Hull Ch 6 Interest rate futures

[emilykychow]

Hi David

I have the following queries about Hull Chapter 6:

(1) Question 6.14 at the back of the chapter - Would it be necessary to convert the quarterly compounded forward rates from an actual/360 basis into an actual/365 basis before they are translated into continuously compounded rates for the purpose of working out the LIBOR zero rates?

(2) Quoting page 142:"When interest rates go up, an interest rate futures price goes down. When interest rates go down, the reverse happens, and the interest rate futures price goes up. Thus, a company in a position to lose money if interest rates drop should hedge by taking a long position. Similarly, a company in a position to lose money if interest rates rise should hedge by taking a short futures position." Yet the example underneath this quote seems to suggest otherwise. A gain was made by shorting the futures contracts because the futures price rose with the interest rate?

(3) Page 141: Using the duration-based hedge ratio "has the effect of making the duration of the entire position zero". Please explain.

(4) Page 135: Why does the conversion factor system tend to favor the delivery of low-coupon long-maturity bonds when bond yields are in excess of 6% and the delivery of high-coupon short-maturity bonds when bond yields are less than 6%?


Many thanks
Emily

 

[David Harper CFA FRM]

Hi Emily,

(prior 1)
You could use either the continuous or the semiannual to discount. That's a *great* question that reveals the equivalency; e.g., 12% semiannual is exchangeable with (can be substituted for) 2*LN(1+12%/2) as long as you use them consistently. That is, I think it helps to keep in mind:

since 12% semiannual = 2*LN(1+12%/2) continuous, it follows that
the 1-year semiannual discount factor 1/(1+6%)^2 must equal (=) the 1-year continuous discount factor EXP[-(2*LN(1+12%/2))], if the corresponding rates are used

so, you can use either rate: the key is, if you use the semiannual as given, your discounting must reflect that, and if continuous, discount must reflect that

B/C the problem is so tedious, I entered here into Excel/Zoho (note: we have this in learning XLS under 3.a.9. interest rate futures set)
http://sheet.zoho.com/public/btzoho/6-11-1

note: cell D17 (green) follows the solutions guide (i.e., converts the semi to continuous) but Cell E17 shows an alternative (red) where the semi annual is used "as is;" In which case, the discount must reflect semiannual and the PV, by definition, must be the same. Either is okay, since they must be equivalent.

(new 1)
Yes, totally agree! Kudos to your precision here. (I realize we are only on Ch 1 of entering the Hull questions but we will get these entered ASA. This is a good stretch question b/c you have to do the conversion first).

(2) Please note sentence @ end of first (whole) paragraph: "similarly a company in a position to lose money if interest rates rise should hedge by taking a short futures position."

So, if we start with the underlying exposure, the manager who is long a bond has typical exposure:

long bond =
i rate up -> (higher discount rate) -> bond price down
i rate down -> (lower discount rate) -> bond price up

To hedge, the interest rate future acts directionally like the bond
i rate futures =
i rate up -> lower T-bond price -> lower T-bond futures price
i rate down -> higher T-bond price -> higher T-bond futures price

So, if the investor is long both instrument, he/she is "doubling down:"
i rate up > loss on bond + lower T-bond futures price

...but to hedge, investor must take a short position on futures:
i rate up > loss on bond + gain on T-bond futures = offset (i.e., hedge)

this is much like: if investor is long a bond, he/she can hedge by taking a short position in similar bond

(3) It helps to note the difference between modified duration (a percentage % sensitivity) and dollar duration (i.e., the dollar change given a % change in rate). Although modified duration is a focus of study etc, we really need dollar duration to hedge, as in:

if hold a bond position, given 1% increase in rates:
bond position drops by $X, so a "perfect hedge" is one that simply
increases by $X given the same 1% rate increase

So, this number of contracts is simply solving for an equality:

dollar duration of underlying = dollar duration of hedge instrument
Value of underlying portfolio * Duration of portfolio = [Number of futures contracts * Price/contract] * Duration of Futures
P*Dp = N*F*Df

the point is: solving for N implies the dollar change will be equal for both underlying and hedge (subject to the 1st derivative caveat: only for small changes and parallel shift in yield curve)

(4) That might involve a few factors but I *think* it can be reduced to: the short selects CTD because that maximizes the difference between the benefit (i.e., the bond sold to the long position) and the cost (i.e., the purchase of the delivered bond). The short receives settlement * conversion factor, and the discrepancy arises because the CF is standardized on a 6% yield assumption (the bonds are hypothetically priced "as if" yield were 6%).

So, the short is exploiting a difference between the market price and the hypothetical price (if yields are 6%)
if yield are low (< 6%), the market price > par, and the market price to acquire will be expensive (relative to the 6% assumption). In this case, the "mis pricing" works against the short, and he/she will want to minimize the difference with a short-maturity bond

if yields are high, the market price (to purchase) < par, and relative to the 6% hypo price, the short is getting a bargain. This is a good situation, and the short will want to exploit that by maximizing the difference (i.e., a long maturity bond)

in other words, sorry i am thinking this out load, in any case, the short is receiving proceeds per the conversion factor which is based on 6% yield assumption. So the receipt is (sort of) a given. The cost to acquire the delivered bond is the market price, the varies with yield. If yield > 6%, market prices (costs) are higher and the short does not want to exploit this discrepancy (short maturity, high coupon minimizes the discrepancy just as it minimizes the bond's duration). If yield < 6%, market price (cost to aquire) is lower and now he/she wants to exploit the difference (so seek long-duration bonds)....I am not sure I've considered all variables...

Hope this helps, David

 

[emilykychow]

Thanks David for your prompt response. Sorry I updated my queries before you post your reply. My original question 1 refers to Q6.11 of Hull and I replaced it with query about Q6.14. I also added a new question (4). Would it be OK for you to have a look at them as well?

 

[David Harper CFA FRM]

Hi Emily,

no worries, i edited above to incorporate
#4 gives me a mental challenge, I *think* I captured the intuition but my analysis may be incomplete. FYI, I do have an XLS that computes the CF
http://www.bionicturtle.com/premium/spreadsheet/3.a.10_ctd_tbonds/
(although it's more than you will need for exam, frankly)

David

 

[hsuwang]

Hello David,
Above you mentioned:

"If yield > 6%, market prices (costs) are higher and the short does not want to exploit this discrepancy (short maturity, high coupon minimizes the discrepancy just as it minimizes the bond’s duration). If yield < 6%, market price (cost to aquire) is lower and now he/she wants to exploit the difference (so seek long-duration bonds)"

I'm not sure if in the first sentence you meant if yield "<6%", market prices are higher, and >6% for lower market price in the following sentence. (It seems to me that they are reversed) Also, I'm not getting the first part where if market prices are higher, the short position will want to deliver short maturity, high coupon bond. In this case, what is the difference between a short maturity, high coupon bond and a long maturity, low coupon bond? It seems to me that one does not necessarily have a lower price (cost) than the other, because high coupon implies lower bond price but longer maturity also lowers the bond's price.

Thanks!

 

[notjusttp]

Hi Jack,

1) I agree with you there is a typo error from David and what you are saying is correct. A yield > 6% would necessarily discount the Cash flows to a larger extent and costs to acquire would be low and vice versa.

2) I am not sure if i can convince you as much as David does but for your second point I guess that a short maturity bond ( though higher coupon) would have less number of cash flows as compared to the Long maturity ( lower coupon) bond and hence as David states since the receipt is SORT OF GIVEN, the objective of short being to reduce his cost of acquisition he would go for a short maturity bond which has though higher amount but less number of cash flows.

3) The only doubt i have in my own argument is the fact that since both the bonds are bonds above 15 yrs mat ( since CTD option is only for such bonds) in case the short maturity high coupon is extravagantly larger than the Long maturity bonds low coupon then though the no of cashflows are less their discounted price might be > then the discounted price of the Long mat bonds low coupon cash flow thus negating the objective of the short.

This concept though interesting is not devoid of a grey area.

Cheers
Amit

 

[David Harper CFA FRM]

Yes, Jack apologies for delay but, yes, agreed I did typo/transpose above... I frankly remain stuck in Amit's grey area on this phenomenon...my explanation above leans really on the pull to par phenomenon (but, i don't feel 100% confident that it fully explains), what I mean is:

Take a $100 6% coupon bond
The CTD assumes 6% yield, so would price this bond at par, such that CF = 1.0

Now if yields are 8%, then *purchase price* is $82.71 (term = 15 years, semiannual pay)
If yields are 4%, then purchase price is $122.40

The issue is the *divergence* between the purchase price and the CTD price...
if yields are 8%, due to pull to par, making this bond longer in maturity lowers it's price! If you are buying this price , which the short is, you want longer maturity (see how this is the pull to par effect in reverse?)
If yields are 4%, making the bond longer in maturity has the oppositive effect, it increases your purchase price, so you want none of that.

I think the coupon effect is more complicated...

David