SOA #100 (Things I don't get about bonds)

[Not Paul]

An investor owns a bond that is redeemable for 300 in seven years. The investor has just received a coupon of 22.50 and each subsequent semiannual coupon will be X more than the preceding coupon. The present value of this bond immediately after the payment of the coupon is 1050.50 assuming an annual nominal yield rate of 6% convertible semiannually.

Calculate X.

(A)7.54 (B)10.04 (C) 22.37 (D)34.49 (E)43.98

I am not sure what to do with the 22.50. I presume I can use it to find out which coupon was just received. I also sense some amortization of premium and/or an arithmetically increasing coupon structure. But I am not sure where to begin. As I solved I found myself doing too much algebra. And then I ran out of time. Will someone point me in the right direction?

[Sullinator]

Originally Posted by Not Paul

An investor owns a bond that is redeemable for 300 in seven years. The investor has just received a coupon of 22.50 and each subsequent semiannual coupon will be X more than the preceding coupon. The present value of this bond immediately after the payment of the coupon is 1050.50 assuming an annual nominal yield rate of 6% convertible semiannually.

Calculate X.

(A)7.54 (B)10.04 (C) 22.37 (D)34.49 (E)43.98

I am not sure what to do with the 22.50. I presume I can use it to find out which coupon was just received. I also sense some amortization of premium and/or an arithmetically increasing coupon structure. But I am not sure where to begin. As I solved I found myself doing too much algebra. And then I ran out of time. Will someone point me in the right direction?

The first coupon 22.5 will be at time 0 a long with the price of 1050.50 but ignore the 22.5 at time 0 because it says the present value of the bond immediately after the coupon payment, at semiannual payment 1 it’s x+22.5, then at semiannual payment time 2 it’s 2x+22.5, then at semiannual payment time 3 it’s 22.5+3x.... and so fourth until you get to semiannual time 14 which is 22.5+14x and time 14 you also have a redemption value of 300, once you put that on the time line you can see it’s arithmetically increasing.

[Not Paul]

Originally Posted by Sullinator

The first coupon 22.5 will be at time 0 a long with the price of 1050.50 but ignore the 22.5 at time 0 because it says the present value of the bond immediately after the coupon payment, at semiannual payment 1 it’s x+22.5, then at semiannual payment time 2 it’s 2x+22.5, then at semiannual payment time 3 it’s 22.5+3x.... and so fourth until you get to semiannual time 14 which is 22.5+14x and time 14 you also have a redemption value of 300, once you put that on the time line you can see it’s arithmetically increasing.

I can see the arithmetically increasing coupon payment, but I am not confident that 22.50 is the first coupon payment. It could be the fifth, or the tenth, and the present value is being calculated at that point in time. Treating 22.50 as the first coupon payment, I get X = 12.08ish using the increasing annuity formula.

*Edit

Okay, clearly we have got fourteen coupon payments to go, I just didn't see the inference in the first sentence. I think I've got it now.

[Sullinator]

Originally Posted by Not Paul

I can see the arithmetically increasing coupon payment, but I am not confident that 22.50 is the first coupon payment. It could be the fifth, or the tenth, and the present value is being calculated at that point in time. Treating 22.50 as the first coupon payment, I get X = 12.08ish using the increasing annuity formula.

*Edit

Okay, clearly we have got fourteen coupon payments to go, I just didn't see the inference in the first sentence. Using this increasing annuity formula with n = 14 and i = .03 I get 1050.5 = 79.31X + 300v^14 => X = 10.74. Close, but I'm still missing something.

You are close, x(Ia)angle14 + 22.5a-angle14 +300v^24=1050.50, if you draw out a timeline you can see more clearly as to what is happening. When it says just received means right now at this time which is time 0.

[Not Paul]

I missed the 14 payments of 22.5. Thanks for your input.

[Sullinator]

Originally Posted by Not Paul

I missed the 14 payments of 22.5. Thanks for your input.

No problem, you should get choice A.

[Not Paul]

Originally Posted by Sullinator

No problem, you should get choice A.

I did.