Calculation of change in a bond’s price given duration, convexity, and a change in interest rates

[gargi.adhikari]

https://learn.bionicturtle.com/topic/instructional-video-hull-chapter-4/

In reference to the chapter above, Please refer the screenshot below (also attached). Can someone breakdown the calculation of the Discount Factor Column for me -circled in red. I tried to plug in values into both e ^ -(rt) as well as 1/ ( 1 + r/m ) ^ mt
I am getting values of the discount factor as .9753 instead of a .942....am doing something wrong here ... Appreciate all the help on this...

 

[Deepak Chitnis]

Hi @gargi.adhikari try, e^-12%*0.5=0.94176 (I have 5 decimals, so there maybe some rounding error but it is as good as 0.942), then again e^-12%*1=0.88692, then, for next period e^-12%*1.5=0.83527, and so on try solving remaining by yourself, it will help you understand it. Hope that helps!
Thank you

 

[ShaktiRathore]

FOr all values:
period(t) Formula
0.5 EXP(-12%*.5) =0.942
1 EXP(-12%*1) =0.887
1.5 EXP(-12%*1.5)= 0.835
2 EXP(-12%*2) =0.787
2.5 EXP(-12%*2.5) =0.741
3 EXP(-12%*3) =0.698
thanks

 

[gargi.adhikari]

@ShaktiRathore @Deepak Chitnis Thank you so so very much!

 

[gargi.adhikari]

A quick follow up question on this:-
@ShaktiRathore @Deepak Chitnis
To calculate the discount factors: e^ (-RT) , why do we use the Yield(12%) and not the Coupon (10%) of the bond here ? The Yield of the Bond can change as the price of the bond changes and vice versa....But the coupon of the bond would be fixed...It would be always be paying out 10% ( of the Par Value) ...? All help and inputs on this would be much appreciated...

 

[ShaktiRathore]

Hi
The coupon are fixed at 10% is the cash flow in terms of coupon that is paid from the Bond,the coupon rate is used to find the coupon that is being paid by the Bond that is it.Yield to maturity is very different from the coupon rate it captures both returns from the coupon pays and the price changes of the bonds that is why we discount the Bond cash flows with these yield in order to calculate the present value of the Bonds future cash flows.This is how you calculate present value of cash flow 100 in time T in an account earning 8% per annum as 100/(1.08)^T.Similarly the Bond is earning the yield on both the coupon and the price changes ,such that cash flows CF at t time in to the future would be discounted as CF/(1+yield)^t to get the present value or the price of the Bond.If i have z-coupon bond then there is no coupon pay i would receive a yield on the price change only so in order to calculate the present value of these bond divide the face value of bond with (1+yield)^maturity where yield is the return i earn due to the price changes.
Thanks

 

[gargi.adhikari]

@ShaktiRathore - Thanks so very much . Have a follow up question though...
In reference to P1.T3. Hull, Chapter 4
https://learn.bionicturtle.com/topic/instructional-video-hull-chapter-4/
In cases where we are spot rates(as below), we have used the spot rates to calculate the Discount Factor and the PV.
So in cases where say we are given both Spot rates and as well as the Yield , which one should be used to calculate the Discount Factors and the PVs..?

 

[brian.field]

If you are given spot rates, you can use them assuming they correspond to the appropriate times until each cash flow. For example, consider a 2 year bond that pays coupons every 6 months. You can use a 6 month zero, a 1 year zero, and a 1.5 year zero to determine the corresponding 6 month forward commencing in 1.5 years or you can also determine the 2 year zero if given a bond price. A "yield" is the rate that produces the same price that is produced by using the zeros but the yield is used as a discount rate for all cash flows.. You can also think of a yield as a "zero" rate that is "constant" across all terms, i.e., for all cash flows so that the price remains the same. I.e., assume the price is 98.50 when pricing with zeros. The yield is the "single" rate that when applied to all cash flows also generates a price of 98.50!

 

[gargi.adhikari]

@brian.field Thanks so very much for your inputs on this. I was stuck on this for a couple of days. So Thank You
So to confirm, if we have both the Spot Rates and as well as the Yield given, we can use either to compute the discount factors and the current market price of the Bond?

 

[brian.field]

I think so - your phrasing is a little confusing to me and the fact that this question is under "calculation of change in a bond's price ... " is also a bit confusing.

To restate my point. If you are given a set of zero rates, you can determine a bond price. (You needn't know anything about the bond's yield.)

Similarly, if you are told the bond's yield, you needn't know anything about the zero rates for each cash flow.

If you are given some zeros and a yield, then you may be able to determine the zero rates that were not already provided. Moreover, you may be able to determine forward rates as well.

Does that clarify?

 

[gargi.adhikari]

My bad The phrasing was extracted from the topic I was studying... sorry about that...

i guess to be clear..my question was....if we are given both the Spot Rates and as well as the Yield, can we use either the Spot Rate or the Yield to compute the Discount Factors and hence the current market price of the Bond?

Discount Factor= e ^ (-RT)
we can use either the Spot Rate or the Yield as values for R in the above formula ?

 

[brian.field]

The value of R will be constant if using a yield and R would change for each zero....but you will arrive at the same market price in either case.

Consider a 2 year bond with an 8% coupon convertible semiannually. Assume the yield is also 8%. Then we would have a market price of 100 and we could solve for 100 = 4/(1 + y/2)^.5. + 4/(1 +y/2)^2 + 4/(1+y/2)^3 + 104/(1+y/2)^4.

Solving for y gives y=8%. Since we were given the yield to begin with, we could have solved for the market price of 100.

Now, assume the 6 month zero is 5%, the 1 year zero is 7%, the 1.5 year zero is 8%. We can solve for the 2 year zero as follows. Recall that we know the price is 100, so we have

100=4/(1+.05/2)^(1/2) + 4/(1+.07/2)^(2/2)+ 4/(1+.08/2)^(3/2 + 104/(1+x/2)^(4/2).

X is the 2 year zero rate.

 

[brian.field]

I might be slightly off (since I'm typing in a phone) but hopefully you get the gist...

 

[gargi.adhikari]

@brian.field Got it Thank You !