Value of subordinated debt

[Tipo]

I have difficulties understanding the value of subordinated debt portion.

R44.P2.T6.Stulz, page 11
V = D(V, F,T,t) + SD(V,U,T,t) + S(V,U + F,T,t)

How does this work out as call options?

In R43.P2.T6.deservigny.pdf, it was mentioned in your video that the holders of the equity have the option to pay off all the debt(U+F), which results in them owning all the assets of the firm, thus it can be priced like a call option. Im fine with this for the equity portion

1)How does this work out for Debt? What does it mean when you have D as strike of F and SD as strike of U?

2)Value of subordinated debt = value of the firm (V) minus value of the equity minus the value of the senior debt
Shouldnt it simply be SD = V - S - D ?
How did we even get to c(V , F ,T , t) - c(V , F - U ,T , t)

3)This is related to 2), but how is the value of senior debt V - c(V , F ,T , t). Shouldnt it simply be c(V , F ,T , t)

 

[David Harper CFA FRM]

Hi @Tipo

I will use the first column in https://www.dropbox.com/s/nvkqqtqnkd14ph3/stulz_subordinate_2.xls?dl=0 as an example (Stulz' high value firm). The assumptions are:

• Firm value (V) = 200
• Face value of senior debt (F) = 100
• Face value of subord debt (U) = 50

As you note, V = D(V, F,T,t) + SD(V,U,T,t) + S(V,U + F,T,t), where D(.) is present value of senior debt, SD(.) is value of subord debt, and S(.) is value of equity.

• The value of equity as a call option = 64.60 = S(200, 150, 1.0, 0 ) = S(V = 200, U + F = 150, ...) where the "strike price" is 150 of total debt; as usual.
  
• The value of senior debt = 90.48 = D(V, F, T,t) = V - c(V, F, T,t) = 200 - c(200, 100, 1.0, 0) = 200 - 109.52, because c(V, F, T,t) is the option value at a strike of 100, so it's the option value for both equity and subordinate holders (ie, as if there were no subordinated holders).
• The value of subordinate holders, as shown by Stulz 18.13, SD(V, U, T, t) = V - c(V, F+U, T, t) - [V - c(V, F,T,t)] = value of firm - value of equity - value of senior debt = 200 - 64.60 - 90.48 = 44.91, but this reduces to --> SD(V, U, T, t) = c(V, F,T,t)] - c(V, F+U, T, t) = 109.52 - 64.60. I hope that explains!

 

[laboheme]

Are pref shares then valued using the same method as subordinated debt since they also get what is "left over" the residual after the senior are paid off?

 

[laboheme]

I tried the excel model. If you take a shell company and it has $2 of cash and $1 of senior debt and $1 of junior debt its Firm Value is $0. The excel model gives NA for the debt value.
https://prnt.sc/vv935r