I don't get how to do this one, is a bond similar to compounding interest?
Asked
Active
Viewed 67 times
0
1 Answers
1
Perhaps I am missing something, but it would see to me that you could buy ${8 \over 14}$ of the #1 bond and sell ${6 \over 14}$ of the #2 bond for a net cost of ${8 \over 14} 95 - {6 \over 14}97 \approx 12.71$. Then the semi-annual payments would be zero (since ${1 \over 2} ({8 \over 14} 0.06 - {6 \over 14}0.08) = 0$), and the value after 5 years would be ${8 \over 14} 100 - {6 \over 14}100 \approx 14.29$.
Then solving $(1+r)^5 = {14.29 \over 12.71}$ gives $r \approx 2.4$%.
(This assumes that you can short the bond, of course.)
The following table illustrates the idea, I have used integral units of stock to simplify. \begin{array}{c|ccccccccccc} \text{cash flow}& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \\ \text{buy 8 #1} & -760 & & & & & & & & & & 800 \\ & & 24 & 24 & 24 & 24 & 24 & 24 & 24 & 24 & 24 & 24 \\ \text{sell 6 #2} & 582 & & & & & & & & & & -600 \\ & & -24 & -24 & -24 & -24 & -24 & -24 & -24 & -24 & -24 & -24 \\ \hline \\ \text{net} & -178 & & & & & & & & & & 200 \\ \end{array}
Hence $r = \sqrt[5]{{200 \over 178}}-1 \approx 2.35\%$.
copper.hat
- 172,524
-
-
-
The #1 bond pays 6% pa. and the #2 bond pays 8% pa. To get a zero coupon payout, I must (in this case) buy an appropriate quantity of the #1 bond and sell an appropriate quantity of the #2 bond so that the payments cancel. If I buy 8 #1 bonds, my yearly coupon will be $$8 \cdot 6$ and if I sell 6 #2 bonds, I will need to pay out a yearly coupon of $$6 \cdot 8$, so they match. Then the net transactions are the initial buying/selling of the bonds and the final payout. – copper.hat Feb 02 '14 at 18:05
-
-
The question also says 'per annum, paid semi-annually' (fairly typical for bonds). In this case it doesn't matter, as we are matching the net coupon payments, so the rates are used purely to compute the matching coupon payments. – copper.hat Feb 02 '14 at 19:26
-
I would suggest writing out a time line with $0,...,10$ as the axis, and write out the number for buying 8 #1 bonds and selling 6 #2 bonds on separate lines. Then add the columns to get the net payments. – copper.hat Feb 02 '14 at 19:30
-
this is a lot more complicated than i expected, i find it funny that im better at abstract algebra than this. i will practice. – terrible at math Feb 02 '14 at 19:33
-
I added a table above to illustrate the idea. I find the ideas straightforward, but the nomenclature confusing. – copper.hat Feb 02 '14 at 19:52
-
thanks a lot for this, i am sure i will figure it out now, that is fantastic. – terrible at math Feb 02 '14 at 19:58
-
The coupon payment is off. For 8 units of bond at rate 6% ( quoted in 'per annum' convention) paying coupon semiannually. The coupon paid at end of each period is $$ 8 \times $100 \times 6% \times 1/2 = $24 $$ – achille hui Feb 02 '14 at 20:04
-
@achillehui: You are correct, thanks for catching that. I will fix it. I had a few other related issues, all because I know the net coupon payment is zero anyway. – copper.hat Feb 02 '14 at 20:17
-
@copper.hat hey man thanks so much, i did it myself and got it and understood it :) could you provide the formula that you used to calculate the zero rate though? its familiar but im forgetting – terrible at math Feb 02 '14 at 21:23
-
It you start with principal $P_0$ and invest it for $n$ years at a fixed annual rate $r$ then after $n$ years it will have value $P_n=P_0(1+r)^n$. To given $P_0,P_n$, you can figure out an equivalent yearly rate by solving the equation for $r$, that is, $r = \sqrt[n]{{P_n \over P_0}} -1$. – copper.hat Feb 02 '14 at 21:32
-
I see, so it's the familiar form that I'm used to, just this $P_n$ that's new. We don't care about the negative sign on our $P_0$ in this case? – terrible at math Feb 02 '14 at 22:00
-
Well, no, that's cash flow, so you purchase the bond (money leaves your wallet) and at the end you receive a payment (money enters your wallet). So $P_0 = |\text{outgoing}|$ and $P_n = |\text{ eventual return}|$. – copper.hat Feb 02 '14 at 22:07
-
I see. We also sold bonds in this example, so that means we can owe money too, right? – terrible at math Feb 02 '14 at 22:44
-
Well, you could, but you would not choose a portfolio if the net return was negative. In this example, the net return is 2.35% after the final sell/buy, so this is a gain. – copper.hat Feb 03 '14 at 03:56
probably not, so ive been thinking about it some more..
it pays 6 dollars (6% of 100, its FV) 2 times a year, so in 5 years it will give100 + (10*6) = 160
how do i use the selling price though? (95 dollars)
– terrible at math Feb 02 '14 at 08:33The hint it gives makes little sense to me.. "receives no payments until five years from now"
why would we ignore the coupon though?
– terrible at math Feb 02 '14 at 08:58