Forward rate

[Deepak Chitnis]

Hi David,
There is a GARP's question on forward rate,
Below is a table of term structure of swap rates:
Maturity in years swap rate
1 2.50%
2 3.00%
3 3.50%
4 4.00%
5 4.50%
The 2-year forward swap rate starting in three years is closest to:
A.3.50%
B.4.50%
C.5.51%
D.6.02%
Please explain this,
Thank you,

 

[David Harper CFA FRM]

Hi @Deepak Chitnis

The key (imo) is to grasp the no-arbitrage assumption that investing over five years must have the same expected return as investing over three years then "rolling over" into the F(3,5) forward; both are five year horizons, the forward rate prices to make us indifferent. So we want this equality:

(1 + 3.5%)^3*(1 + f[3,5])^2 = (1 + 4.5%)^5; such that:
f[3,5] = sqrt[(1 + 4.5%)^5/(1 + 3.5%)^3 - 1] = 0.060181. I hope that helps.

 

[Deepak Chitnis]

Thank you David!

 

[Deepak Chitnis]

Hi @David Harper CFA FRM CIPM, Please explain the belows question:
Suppose you observe a 1 year (zero-coupon) treasury security trading at a yield to maturity of 5% (price of 95.2381% of par). You also observe a 2 year T note with a 6% coupon trading at yield to maturity 5.5% (price of 100.9232). And finally, you observe a 3 year T note with a 7% coupon trading at a yield to maturity of 6.0%(price of 102.6730). Assume annual coupon payments and discrete compounding. Use bootstrapping method to determine 2-year and 3-year spot rates.
2 year spot rate 3 year spot rates
A.5.51% 5.92%
B.5.46% 5.92%
C.5.51% 6.05%
D.5.46% 6.05%
Please help in these question, and does GARP always provide that rates are discrete or continuous?
Thank you

 

[Dr. Jayanthi Sankaran]

Hi Deepak,

I get the answer to be (C) 2 year spot rate = 5.51%, 3 year spot rate = 6.05%

Bond 1
1 year (zero-coupon) treasury security, YTM = 5%, Price = 95.2381% of par

$95.2381 = $100*d(1.0)
Solving for d(1.0) we get d(1.0) = 0.952381
To compute 1 year spot rate r(1.0): 1 + r(1.0) = 1/0.952381 = 1.05, r(1.0) = 5%. Interestingly, YTM = r(1.0) = 5% which is what one would expect for a zero coupon bond.....

Bond 2
2- year T note 6% coupon trading at YTM 5.5%, Price of 100.9232% of par because coupon rate 6% > YTM 5.5%
Solving for d(2.0)
$100.9232 = $6*d(1.0) + $106*d(2.0)
$100.9232 = $6*0.952381) + $106*d(2.0)
Solving for d(2.0) we get d(2.0) = 0.898197
To compute 2 year spot rate: d(2) = 1/((1 + r(2))^2, ((1 + r(2))^2 = 1/0.898197 = 1.113342, r(2) = 5.515%

Bond 3
3-year T-note 7% coupon trading at YTM 6%, price of 102.6730 of par because coupon rate 7% > YTM 6%
Solving for d(3.0)
$102.6730 = $7*d(1.0) + $7*d(2.0) + $107*d(3.0)
102.6730 = $7*(0.952381) + $7*(0.898197) + $107*d(3.0)
Solving for d(3.0) we get d(3.0) = 0.838495
To compute 3 year spot rate: d(3.0) = 1/((1+r(3))^3, (1 + r(3.0))^3.0 = 1/0.838495 = 1.192613, r(3.0) = 6.047%

Thanks!
Jayanthi

 

[Deepak Chitnis]

Hi @Jayanthi Sankaran, Thank you so much! Got stucked with that one. Thanks a lot.

 

[Dr. Jayanthi Sankaran]

You are very welcome, Deepak

Jayanthi

 

[Dr. Jayanthi Sankaran]

Thanks for the star, Nicole - appreciate it

Jayanthi