CT1 April 2005 Q7

[rajiv_p_lk_2001]

This is a standard question with Capital Gains Test. But can some one explain why do we assume the the loan is redeemed as late as possible?

Given that the redemption is at the option of the borrower. Why cannot we use the earliest redemption date? I know this would give the maxmium price for the borrower, but the question does not mention if its maximum or minimum price. It only states a minimum return of 4% p.a.

Working with a term of 15 years we can still calculate a price which would give the minimum effective yield of 4%.

Can anyone please explain if I am wrong? Or is it correct to use either term?

 

[joxer]

Hi rajiv,

It because we are asked for the price that gives the investor a yield of at least 4% p.a.

Since there is a capital gain in this example

i(p) > (1-t)g​

the yeild will be minimised if the loan is redeemed as late as posibble. If its redeemed before the 20 years then the yield will exceed 4% p.a.

I hope this reasoning is correct, If you make a gain then the redemption payment makes up the majority of your return.

Let R = redemption payment and P = Purchase Price.

If this is recieved after 15 years then the yield is approx (R/P)^(1/15).
If its recieved after 20 years then the yield is approx (R/P)^(1/20).

if R and P are the same for redemption after 15 and 20 years then clearly the yeild is greatest if we redeem after 15 years and smallest if redemption is after 20 years.

So we find the price based on achieving a yield of 4% for redemption after 20 years and if redemption occurs before this then our yield will exceed 4%.

Ultimately it dosen't matter who has the option of redemption in this example.

 

[rajiv_p_lk_2001]

I am still not convinced...

Hi joxer,

The question asks for a yield of atleast 4%, which means it can be more than 4% and as you have correctly mentioned:

If its redeemed before the 20 years then the yield will exceed 4% p.a.

OR if u hold the yield at 4% the price of the loan will increase. So why cant we use 15 years? The question does not ask to calculate the MINIMUM price.

I am sorry if I am missing your point, but I still dont understant why we cant use 15 years. The only difference will be that the price will be higher. (I think).

 

[joxer]

Hi rajiv,

Try this.

Take the price you found for redemption at 15 years, using this price write down the equation of value for redemption after 20 years.

As the term is now longer but the present value is still the same, the value of i that will solve this equation must be less than 4% p.a., do you agree?

What you are being asked for is the price that will garuntee the investor a return of 4% (or greater) p.a.

Now find the price that gives the investor a yield of 4% p.a. for redemption at 20 years, using this price, write down the equation of value for redemption after 15 years.

As the term is now shorter but the present value is still the same, the value of i that will solve this equation must be greater than 4% p.a., do you agree?

Hope this helps!

 

[Phil]

Hi Rajiv

Here's another possible explanation that I'd just finished calculating when I noticed Joxer had already got back to you


There's a capital gain, you want it as soon as possible
If you could be certain that the bond would be redeemed after 15 years you would be happy to pay £102,219 for it
If you could be certain that the bond would be redeemed after 16 years you would be happy to pay £102,187 for it
If you could be certain that the bond would be redeemed after 17 years you would be happy to pay £102,156 for it
If you could be certain that the bond would be redeemed after 18 years you would be happy to pay £102,126 for it
If you could be certain that the bond would be redeemed after 19 years you would be happy to pay £102,099 for it
If you could be certain that the bond would be redeemed after 20 years you would be happy to pay £102,072 for it
The bond becomes less attractive to you the later it is redeemed/the later you get your capital gain, so the later the redemption date the less you are prepared to pay for it.

So if as an investor you made the mistake of assuming the borrower (the goverment) would redeem it after 15 years when you worked out how much you would pay for it and so agreed to pay £102,219 and it turned out the goverment didn't redeem it until after 20 years.........you will have paid £147 more than you should have done (£102,219 - £102,072). The only way to be sure you won't have ended up overpaying is to assume the borrower gives you this capital gain as late as possible i.e. after 20 years


Another way of thinking of it:
Just accept/remember that in financial services work we often have to assume worst case scenarios when pricing something or setting assumptions. So here too, when pricing a bond that gives us a capital gain, think what the worst is that can happen to us i.e. it's redeemed as late as possible.

Hope that helps

 

[rajiv_p_lk_2001]

Thanks Phil and Joxer

Thanks guys, I think I get it now. But to be on the safe side can I make a general assumption that:

If there is a capital gains, use the longest term in order to meet the minimum yield. Or if there is no capital gains, the use the shortest term.

Regardless of who as the option to redeem?

 

[Phil]

I'm afraid not. It depends who has the option to redeem

If investor has the option to redeem a bond with a capital gain, then assume he redeems it at the earliest possible date

If investor has the option to redeem a bond with a capital loss, then assume he redeems it at the latest possible date

If borrower has the option to redeem a bond that gives the investor a capital gain, then assume the borrower redeems it at the latest possible date

If borrower has the option to redeem a bond that gives the investor a capital loss, then assume the borrower redeems it at the earliest possible date


So maybe you should ignore the last paragraph in my previous post - apologies if that confused you

 

[joxer]

Hi Phil,

I'm not so sure about this!

Does it really matter who has the option to redeem or is it more important to consider if the investor wishes to make a yield of at least x% or just x%?


Dealing with an example where there is a capital gain:

At least x%, then assume redemption as late as possibble.

For just x%, then assume redemption as early as possibble.

I'm sure I'm missing something!!

 

[Mark Mitchell]

More clarification (hopefully!)....

There are two ways in which these questions can be worded.

1. You may be told to calculate the price so that the investor makes a yield of AT LEAST x%. In this case, always work out the worst case for the investor.
So, as joxer rightly says, if there's a capital gain to the investor, the worst thing is that it occurs as late as possible.
In this type of question you can effectively ignore who has the option, and look at everything from the investor's point of view and what would be worst for him.

2. You may be told to calculate the price so that the investor makes a yield of x% (no appearance of "AT LEAST" or "MINIMUM"). In this case, it does matter who has the option on when redemption occurs and Phil's summary of the four possiblilities is excellent.

 

[joxer]

Thanks Phil, thanks Mark