Zero Rate from Zero Coupon Bonds 10 Year

[rahul.goyl]

Hi David,
Could you please throw some light, how to go about in the below Question?

A 10-year, 8% coupon bond currently sells for $90. A 10-year, 4% coupon bond currently sells for $80. What is the 10-year zero rate?
Choose one answer.
a. 7.3%
b. 6.4%
c. 5.7%
d. 3.57%

Thanks
Rahul

 

[David Harper CFA FRM]

Hi Rahul,

I love this question, can i trouble you for the source?

the question invites you to replicate the zero with the other bonds, and you only need to synch the coupons to see that 2*long the 4% coupon = 1*short the 8% coupon, if you do those trades:

today, you pay 2* 80 and receive 1*90 = pay 70
...in the meantime your coupons "net" to zero
...and in 10 years, you receive $200 par on the long but pay $100 par on the short (i.e., recieve $100).
So, you have replicated a zero: pay 70 today, no coupons, recieve 100 in future

since 70*(1+rate/2)^2 = 100, rate = [(100/70)^(1/20)-1]*2 = 3.5987%
(we would typically assume semi-annual unless instructed otherwise; the question does the same thing but uses continuous discounting ... at least for FRM, it should specify continuous b/c we would default to semiannual)

David

 

[rahul.goyl]

Hi David,
Thanks for your Bullet respond,
Sorry to diappoint you, I myself don't know what the excat source.. I just picked up from one of the coaching provider Q Bank..
However I don't understand how did you think about going long & short
Looking at the question, I was blank how to go about ....

Can you throw some light how we decide on going long 1 bond & shorting another...

Thanks a Lot for your efforts..

Rahul

 

[recalcitrant]

I think this is the way to go about it... :

the first bond has following cash flows..:
2d1 + 2d2 + 2d3 + ..... +102d20 = 80 .... eqn. (i)

second bond has the following cash flows:
4d1 + 4d2 + 4d3 + .... + 104d20 = 90 ..... eqn (ii)


where d1, d2, d3.... d20 are the corresponding discount factors for the 20 six-months periods..


we can get the rates by following..

2 X 80 - 2X102Xd20 - 90 + 104Xd20 =0

d20 is the discount factor associated with the last coupon flow.. i.e. 10 yrs...

if taken in continuous terms, d20 = exp(-r*10)

this will give the answer as 3.57%

 

[David Harper CFA FRM]

Ujjwal,

Thanks, I like your approach better: by setting up with discount factors (d1...), it's more obvious/robust to finding that you need 2x on the first bond...that's better than my way...we end up same place as your continuous 3.57% = 2*LN(1+3.5987% discrete/2)....thanks for sharing your thought process! David

 

[notjusttp]

Hi David,

Just to update you on the source, Its Q 4.16 from John Hull..Cheers..Amit

 

[srinu2singaraju]

Hi Ujjwal,

can you please send any spread sheet for the above calculation.
I am not able to get this.

thanks
srinivas

 

[structurer]

Given that the PV of 8% coupons for 10y is twice the PV of the 4% coupons over 10y, one can simply create a system of two equations and two unknowns:

Given C = PV of 4% coupons over bond's 10y life, then

1) C + PV Par = 80
2) 2C + PV Par = 90

C = 80 - PV Par
2(80 - PV Par) + PV Par = 90
160-90=PV Par
PV Par = 70

then, assuming continuous rates, e(-r*10) = 0.70
r= ln(o.70)/-10 = 3.5667%

 

[dthigale]

Hi David,

I also liked this problem and your approach. I was trying to find out the corrspnding yieds for two bonds and then find out yield for zero coupon but with different coupons it may not work.

*** I have difficulty understanding how the zero rate corresponds to the discount rate d20.

*** Could you pl. give a quick comment?

-D