Hi all,I am trying to resolve an example in the 5th edition of hull's book (Options, Futures and other derivatives) Chapter 27, Example 27.1 , page 640.I would like to understand how u(t1) = 0.9637 is calculated since I do not see the difference between u(ti) and v(ti).u (t):"Present value of payments at the rate of $1 per year on payment dates between time zero and time t"The part of example required is the following : QuoteSuppose that the risk-free rate is 5% per annum with semiannual compounding and that, in five-year credit default swap where payments are made semiannually, defaults can take place at the end of years 1,2,3,4 and 5. The reference obligation is a five-year bond that pays a coupon semiannually of 10% per year. Default times are immediately before coupon payment dates on this bond.Thanks for your help.

This book is more than 11 years old, is superseded by three later editions, and you have picked a chapter that describes a market that has completely changed more than once since it was written. Please, do yourself a favor and upgrade to a slightly more current text.

Not wishing to diminish sales of Dr Hull's worthy tome but I think I can clarify this confusion.v(t) is the present value of a payment of $1 at time t. u(t) is the present value of payments at a rate of $1 per annum. hence u(t) is the integral of the instantaneous discount factors between 0 and t. you can show that for a constant yield of r that u(t) = (1-b(t))/r where b(t) is the discount factor at time t. In this example, with t1 = 1 and r = 5%, b(t1) = 0.951814 and u(t1) = 0.9637121.hope this helps edited to correct u instead of v

QuoteOriginally posted by: daveangelNot wishing to diminish sales of Dr Hull's worthy tome but I think I can clarify this confusion.v(t) is the present value of a payment of $1 at time t. u(t) is the present value of payments at a rate of $1 per annum. hence v(t) is the integral of the instantaneous discount factors between 0 and t. you can show that for a constant yield of r that v(t) = (1-b(t))/r where b(t) is the discount factor at time t. In this example, with t1 = 1 and r = 5%, b(t1) = 0.951814 and v(t1) = 0.9637121.hope this helpsThanks for your help but it seems to have a confusion between the discount factor b(t) and v(t).v(t) : Present value of $1 received at time t u (t): Present value of payments at the rate of $1 per year on payment dates between time zero and time tI am a bit confused since Hull solution is (v(t) seem to correspond instead to your discount factor b(t)):v(t1)=0.918, v(t2) = 0.9060 while u(t1)=0.9637, u(t2) = 1.8810

QuoteOriginally posted by: bearishThis book is more than 11 years old, is superseded by three later editions, and you have picked a chapter that describes a market that has completely changed more than once since it was written. Please, do yourself a favor and upgrade to a slightly more current text.I have 8th edition too but this formula seems to have been removed.

QuoteOriginally posted by: MoonDragonQuoteOriginally posted by: daveangelNot wishing to diminish sales of Dr Hull's worthy tome but I think I can clarify this confusion.v(t) is the present value of a payment of $1 at time t. u(t) is the present value of payments at a rate of $1 per annum. hence v(t) is the integral of the instantaneous discount factors between 0 and t. you can show that for a constant yield of r that v(t) = (1-b(t))/r where b(t) is the discount factor at time t. In this example, with t1 = 1 and r = 5%, b(t1) = 0.951814 and v(t1) = 0.9637121.hope this helpsThanks for your help but it seems to have a confusion between the discount factor b(t) and v(t).v(t) : Present value of $1 received at time t u (t): Present value of payments at the rate of $1 per year on payment dates between time zero and time tI am a bit confused since Hull solution is (v(t) seem to correspond instead to your discount factor b(t)):v(t1)=0.918, v(t2) = 0.9060 while u(t1)=0.9637, u(t2) = 1.8810that should be u(t) = (1-b(t))/r where b(t)= v(t)sorry for the confusion.

Thanks a lot Dave,This mean that Hull should have written the variables in this more easy way :v(t) : Present value of $1 received at time t = Discount factor at time tu (t): Present value of payments at the rate of $1 per year on payment dates between time zero and time t = (1-b(t))/rThat really make me things more clearer.Thanks again.

QuoteOriginally posted by: MoonDragonThanks a lot Dave,This mean that Hull should have written the variables in this more easy way :v(t) : Present value of $1 received at time t = Discount factor at time tu (t): Present value of payments at the rate of $1 per year on payment dates between time zero and time t = (1-b(t))/rThat really make me things more clearer.Thanks again.the expression for u(t) in terms of b(t) above is only true for constant r. In the general case, you would need to evaluate he integral numerically because rates are by no means constant.

QuoteOriginally posted by: daveangelQuoteOriginally posted by: MoonDragonThanks a lot Dave,This mean that Hull should have written the variables in this more easy way :v(t) : Present value of $1 received at time t = Discount factor at time tu (t): Present value of payments at the rate of $1 per year on payment dates between time zero and time t = (1-b(t))/rThat really make me things more clearer.Thanks again.the expression for u(t) in terms of b(t) above is only true for constant r. In the general case, you would need to evaluate he integral numerically because rates are by no means constant.If I am not wrong the general formula of the discoun as following :[$]v(t)=b(t)= B(0,t) = e^{-\int_0^t r(s)ds} [$]For u(t), I still confuse. Even for r (risk free rate) being constant, I still do not really why we have $1 = b(t) + u(t)*r.

QuoteOriginally posted by: MoonDragonQuoteOriginally posted by: daveangelQuoteOriginally posted by: MoonDragonThanks a lot Dave,This mean that Hull should have written the variables in this more easy way :v(t) : Present value of $1 received at time t = Discount factor at time tu (t): Present value of payments at the rate of $1 per year on payment dates between time zero and time t = (1-b(t))/rThat really make me things more clearer.Thanks again.the expression for u(t) in terms of b(t) above is only true for constant r. In the general case, you would need to evaluate he integral numerically because rates are by no means constant.If I am not wrong the general formula of the discoun as following :[$]v(t)=b(t)= B(0,t) = e^{-\int_0^t r(s)ds} [$]For u(t), I still confuse. Even for r (risk free rate) being constant, I still do not really why we have $1 = b(t) + u(t)*r.for a small time ds, the payment is $1.ds. the present value of this is $1.ds.b(s) where b(s) is the the discount factor for time s. there fore u(t) which is the present value of payments at the rate of $1 per annum between 0 and t is given by[$]u(t)=\int_0^t 1.ds.b(s) [$]if you substitute[$]b(s)=e^{-r.s}[$]then [$]u(t)=\int_0^t 1e^{-r.s}ds [$]which is then[$] u(t) = -e^{-rt}/r + e^{0}/r [$] or[$] u(t) = 1/r - e^{-rt}/r [$]or[$] $1 =b(t) + r*u(t) [$]which basically states that a $1 today is the same as a 1$ at time t (b(t) ) plus the interest (r*u(t)). u(t) is effectively the measure of time or duration.

I have just found my third hull book (6th edition) in which apparently I previously made this exercise without any issue (more than 3 years ago).The formula I used was :v(t1)=1/(1+0.025)^2v(t2)=1/(1+0.025)^4v(t3)=1/(1+0.025)^6....u(t1) = 0.5/(1.025)+0.5/(1.025)^2u(t2) = 0.5/(1.025)+0.5/(1.025)^2+0.5/(1.025)^3+0.5/(1.025)^4u(t3) = 0.5/(1.025)+0.5/(1.025)^2+0.5/(1.025)^3+0.5/(1.025)^4+0.5/(1.025)^5+0.5/(1.025)^6....I believe this is the right way to make these calculation.It is really not help to change books according to new edition in addition of travel.

QuoteOriginally posted by: MoonDragonI have just found my third hull book (6th edition) in which apparently I previously made this exercise without any issue (more than 3 years ago).The formula I used was :u(t1) = 0.5/(1.025)+0.5/(1.025)^2u(t2) = 0.5/(1.025)+0.5/(1.025)^2+0.5/(1.025)^3+0.5/(1.025)^4u(t3) = 0.5/(1.025)+0.5/(1.025)^2+0.5/(1.025)^3+0.5/(1.025)^4+0.5/(1.025)^5+0.5/(1.025)^6....I believe this is the right way to make these calculation.You should be able to see that the above are approximations to what I have given you as the general case.

QuoteOriginally posted by: daveangelQuoteOriginally posted by: MoonDragonI have just found my third hull book (6th edition) in which apparently I previously made this exercise without any issue (more than 3 years ago).The formula I used was :u(t1) = 0.5/(1.025)+0.5/(1.025)^2u(t2) = 0.5/(1.025)+0.5/(1.025)^2+0.5/(1.025)^3+0.5/(1.025)^4u(t3) = 0.5/(1.025)+0.5/(1.025)^2+0.5/(1.025)^3+0.5/(1.025)^4+0.5/(1.025)^5+0.5/(1.025)^6....I believe this is the right way to make these calculation.You should be able to see that the above are approximations to what I have given you as the general case.In your formula I don't see how you take into account the coupons (here 10% per year) while for example.

QuoteOriginally posted by: MoonDragonQuoteOriginally posted by: daveangelQuoteOriginally posted by: MoonDragonI have just found my third hull book (6th edition) in which apparently I previously made this exercise without any issue (more than 3 years ago).The formula I used was :u(t1) = 0.5/(1.025)+0.5/(1.025)^2u(t2) = 0.5/(1.025)+0.5/(1.025)^2+0.5/(1.025)^3+0.5/(1.025)^4u(t3) = 0.5/(1.025)+0.5/(1.025)^2+0.5/(1.025)^3+0.5/(1.025)^4+0.5/(1.025)^5+0.5/(1.025)^6....I believe this is the right way to make these calculation.You should be able to see that the above are approximations to what I have given you as the general case.In your formula I don't see how you take into account the coupons (here 10% per year) while for example.where does the 10% come in ?the 0.5 in you formulae are the fiddle factors of numerical integration.the general solution for u() is[$]u(t)=\int_0^t b(s) ds [$]we can evaluate numerically for example using the trapezoidal rule. for example if we use 3 vertices at 0, 0.5 and 1 then the discount factors are 1, 0.975309912 and 0.951229425. applying the trapezoid rule gives usu(1) = (1+2*0.97531+0.951229)/(2*2) = 0.975462and if we use 5 vertices then the discount factors are 1, 0.9875778, 0.975309912, 0.963194418, 0.951229425 and the trapezoid rules gives usu(1) = ( 1 + 1.975155601 + 1.950619824 + 1.926388835 + 0.951229425 ) / ( 2 * 4) = 0.975424the exact solution assuming that r is constant is (1 - b(1))/r = ( 1 - 0.955122942) / 0.05 = 0.975412if we want to calculate u(2) numerically using 4 trapezoids then the discount factors are 1,0.975309912,0.951229425,0.927743486,0.904837418and the trapezoid rule gives us = 2/(2*4) * (1 + 1.950619824 + 1.902458849 + 1.855486973 + 0.904837418) = 1.903351. And the exact form is 1.903252.

Then, this mean the 10% coupon provided in the book was in fact irrelevant for the exercise?I found this example a bit misleading due to lack of explanation in the Hull's book. I start seriously consider bearish implicit advice...but at the same time I am sure that I am not the only struggling with this example.