Interest Rates

[puneet_]

Hi, I could not find a easy way to solve this question for Interest rates using bootstrap? can someone please help me?

Find coupon rate for 2nd year (X in below table)

time -0.5- 1- 1.5- 2
Bond price 99- 98- 97- 95
Coupon rate 2.0000%- 3.0000%- 3.5000%- X.0000%
Spot rate 4.0001%- 5.0208%- 5.5514%- 7.6993%

 

[puneet_]

different numbers but same problem. how to find out coupon rate at time 2.5 (empty box)? can someone please help? @ShaktiRathore

 

[ShaktiRathore]

Hi
I think the Bond prices shown are rounded.
Let coupon rate be X%.
97=((X*100)/2)/(1+8.0097%/2)+((X*100)/2)/(1+8.3092%/2)^2+((X*100)/2)/(1+10.2882%/2)^3+((X*100)/2)/(1+10.0353%/2)^4+(100+(X*100)/2)/(1+10.1966%/2)^5
=> (100*X/2)*(1/(1+8.0097%/2)+1/(1+8.3092%/2)^2+1/(1+10.2882%/2)^3+1/(1+10.0353%/2)^4+1/(1+10.1966%/2)^5)=97-(100/(1+10.1966%/2)^5)
=>(100*X/2)*4.34561630525198=19.013120381231
=>X=(19.013120381231/4.34561630525198)*(2/100)
=>X=8.7505
Thus coupon rate is 8.7505%.
thanks

 

[puneet_]

the answer that i got is 9.1024. is it with continuous compounding? so do we need to write long equation with ln()? i can not imagine :-( is there any shortcut?

 

[ShaktiRathore]

Hi yes there is continuous compounding,u didn't mentioned that
97=((X*100)/2)*exp(-8.0097%*0.5)+((X*100)/2)*exp(-8.3092%*1)+((X*100)/2)*exp(-10.2882%*1.5)+((X*100)/2)*exp(-10.0353%*2)+(100+(X*100)/2)*exp(-10.1966%*2.5)
=>(100*X/2)*(exp(-8.0097%*0.5)+exp(-8.3092%*1)+exp(-10.2882%*1.5)+exp(-10.0353%*2)+exp(-10.1966%*2.5))=97-100*exp(-10.1966%*2.5)
=>X*(exp(-8.0097%*0.5)+exp(-8.3092%*1)+exp(-10.2882%*1.5)+exp(-10.0353%*2)+exp(-10.1966%*2.5))=(97-100*exp(-10.1966%*2.5))*(2/100)
=>X*4.33114=.39004
=>X=0.39004/4.33114
=>X=0.09005=9.005%
there is no shortcut you need to solve the eqaution for X i think,with excel you can get this pretty fast.
thanks

 

[puneet_]

Thanks Shakthi. you are the best!

 

[David Harper CFA FRM]

This is a good exercise, it is quick in Excel, but @ShaktiRathore I get a different result yet, given our approaches look the same, I don't see why. Please note I get 10.97% for the coupon rate
(the spot rates imply continuous compounding; my 2nd tab assumes semi-annual compounding for a slightly different result).
Sheet is here @ https://www.dropbox.com/s/ttrvssonbk1x4l9/0426-find-coupon.xlsx?dl=0

 

[desh]

John Hull 4.33. Portfolio A consists of a 1-year zero-coupon bond with a face value of $2,000 and a
10-year zero-coupon bond with a face value of $6,000. Portfolio B consists of a 5.95-year
zero-coupon bond with a face value of $5,000. The current yield on all bonds is 10% per
annum.
(a) Show that both portfolios have the same duration.
Answer:
Time single cash flow PV weight time x Weight
1 Yr 2000 @10% 1809.67 0.45 0.45 = 04.5
10 Yr 6000 @10% 2207.27 0.55 0.55x10=5.50
Total: 8000 4016.94 1 5.95 => duration is equal to 5.95 yrs for second portfolio.
(b) Show that the percentage changes in the values of the two portfolios for a 0.1% per
annum increase in yields are the same.
Answer: Delta B =-BD delta Y
for portfolio one Delta B = -4016.94x5.95x0.1% = -23.90 ,
% change = 0.595
For second portfolio
5.95 Yrs 5000 @ 10% PV 2757.80 weight 1 duration 5.95
Delta B = -2757.81 x 5.95 x 0.1% = -16.141
% change =16.141 x 100/2757.80 =.585
Then How is this correct ??? The answer is coming correct by considering six year instead of 5.95 for discounting ????



(c) What are the percentage changes in the values of the two portfolios for a 5% per
annum increase in yields?
Answer: same as in (b) considering Delta Y =0.5 we can solve it ! is it so ?

Please guide ... whether to consider six year or 5.95 years or my calculations are wrong ?... Thanks is advance

 

[desh]

Hi yes there is continuous compounding,u didn't mentioned that
97=((X*100)/2)*exp(-8.0097%*0.5)+((X*100)/2)*exp(-8.3092%*1)+((X*100)/2)*exp(-10.2882%*1.5)+((X*100)/2)*exp(-10.0353%*2)+(100+(X*100)/2)*exp(-10.1966%*2.5)
=>(100*X/2)*(exp(-8.0097%*0.5)+exp(-8.3092%*1)+exp(-10.2882%*1.5)+exp(-10.0353%*2)+exp(-10.1966%*2.5))=97-100*exp(-10.1966%*2.5)
=>X*(exp(-8.0097%*0.5)+exp(-8.3092%*1)+exp(-10.2882%*1.5)+exp(-10.0353%*2)+exp(-10.1966%*2.5))=(97-100*exp(-10.1966%*2.5))*(2/100)
=>X*4.33114=.39004
=>X=0.39004/4.33114
=>X=0.09005=9.005%
there is no shortcut you need to solve the eqaution for X i think,with excel you can get this pretty fast.
thanks

@ShaktiRathore ....... Instead of X*100/2 ... we may consider coupon as 50X => 50 X *(DF1+DF2+DF3+DF4) + (100+50X)DF5 =97
(as coupon for 4 half years is same 100*X/2)
DF is Discount Factor... By solving this I am getting answer X = 8.77% ........Advise me where I am wrong in my calculations ? I think I am not !!

 

[David Harper CFA FRM]

Hi @desh

(b) doesn't discount, both use the same 5.95 duration (because the 10.0% yield is continuous, mod duration = mac duration in this problem, which is convenient!). Your second calculation, which produces 0.585, appears to contain an error because its tautological (under this approach). Due to ΔB = -BDΔy --> ΔB/B = -DΔy; i.e., the percentage bond price change given by ΔB/B is approximated by -DΔy such that, given both portfolios have duration of 5.95 years, both of there estimated price changes are equivalent and given by -DΔy = -5.95*0.10% = -0.595%. Notice your calcs are including the price then just dividing it out again in an extra step.

This question (b) doesn't make sense if Hull wants you to simply apply the duration estimate in this tautological way. I think he really wants you to show that, given the durations are equivalent, how do these approximations compare to the actual price difference. Specifically, the price of portfolio B will change to 5000*e^(-10.10%*5.95) = 2741.45 which is a 16.36/2748 =0.5932% price drop; i.e., not exactly 0.5950% but we don't expect the exact 0.5932% to equal the approximate 0.5950% because convexity is omitted. Portfolio (A) will be different but similarly near to the approximation. I hope that's helpful!