Zero Rate VS Spot Rate VS Par Yield

[skoh]

Hi David,

Is spot rate always equals to zero rate? I'm looking at question 158.2 and 158.3 in P1.T3. Markets & Products: Hull Chapters 4, 5 and 6.

For question 158.3, shouldn't it be Par yield = [1 + 1*exp(-0.05*2.0)]*2/3.80710 instead of [1 - 1*exp(-0.05*2.0)]*2/3.80710? I'm confused by the formula.

Thanks!

 

[skoh]

Also I have a question from Hull as such:

The 6month and 1 year zero rates are both 10% per annum. For a bond that has a life of 18 months and pays a coupon of 8% per annum (with semiannual payments and one having just been made). the yield is 10.4% per annum.All rates are quoted with semiannual compounding.

a) What is the bond price?

b) What is the 18-month zero rate?

My answer for (a) is $101.1296. Derived by 4e^(-10.4% x 0.5) +108e^(-10.4%). But I am lost at (b) because the term zero price is not given. How should we solve this?

Thanks

 

[Aleksander Hansen]

a) 4/1.052 + 4/(1.052)^2 + 104/(1.052)^3 = $96.74

b) 4/1.05 + 4/(1.05)^2 + 104/(1+ R/2)^3 = $96.74 => R = 10.42%

 

[skoh]

Hi Aleksander,

May I know how can I derive R = 10.42%?

Many thanks!

 

[Aleksander Hansen]

3.809 + 3.628 +104/(1+R/2)^3 = 96.74, so
104/(1+R/2)^3 =89.303
104/89.303 = (1+R/2)^3
1.16457 = (1+R/2)^3
take the cube root,
1.052099 = 1+R/2
0.052099 = R/2
So R must be 0.1042

 

[Aleksander Hansen]

Is spot rate always equals to zero rate?

I overlooked your initial question.
Spot rates are directly observable in the market. Zero rates are extracted from some asset to serve as a proxy for the spot rate. So, they are the same for pretty much all purposes, but there is that distinction.

 

[David Harper CFA FRM]

@Aleks - that is great to know about spot/zero, I actually did not realize that subtle distinction. Very cool, thanks!

 

[skoh]

Thank you!

 

[skoh]

To think of it, is zero rate always discrete?

 

[Backwardation]

I overlooked your initial question.
Spot rates are directly observable in the market. Zero rates are extracted from some asset to serve as a proxy for the spot rate. So, they are the same for pretty much all purposes, but there is that distinction.

Hi,

nice distinction. I basically know spot rates as a synonym for zero bond investment rates.

Another nice relation is that between zero rates and the par yield, since the par yield can be seen as a weighted spot rate over the horizon.

And what is the difference between swap rate and par yield curve?