2 bonds

[mathsolver123]

I'm stuck with this practice problem comparing 2 bonds:
- Bond A is a 10-year bond, redeemed at par, with face amount of 100 and coupons payable semiannually at an effective interest rate of r per 6-month period and is currently selling at par. The nominal yield rate convertible semiannually is j.
- Bond B is a 10-year bond, with face amount of 300, redemption amount of C and coupons payable semiannually at an effective interest rate of 3r per 6-month period and is currently selling at par. The nominal yield rate convertible semiannually is j/4
i'm stuck on how to start this problem after writing the data
Bond A: n = 20, F = 100 = C, coupon rate = r/6months, and yield rate = j/2
Bond B: n = 20, F = 300, C = ?, coupon rate = 3r, yield rate = (J/4)/2 = J/2

 

[tkhunny]

Coupon Amount?

 

[mathsolver123]

Coupon amount is Face value * coupon rate:
A: has face value of 100 and coupon rate of r per 6 months (not given in question)
B: has face value of 300 and coupon rate of 3r per 6 months

 

[tkhunny]

Coupon amount is Face value * coupon rate:
A: has face value of 100 and coupon rate of r per 6 months (not given in question)
B: has face value of 300 and coupon rate of 3r per 6 months

Right. So, write an expression, in terms of what is known, for the coupon amount.

 

[mathsolver123]

it's useless i do it and i get too many unknowns which is why i asked the question in the first place to get some insight on a clue to get started:

A:100r*An (n= 10*2=20) + 100v^20 (the redemption amount discounted for the whole period)
B: 300(3r)*An (n=20) + Cv^20

yield rate for both bonds are the same.

I end up with their prices, yield rate, coupon rate and C unknown

 

[JeffM]

Hmm. I am not 100% certain that you can get a numerical result, but you do not need one. It is sufficient in a comparison to get ratios. There may not be a closed form, but something like Newton's Method should work. In any case, you should be able to get the redemption value of Bond B, at least to any desired degree of accuracy, in terms of rate or yield. Thus, it seems likely that you can get a good enough numerical result.

I do see an error, namely

[MATH]\dfrac{\dfrac{j}{4}}{2} = \dfrac{\dfrac{j}{4}}{\dfrac{2}{1}} = \dfrac{j}{4} * \dfrac{1}{2} = \dfrac{j}{8}.[/MATH]
I am not going to try to solve this, partly because I have no idea what you mean by a convertible yield: convertible into what, yen, swiss francs, or something else that you have not disclosed. The technical definitions of rate and yield must be specified, and you have not done so. US banks work with both TIL and TIS definitions, and, if I remember correctly, there are slight differences between them. The US bond market may use still yet a different definition. Some economists use a continuous compounding formula. I am not going to guess what you mean by your terms.

 

[tkhunny]

it's useless i do it and i get too many unknowns which is why i asked the question in the first place to get some insight on a clue to get started:

Algebra requires abstraction. Let it help you. It doesn't matter if you have no numerical exression.

F = C = 100
coupons payable semiannually at an effective interest rate of r per 6-month period

r, the effective rate, is annual. 6-months is the compounding. Thus, [(1 + r)^(1/2) - 1]* 100 gives the coupon.

F = C = 300
coupons payable semiannually at an effective interest rate of 3r per 6-month period

3r, the effective rate, is annual. 6-months is the compounding. Thus, [(1 + 3r)^(1/2) - 1]*300 gives the coupon.

At least, that's one way to interpret what those interest rates mean. Can you think of another way to interpret, based on your text or other course work? There may be a specific methodology prescribed.

 

[JeffM]

Algebra requires abstraction. Let it help you. It doesn't matter if you have no numerical exression.

F = C = 100
coupons payable semiannually at an effective interest rate of r per 6-month period

r, the effective rate, is annual. 6-months is the compounding. Thus, [(1 + r)^(1/2) - 1]* 100 gives the coupon.

F = C = 300
coupons payable semiannually at an effective interest rate of 3r per 6-month period

3r, the effective rate, is annual. 6-months is the compounding. Thus, [(1 + 3r)^(1/2) - 1]*300 gives the coupon.

At least, that's one way to interpret what those interest rates mean. Can you think of another way to interpret, based on your text or other course work? There may be a specific methodology prescribed.

Actually, I do not think the student has seriously tried any algebra. No equations are given. I suspect, if we were given a comprehensible version of the problem, that we might be able to use the information on Bond A to compute yield numerically. Then, on the assumption that both bonds fall in the same risk class, we could apply that yield to Bond B to determine numerically the redemtion value on Bond B. Mathematically, it may be nothing more than a problem in two simultaneous non-linear equations solvable by substitution.

Notice that the student has not specified any equation. Hard to find solutions to equations if you do not set any up.