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# Calculate the bond price using forward rates/Par rate

#### mastvikas

##### Member
Hi David

On notes page 98 and 99 .

We still start with the cash flows. But instead of spot rates, we discount will forward rates.
The key here is to keep your “raise to powers” consistent.

Price $3/(1+0.015/2)^1 + 103/1+0.015/2)^1 * (1+0.025/2)^1 how can i check for bond prices using forward rates such as 2.75 , 3.25 and 3,75 How can i calculate Bond price using Par rates . kindly help me Vikas #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber Hi Vikas, if it's a 1.5 year bond, you just extend:$3/(1+1.5%/2) + $3/[(1+1.5%/2)*(1+2.5%/2)] +$103/[(1+1.5%/2)*(1+2.5%/2)*(1+2.75%/2)]

keep in mind this is just replacing spot rate with forward rates:
(1+2.25%/2)^3 ~= (1+1.5%/2)*(1+2.5%/2)*(1+2.75%/2; i.e., we got the forward rates by assuming that invest for 1.5 years at spot of 2.25% (left side) must equal roll-over at the forward rates

Re: par rates: the AIM doesn't ask us to price a bond with par rates. I don't know what that means. The par rate finds the coupon rate that prices to face value, so I'm not sure how the par rates would be used as pricing inputs (I can see how you do it, but i don't know what it means), thanks,

#### mastvikas

##### Member
Hi David ,

thanks for the explanation

i am not getting the answer as 103.95 . i may have done wrong calculation.
2.9776 + 2.9409+ 99.6015.

Also for Par rate , i have an example which i will post later

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi mastvikas,

Yours is correct for a 1.5 year bond, under the given spot rate term structure on page 98 (i.e., 1.5% @ 0.5 years, 2.0% @ 1.0 years, 2.25% @ 1.5 years), the 6.0% semi-annual coupon bond price = $2.98 +$2.94 + $99.60 =$105.52

Compare to the bond illustrated on page 98, which is a 1.0 year bond with price = $2.98 +$100.97 = $103.95 Note that, as expected the bond is "pulling to par:" as the maturity decreases the bond price (for the same given coupon rate of 6.0%) is tending toward the par value of$100. Thanks,

#### mastvikas

##### Member
Hi David,
Thank you for the explanation.

regarding using Par rates for calculating bond price refer below-:

Ct/2*At+D(t)=Bond price ----- At is the Annuity factor.
Maturity (Y) D(t) Spot rate 6month forward Par rates
0.5 0.992556 1.50% 1,50% 1.5000
1. 0.978842 2.15 2.80% 2.1465
1.5 0.962990 2.53 3.29% 2.5225
2. 0.943299 2.94 4.18% 2.9245

Bond Price using discount factors
2*0.992556 + 102*0.978842 =101.83

using spot and forward also we get 101.83

using Par rates

Bond price =$100+[$2-{2.1465/2}]*[0.992556+0.978842]=101.83

similarly for 1.5 years we get as 102.168

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi mastvikas, briefly, what is the problem (I haven't checked the computations) as you are getting 101.83 with consistency? this is a different term structure than p 98, yes?, what is source? Thanks,

#### mastvikas

##### Member
Hi David,
yes i am getting the answer with consistency . yes this is a different term structure . source - this example is from schweser . here i just showed the bond price using Par rates .

#### vasan_mnm

##### New Member
If par rates require discount rates to arrive at the bond price then why not calculate bond price directly from discount rates?