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Hello.
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So to find the four -year spot rate, we can label this as s -sub -4, we need to determine the present value of bond d using the spot rates from years 1 through 4.
00:11
The spot rate, we can label this as just s of n.
00:15
This is any random spot rate for an n amount of years.
00:19
Bond is the yield on a zero -coupon bond maturing in n years.
00:23
So we have some given information here.
00:25
We have the maturity equaling four years.
00:27
Our coupon rate at 12 percent.
00:28
Our yield to maturity at 5.
00:30
Point seven eight three percent and our face value is at a thousand percent our annual coupon payment is point twelve because of the twelve percent multiplied by a thousand our face value giving us 120 and our market price of bond given by the present value of its cash flows discounted at the yield to maturity is p sub d so the market price of bond d is equal to 120 over one plus s one plus 120 over 1 plus s2 squared and etc plus 120 over 1 plus s3 cubed and so forth until we well i might as will just say the last one and then 120 over 1 plus s 4 to the power of 4 right so first we need to compute the bond prices using the given spot rates so first calculate the bond values of each cash flow using the spot rates we have bond a the zero coupon bond given to us as p sub a is equal to 1 ,000 over 1 .02 to the power of 1, which is simply 980 .39.
01:34
And we have s1 at a 2%.
01:38
Now, bond b, coupon bond with a 10 % coupon rate, we can solve for s2.
01:48
So we have 1 ,000 is equal to 100 over 1 .02 to the power of.
01:57
Of 1 plus 1100 over 1 plus s sub 2 squared.
02:07
So now we can solve for s2.
02:13
We're left with 1000 equals 98 .04 plus 1100 over 1 plus s sub 2 squared.
02:24
And this simplifies down to 901 .96 is equal to 11001 plus s2 squared.
02:33
And of course, now multiply both sides by 1 plus s sub 2 squared and divide 901 to the other side.
02:42
So you're left with something like this.
02:44
1 plus s sub 2 squared is equal to 1100 over 901 .96.
02:57
And then simplifying everything, of course, you take the square root both sides.
03:02
You're going to get 1 .1042.
03:07
And s2 is equal to 10 .42%.
03:11
Now, we need to find bond c with a coupon bond with 6 % coupon rate.
03:19
So we're actually going to slide down here because we're going to need more space.
03:22
I'll leave the answer to s, or excuse me, pa, s1 and s2...