## Sunday, December 15, 2013

### Sommers'D - two dirty implementations: R vs F#

A couple of days ago I started playing with F#.

Although I'm VERY far from being skillful F# programmer, I am slowly moving forward.

Not far ago I implemented one of the measures of association in R - Somers' D.

Somers' D is sometimes used for testing concordance of internal and external risk scores / rankings.

I had some problems with finding the right formula for Somers' D Asymptotic Standard Error, and when I finally found the solution I didn't have much energy to clean up my R code ;)

I thought, using this dirty code as a base for Somers' D implementation in F# may bring interesting results. My intention was not to give R too large a head start over F#.

Still, the differences are visible in many places...

First of all, I have been pretty surprised that basic matrix operations are not available in the core F# version.

It is necessary to add F# PowerPack to work with matrices.

Even then, working with matrices in F# does not seem so natural as in R (or Matlab). Or, probably, I still know too little about F#.

A couple of examples:

constructing matrix in R:

1. PD    <- c(0.05,0.10,0.50,1,2,5,25)/100
2. total <- c(5,10,20,25,20,15,5)/100
3.
4. defaulted    <- total*PD
5. nondefaulted <- total*(1-PD)
6.
7. <- sum(total)
8.
9. portfolio <- rbind(defaulted,nondefaulted)/n

constructing matrix in F#:

1. #r "FSharp.PowerPack.dll"
2.
3. let PD             = [ 0.05; 0.10; 0.50; 1.00; 2.00; 5.00; 25.00 ]
4. let counterparties = [ 5.; 10.; 20.; 25.; 20.; 15.; 5]
5.
6. let groups = PD.Length // risk groups no.
7.
8. let div100 x = x / 100.0
9.
10. let PDprct = [ for x in PD do yield div100 x ]
11. let CPprct = [ for x in counterparties do yield div100 x ]
12.
13. let n = CPprct |> Seq.sum
14.
15. let defaulted    = [ for i in 1..groups do yield CPprct.[i-1]*PDprct.[i-1]/]
16. let nondefaulted = [ for i in 1..groups do yield CPprct.[i-1]*(1.0-PDprct.[i-1])/]
17.
18. let x = matrix [ defaulted; nondefaulted ]

calculating WR/DR in R:

1. wr <- n^2-sum(sapply(1:nrow(x)function(i) sum(x[i,])^2))

calculating WR/DR in F#:

1. let xr = x.NumRows
2.
3. let rowSum (x : matrix) (i : int) = Array.sum (RowVector.toArray (x.Row(i-1)))
4.
5. // WR / DR
6.
7. let wr =
8.
9.     let mutable mat_sum = 0.0
10.
11.     for i in 1..xr do
12.         let row2  = rowSum x i ** 2.0
13.         mat_sum   <- mat_sum + row2
14.
15.     n ** 2.0 - mat_sum

Later it gets a little better, but...

'A' function in R:

1. <- function(x,i,j) {
2.
3.   xr <- nrow(x)
4.   xc <- ncol(x)
5.
6.   sum(x[1:xr>i,1:xc>j])+sum(x[1:xr<i,1:xc<j])
7.
8. }

'A' function in F#:

1. let A (x : matrix) i j =
2.
3.     let xr = x.NumRows
4.     let xc = x.NumCols
5.
6.     let rowIdx1 = List.filter (fun x -> x>i) [ 1..xr ]
7.     let colIdx1 = List.filter (fun x -> x>j) [ 1..xc ]
8.
9.     let rowIdx2 = List.filter (fun x -> x<i) [ 1..xr ]
10.     let colIdx2 = List.filter (fun x -> x<j) [ 1..xc ]
11.
12.     let mutable Asum = 0.0
13.
14.     for r_i in rowIdx1 do
15.         for r_j in colIdx1 do
16.             Asum <- Asum + x.[r_i-1,r_j-1]
17.
18.     for r_i in rowIdx2 do
19.         for r_j in colIdx2 do
20.             Asum <- Asum + x.[r_i-1,r_j-1]
21.
22.     Asum

As I've mentioned at the beginning of the post - both codes are "dirty". Also, I definitely know R better than F# (even if it may not be apparent from the R code above ;)

Still, F# seems to require more coding and many "simple" operations (matrices...) may not be so easy in F# in comparison to R.

I'm still to find where F# excels :)

[ dirty R code ]

[ dirty F# code ]

## Wednesday, December 11, 2013

### F# - even the longest journey begins with a single step. And there may be bumps along the way

Don't take me wrong. I have just started familiarizing myself with F# - a fairly new functional programming language developed with heavy involvement of Microsoft.

My intention has been to examine, whether F# can be used for various tasks I usually perform with R (http://www.r-project.org/).

As for now, F# looks pretty strange.

It is different in many ways from standard programming languages like C/C+. It is also different from R.

Learning it seems like solving a series of logic puzzles, at this stage.

My (very early) F# code is definitely not optimal, but it may give a hint of what may come later.

Take for example a simple function for calculating return on investment in a bond, used in my previous post.

In R, the function looks like that:

1. # expected (discounted) return
2. pv <- function(fa,n,cr,rf) {
3.   -fa+sum(sapply(1:n, function(i) (fa*cr)/(1+rf)^i))+fa/(1+rf)^n
4. }

You can see the code in context here: http://pastebin.com/bFEHQQnM

Meanwhile, my F# equivalent is:

At least both functions return the same result :)

The nice thing about F# is that, although Microsoft did not include it in the free Visual Studio Express 2013, there is an online version of the F# available. You can write and test your F# code there.

OK, why F# may look strange? Just a couple of observations:
• calculating power for floats and integers is handled differently - pown for integers and ** for floats
• once a function is used with one type of argument - say int - you cannot use it again with any other type - say float
• separate operations for adding a single element at the beginning of a list (::) and for joining the lists (@)
• some symbol combinations (example: !!!), while it is possible to define the operations they perform, cannot be used between arguments, i.e. !!! 2 3 is fine, while 2 !!! 3 is not
I would like to stress again, that I am at the very beginning of my journey with F#.

The peculiarities of F# have not discouraged me so far. I'd say, it is quite the opposite. They have increased my hunger for learning fore about this bizarre creature ;)

## Tuesday, December 10, 2013

### When interest rates go to infinity

CAUTION: It is my first post about corporate debt so excessive simplifications and mistakes are highly probable. Comments welcome.

***

The standard formula for calculating bond return with 3 year maturity and annual coupon of 5% tells us, that we should expected discounted return of around 13.6%, given the extremely low current "risk free" interest rate.

> pv(fa=100,n=3,cr=0.05,rf)
[1] 13.59618

Do you think it is adequate for the risk we are taking?

Actually it depends :)

Fig.: Probability of Default vs. Interest Rate curve

If we are unlucky and our bond defaults, we may actually lose approx. between 46% and 60% of our investment (assuming RR=37% and RT=1Y, see below).

> de(fa=100,di=0,cr=0.05,rv=0.4,rf,rl=1) # default in year one; no coupon payments
[1] -60.17087
> de(fa=100,di=1,cr=0.05,rv=0.4,rf,rl=1) # default after first coupon payment
[1] -55.36236
> de(fa=100,di=2,cr=0.05,rv=0.4,rf,rl=1) # default after second coupon payment
[1] -50.5744
> de(fa=100,di=3,cr=0.05,rv=0.4,rf,rl=1) # default after third coupon payment
[1] -45.80689

Three critical factors here are Probability of Default (PD), Recovery Rate (RR) and Resolution Time (RT).

The first tells us, how likely we are to lost all or part of our initial investment.

The second - what part of the investment we could get back.

The third - when can we expect some of our money back after the default.

Average may be misleading here. The default rate for speculative bonds surpassed 11% in the period. In addition, intensity of defaults varies between geographies and industries.

According to Moody's, Resolution Time can take between 6 months and more than 3 years.

Let's focus on the Probability of Default - i.e. freeze all the other parameters: bond maturity = 3 years, Recovery Rate = 37%, Resolution Time = 1 year, and risk free (RF) interest rate = 0.429%.

The 5% annual coupon on our bond implies its Probability of Default at around 5.5%.

This estimation method used means that if we would have a large portfolio of identical bonds with equal and constant PD of 5.5% and annual coupon of 5%, we would finish our investment with (discounted) return of zero - i.e. we have treated our coupon as zero profit interest rate.

PD of 5.5% is clearly above the average historical default rate as recorded by Standard&Poor's. Hence if we believe the actual PD will be lower, say 2%, we will make a profit. Zero profit interest rate at PD equal 2% is 2.2%, so the difference (spread) between our coupon and risk level is 2.8 pp.

The table below shows the relation between PD and zero profit interest rates required for PDs between 0% and 10%:

PD    IR
[1,]   0.00  0.005
[2,]   0.01  0.013
[3,]   0.02  0.022
[4,]   0.03  0.031
[5,]   0.04  0.039
[6,]   0.05  0.048
[7,]   0.06  0.057
[8,]   0.07  0.066
[9,]   0.08  0.075
[10,]   0.09  0.085
[11,]   0.10  0.094

Reminder: debt maturity, RR, RT and RF are still frozen

Clearly, when default rate increases, we should ask for the higher interest rate. However, as the chart at the beginning shows, the situation starts to be pretty hilarious after reaching some PD level. Around PD of 60%, we need to ask for 100% interest. And even further the required zero profit interest rate goes into infinity...

[ R code used ]