I recently got fed up trying to understand my mortgage using excel. After twenty minutes guddling with cells and individual values, I felt the need to create higher-level abstractions such as “mortgage” and “payment strategy”. I also wanted to create a list of possible repayment strategies and easily compare them to see how it affects the loan duration and total interest payed. This is possible in excel, but no fun.

So, fast-forward to the end of an evening’s hacking with Haskell. I now have hmortgage, a EDSL for expressing payment strategies and code which will expand out a mortgage into monthly steps, like this:

We are looking at loan of £1000.00 at 5.0% over 10y, which has required monthly payment of £10.58 Baseline: Total interest: £272.97 Total payments: £1272.97 Duration=10y 1m Overpayment scenario "2 pm, 200 initial": Total interest: £132.09 Total payments: £1132.09 Duration=6y 3m Compared to baseline: interest=£-140.88, payments=£-140.88, duration=-3y 10m For month 1, balance: £1000.00 -> £791.58 (interest: £4.16, payment: £212.58) For month 2, balance: £791.58 -> £782.29 (interest: £3.29, payment: £12.58) For month 3, balance: £782.29 -> £772.96 (interest: £3.25, payment: £12.58) For month 4, balance: £772.96 -> £763.60 (interest: £3.22, payment: £12.58)

ie. if you overpay by £2 each month, and pay an initial lump sum of £200, you’ll save about £140 overall and will repay the mortgage nearly 4 years early.

There’s a few points of haskelly interest in this code, mostly inspired by stuff I read a few years ago – behaviors in FRP, and SPJ’s “composing contracts” paper.

### Combinators for payment strategies

I have a few primitive payment strategies, which can be combined into more complex strategies:

- monthlyPaymentsOf (100 Pounds)
- lumpSumOf (100 Pounds)
- lumpSumOf (100 Pounds) `after` (1 Year)
- monthlyPaymentsOf (100 Pound) +. (lumpSumOf (100 Pounds) `after` (1 Year))

### Shallow embedding of DSL

The dsl is a shallow embedding; it represents the monthly payment plan as a function from month-number to the payment amount, ie. Integer -> Currency. There’s a problem with this approach – the only thing you can do with a function is apply it to some arguments. This is fine for finding the payment for a particular month, but I would also like to derive a textual description of the payment plan – which isn’t possible with functions.

From stuff I’ve read previously, I think my two options are:

- Lisp-like: Represent the payment schedule as data (ie. like an AST) and provide an eval function. This allows introspection into structure of the payment schedule. Code is data, data is code.
- Arrow-like: The payment strategy could be a tuple of the function and a textual description. When strategies are combined, the combinator would merge the textual descriptions as well as producing new combined functions. I’m not totally convinced that the english language is ‘compositional’ in this way though – it might end up with really clumsy phrasing.

### Crazy Lennart-inspired postfix operators

Initially, the only way I had to create a ‘Currency’ value was via the ‘pounds’ function. In haskell, the function precedes the argument, hence it looks like “pounds 20”. The source code would read nicer if I could write this as “20 pounds” like we do in english. I didn’t think this was possible in Haskell.

Then I remembered seeing Lennart Augustsson’s crazy embedding of BASIC into Haskell. In particular, he had code which looked like this:

```
runBasic $ do
10 PRINT "HELLO"
20 END
```

How the heck does that parse? It’s using ‘do’ notation, so “20 END” must have a type in the Monad class. But, as I understood things, “x y” means “apply the (function) value x to value y”. And “20” doesn’t look much like a function to me.

Digging into the source, I found this:

```
-- 10 END
instance Num (END -> Expr a) where
fromInteger i c = ...
```

Hmm, interesting. This is saying that (some) function type can be treated as if it is “number like” and provides a mechanism for converting integer literals in source code to that type. I hadn’t fully appreciated this, but the Haskell Report says that numeric literals aren’t quite as literal as I expected – the literal integer value gets passed through ‘fromInteger’ and can therefore be made into any Numeric type.

So this code really says “Hey ghc, if you come across a “42” in the source code, you can turn that into a function if you need to”. In the BASIC example, the next thing on line 20 is “END”, a constructor for the type also called END. So, ghc will be looking to turn “42” into something that can be used as a function taking an argument of type END, and so it’ll call this instance of fromInteger.

Hurrah, I can use the same ‘trick’ to make my currencies look nicer:

```
data MONEY = Pounds | Pence
instance Num (MONEY -> Currency) where
fromInteger i Pounds = C (i * 100)
fromInteger i Pence = C i
```

Now I can say “42 Pounds” or “23 Pence”. The “42” will become a function with type MONEY -> Currency. The “MONEY” type is really just a tag – used to choose the parse but that’s it. The Pounds/Pence tags force the appropriate overloading of fromInteger to be chosen, and this will construct a Currency value (represented as number of pence, and using a simple wrapper constructor called C).

Is this better, or just “clever”? I’m not sure yet. It’s certainly easier to read. But I feel I’ve taken a step away from “pure haskell” into a slightly weird world. Still, if I were writing in lisp, I wouldn’t think twice about doing this kind of thing.

### The actual app

Shocker, I’ve produced an app which is actually useful to me in “teh real world”. I have a big TODO list of stuff which will fit nicely into the app – time-varying interest rates, inflation predictions and NPV calculations. None of which, of course, I will ever actually get around to adding. But it’s still useful in its present state, so a win!

Here’s what the “summary” view says – it omits the montly breakdown and instead reports the overall savings possible via the different payment strategies:

We are looking at loan of £1000.00 at 5.0% over 10y, which has required monthly payment of £10.58 Baseline: Total interest: £272.97 Total payments: £1272.97 Duration=10y 1m Overpayment scenario "2 pm, 200 initial": Total interest: £132.09 Total payments: £1132.09 Duration=6y 3m Compared to baseline: interest=£-140.88, payments=£-140.88, duration=-3y 10m Overpayment scenario "2 pm only": Total interest: £216.50 Total payments: £1216.50 Duration=8y 1m Compared to baseline: interest=£-56.47, payments=£-56.47, duration=-2y Overpayment scenario "200 initial": Total interest: £163.52 Total payments: £1163.52 Duration=7y 8m Compared to baseline: interest=£-109.45, payments=£-109.45, duration=-2y 5m Overpayment scenario "400 initial": Total interest: £87.73 Total payments: £1087.73 Duration=5y 6m Compared to baseline: interest=£-185.24, payments=£-185.24, duration=-4y 7m Overpayment scenario "200 after 2y": Total interest: £191.42 Total payments: £1191.42 Duration=7y 10m Compared to baseline: interest=£-81.55, payments=£-81.55, duration=-2y 3m Overpayment scenario "400 after 2y": Total interest: £137.90 Total payments: £1137.90 Duration=5y 10m Compared to baseline: interest=£-135.07, payments=£-135.07, duration=-4y 3m

Eep, it’s 01:30 .. how did that happen? Stoopid jetlag …

A more straightforward alternative to your “crazy postfix operators”:

> {-# LANGUAGE PostfixOperators #-}

>

> pounds i = C (i * 100)

> pence i = C i

>

> foo = (3 `pounds`)

The PostfixOperators extension allows sections (a `f`) to desugar to f a, even when f a is not of a function type. (As opposed to the Report desugaring: (a `f`) becomes \b -> f a b)

Reid: Cool, thanks for that – I didn’t know that sections could be used in that way. 🙂

“From stuff I’ve read previously, I think my two options are… Lisp-like…Arrow-like.”

You might find another option from Elliott et. al. at http://conal.net/papers/jfp-saig/. If you can represent your DSL as haskell functions that range over term types, then you can apply a literal value to evaluate it, or apply a variable to inspect it. I’ve had good luck with this approach in a DSL of my own.

Note that you can define a closed form for a linear recurrence relation.

So, for example if you had Balance(0) = Initial, Balance(n) = Rate * Balance(n-1) + Repayment then there’s a closed form solution for this Balance(n): Balance(n) = Initial * Rate^n + Repayment * ((1-Rate^n)/(1 – Rate))… Useful if you then want to “solve” for n, Repayment etc. So, my challenge to you is: Write a program to “automatically” solve a linear recurrence relation using the method described here http://en.wikipedia.org/wiki/Recurrence_relation! 😉

Thanks for the explanation of the Basic’s NUM instance. I have been looking at that code for last few days trying to make sense of it, your explanation helped.

I am curious about you comment on the Arrow style for the DSL. Can you elaborate and/or give a reference to the DSL syntax in Arrows.

Thanks

Daryoush

Very interesting, thanks! Will have to see if there are synergies with hledger.