 f2) = StrategyCalc (s1 ++ s2) (f1 * f2)

bestStrategyGood :: Int -> [Investment] -> [Investment]
bestStrategyGood timeInYears invs’ = let StrategyCalc res _ = go !! timeInYears in res
  where invs = investmentLeaveInTheBank : invs’
              bestStrat desiredTime = withMax1 factorStrategyGood (bestInvestmentWithTimeOr1 desiredTime) (allCombinations desiredTime)
             bestInvestmentWithTimeOr1 desiredTime = case withMax factorInvestment $ filter (\i -> time i == desiredTime) invs of
Nothing -> StrategyCalc [] 1
Just i -> StrategyCalc [i] (factorInvestment i)
             — For desiredTime=7 “allCombinations” returns strategies e1 ++ e6, e2 ++ e5 and e3 ++ e4
             allCombinations desiredTime = let halfTheTime = floor (fromIntegral desiredTime / 2) in fmap (\i -> combine (go !! i) (go !! (desiredTime – i))) [1..halfTheTime]
             go :: [StrategyCalc]
             go = StrategyCalc [] 1 : fmap bestStrat [1..]

Take your time to digest this: the list of investments in each StrategyCalc will only be evaluated when the caller needs it to be evaluated. However, the combine function will create a StrategyCalc whose factor is calculated in constant time when combining two strategies. In fact, you could even have the final factor of the optimal strategy without having ever constructed a non empty list. Nice!

A little abstraction \text{(skip to results if you prefer)}

I thought a nice touch to finish this article would be to introduce an abstraction: the Monoid.

A Monoid is just a fancy name for a binary operation that is associative and a value that is an identity for this operation. The Int type, the sum function (+) and the value 0 (zero) form an instance of Monoid, for instance, since any number plus zero equals itself and (a+b)+c=a+(b+c) for any a, b, c of type Int.

The same thing happens with investment strategies! So we can replace the combine function by the Monoidal append:

instance Monoid StrategyCalc where
  mempty = StrategyCalc [] 1
  mappend (StrategyCalc i1 f1) (StrategyCalc i2 f2) = StrategyCalc (i1 ++ i2) (f1 * f2)

Don’t forget that <> is an infix alias for mappend!


When taking the code for bestStrategyGood and the three investments from the beginning of the article, let us devise the best strategy to maximize gains over the next 11 years:

$ ghci
$ :l Investments.hs
ghci> let availableInvestments = [ Investment { name = “Investment 1”, rate = 0.09, taxes = 0.25, time = 1 } , Investment { name = “Investment 2”, rate = 0.08, taxes = 0.15, time = 5 } , Investment { name = “Investment 3”, rate = 0.07, taxes = 0, time = 3 } ]
ghci> fmap name $ bestStrategyGood 11 availableInvestments
[“Investment 3″,”Investment 3″,”Investment 2”]

So it seems that buying investment 3, rebuying it and then buying investment 2 is the best strategy in this case.

That’s it! I hope you liked it, and if it helps, do know that this problem is still solvable with the same algorithm if the tax of each investment is a function of the amount of time since the investment title was purchased and if the time until withdrawal is either an exact time or a minimum time. It is also possible to include inflation-correcting investments if you pass around some estimated inflation; all of this with only minor modifications. Also, feel free to change the time unit to months and get something much more precise for your investments!

One equality operator to rule them all

In a lot of programming languages there are two different concepts for equality: Reference equality (also called Identity) and Value equality.

Reference equality is the same as equality of memory addresses. Two objects are the same if they are the same instance (iif they occupy the same address in memory).

Value equality is semantic equality. Two objects are equal if the values in them mean the same thing. For example: two instances of IPAddress that represent the address “” are equal even if they have been instantiated in different places.

My plan with this article is to convince you that there should only be one equality operator. Let me make my point.

First: There should be a null-safe equality operator.*

This means that a language should have some means of comparing objects which won’t throw when a null reference gets compared. For most languages this is the == operator. The Equals instance method is problematic because you have to null check your object first (and you will end up forgetting to do so eventually).

*: It would be much better if we removed null references from all modern languages for good, but let’s leave it at that for now.

Second: Boxing and two kinds of equality are a dangerous mix.

This is very clear when we realize that 1 == 1 but (object)1 != (object)1. This is just a problem waiting to happen, really.

*: Of course, you should really rethink your code if you’re operating on boxed objects. At least in C#, creating generic (as in polymorphic) code is almost certainly the better approach.

Third: In many cases, reference equality is simply misleading.

In C#, for instance, IPAddress.Parse(“”) != IPAddress.Parse(“”). Any type that conveys meaning will be thought of according to that assigned meaning. Unless you’re tracking instances, reference equality is far more confusing than it is useful.

Fourth: More generally and more importantly, having the same operator mean two different things is dangerous.

Why? Well, what does == really mean?

== means Maybe Reference Equality, Maybe Value Equality. == can mean two distinct things depending on what gets compared, but with one unique form/appearance.

And this can’t be good. + is addition for numbers and concatenating for strings, but the signatures are so different that it is hard to get it wrong. What would you think of + if it were addition for integral number types such as Int and Long but meant addition with rounding for fractionals such as Float and Double (we would make a proper Add method available, obviously)? I’m sure you would think it is an error waiting to happen.

All previous cases would be more or less solved if we created a clear separation of equality operators. Kotlin, for example, has == for value equality and === for reference equality. I would like to convince you, however, that we only need one kind of equality for general use, and that reference equality should only be used to implement this equality operator that I propose. For that, let us explore a more interesting case:


An interesting case: the Socket class

When working with multiple Sockets, it is often necessary to compare them to one another. Now, two sockets connected to the same address in the same port aren’t equal because they are independent communication channels, so wouldn’t value equality spoil that?

No, it wouldn’t. In fact these two sockets use different local ports. Even if the sockets aren’t connected nor have been modified yet, the operating system labels all these sockets to manage them internally. Of course, the operating system could use the memory address of the created socket internally, which would then be the socket’s label.

But even in that case, we only need one equality operator. It just happens to be the case that it may be implemented by comparing memory addresses in this case. Sockets just happen to be a case where it is desirable to make equality be Reference Equality. It is a case where semantics matches identity. So in our implementation of == for Socket we could then use object.ReferenceEquals. So do note that Reference equality would only be used to properly implement our beloved equality operator.

Ok.. These examples are good, but aren’t there exceptions?

I believe not. Although of course I can’t provide a formal proof, there is one thing I do know: there is only one kind of equality in Haskell, and Haskell has been used a lot already. I wouldn’t be surprised if other languages do the same.

How do we fix this in C# ?

Sadly, I don’t believe it is possible without changing the language substantially and breaking existing code. What I propose below is the least invasive set of changes I can think of for the language, and it is certainly not pretty. There are better ways around this problem.*

Proposed breaking change: We could make the == operator non overridable and make its implementation always use the Equals instance method of its left-most non null argument (after proper null checks).

One could argue that choosing the left-most argument would make this implementation asymmetric. However, any correct implementation for equality is necessarily symmetric (if a == b then b == a), so it wouldn’t matter if the implementation picked the left-most argument to call Equals on it. Also, we could go ahead and change == to a generic ==<T>(T a, T b) to make this a non-issue. This would break even more code and I’m not sure how all of this would sound to language designers, though.
Also, there is some concern over efficiency. Calling a virtual method like Equals would involve dynamic dispatch, bringing its associated costs with it. I believe this is unavoidable.*

*: Unless we are willing to go even further with our breaking changes. There is a proposal to implement type classes in C# ( Type classes enable a high level of ad-hoc polymorphism while maintaining static dispatch. They are a great feature and one I vouch for strongly. If you’re curious, read my article on it.
With type classes, we could make == an operator/function of some Equatable<T> type class. We would then make its default implementation like the one described above, while still being overridable so that we can override it for some types to avoid calling Equals.


So.. what are your thoughts on this? Do you have counter-examples or do you disagree on some issue? Can you think of more unintended consequences by changing the language as I proposed (I’m sure there are many!)?


EDIT (2018-01-06 14:40): I’m not sure how I missed this, but the new feature of nullable reference types makes my proposed language changes mostly unnecessary. It is also a breaking change, of course, but not only does it improve the language significantly, it also improves the state of equality. After opting in for this new feature, just make sure you never use == again. Stick to Equals and you should mostly be fine. I say “mostly” because not even Equals is always overridden to mean value equality. In the case of StringBuilder, for instance, its Equals(object) is inherited from Object and tests for reference equality while its IEquatable<StringBuilder>.Equals(StringBuilder) method tests for value equality. I believe this only strengthens the claim that there should be a single general purpose equality operator, keeping reference equality only as a means to implement equality in some cases.