## 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)
 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
 -60.17087
> de(fa=100,di=1,cr=0.05,rv=0.4,rf,rl=1) # default after first coupon payment
 -55.36236
> de(fa=100,di=2,cr=0.05,rv=0.4,rf,rl=1) # default after second coupon payment
 -50.5744
> de(fa=100,di=3,cr=0.05,rv=0.4,rf,rl=1) # default after third coupon payment
 -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 ]