Sponsored Ads
D.W. Simpson
Worldwide Actuarial Recruitment
Life Health Pension Casualty US Asia Australia Europe Salary Apply
Ezra Penland Actuarial Recruiters
Top Actuary Jobs
Salary Surveys Apply Bios Casualty Health Life Pension
Pauline Reimer, ASA, MAAA
Pryor Associates
Actuarial Openings: Life, P&C, Health, Pensions, Finance
Advertise Here
Contact us at actuary@actuary.com or 770-425-8576
Reach top actuarial professionals

Results 1 to 8 of 8

Thread: Arbitrage Question!

  1. #1
    Actuary.com - Level I Poster
    Join Date
    Jun 2007
    Location
    Texas
    Posts
    29

    Arbitrage Question!

    A 1-year bond paying annual coupons of 12% has an annual effective yield of 12%. a 2-year bond paying annual coupons of 10% has an annual effective yield of 14.847%. The bonds have $100 of par value.

    (i) calculate the value of the implied 2-year spot rate, s(2)

    (ii) An investor wants to invest $1000 for 2 years at the spot rate calculated in part (i). Suppose the investor is unable to find an institution willing to accept a 2-year deposit. Describe how the investor's objective can be accomplished through purchase and sale of the 1-year 12% bond and the 2-year 10% bond.

    I thought the first part was simple but I cannot figure out the second one:

    the answer to the first part is 15% and the answers to the 2nd part is that the investor should purchase 12.0227 2-year bonds and sell 1.0735 1-year bonds

    Can someone please explain?

  2. #2
    Actuary.com - Level I Poster
    Join Date
    Oct 2007
    Location
    Colorado
    Posts
    21
    I'll do my best to explain it. By investing 1000 @ spot rate of 15.02%, you want to invest 1000 now and receive 1323.09 at the end of year 2. To me, the approach is similar to that of the exact matching or dedication problems. So you want 1323.09 @ end of year two, nothing else, especailly no coupons at the end of year one. So, to get 1323.09 @ end of year two you need to buy 1323.09/110 worth of 2 year coupon bonds. This is where you get the 12.028 2 year coupon bonds from. In other words, how many 2 year bonds will provide 1323.09 in year 2, 12.028 will. Now, we have to deal with the coupons that the 2 year bond will pay at then end of year one. If you buy the 12.028 2 year bonds, then at time one you have 120.28 worth of coupons coming, and you want to get rid of them, because you want the zero coupon bond rate. So to get rid of that income at time 1, you create an obligation by selling 1 year bonds. A single 1 year bond will provide for 112 at time 1, which is not quite enough. How many do you need to sell, you need to sell 120.28/112=1.074 to make to asset match the liability so to speak. Hope this helps, and if you think it is incorrect, let me know.
    Last edited by halfym; October 29th 2007 at 08:15 AM.

  3. #3
    Actuary.com - Level I Poster
    Join Date
    Oct 2007
    Posts
    15
    Sokura,

    Could you explain how to do part i? Thanks!

  4. #4
    Actuary.com - Level I Poster
    Join Date
    Oct 2007
    Posts
    13
    Price of second bond = 10a_2+100v^2 , where i = .14847

    calculate that.

    Price of second bond = 10/(1.12)+110/(1+s_2)^2

    solve that.

    you then get s_2 = 0.14999 , better known as 15% .


    and great explanation of part II by halfym.
    Last edited by numslock; October 29th 2007 at 09:29 AM.

  5. #5
    Actuary.com - Level I Poster
    Join Date
    Jun 2007
    Location
    Texas
    Posts
    29
    just as numslock stated too

    the price of the 2 year bond can be calculated with the bond price formula:

    P = (100 * .1 * a_2) + (100 / (1 + .14847)^2)

    where a_2 = 1.62888335 = [1 - (1 + .14847)^-2] / .14847

    so P = 92.10480244

    but also because of the formula using spot rates,

    [(100 * .1) / (1 + s1)] + [110/ (1 + s2)^2] = P = 92.10480244

    the spot rate for year 1 is 12% because the 1-year bond produces only one cash flow making the spot rate equal to the yield.

    now you can solve for s2 which is .149997044

  6. #6
    Actuary.com - Level I Poster
    Join Date
    Jun 2007
    Location
    Texas
    Posts
    29
    I guess i had another question as an add on to the previous one. anyway part 2.



    Suppose the investor now finds an institution willing to accept a $1000 deposit for 2 years. To the surprise of the investor, the institution is willing to pay 17% interest on the 2-year deposit. Based on the 2-year spot rate = 15%, the investor realizes that there is an opportunity to earn an arbitrage profit. The investor is unable to find an institution willing to lend at the 2-year spot rate = 15%, but the investor is able to buy and/or sell the 1-year 12% bond and the 2-year 10% bond.

    Describe a strategy that allows the investor to earn an arbitrage profit of $35.08.


    I think i finally understand part 1 thanks to halfym. This part confuses me even more. the answers are supposed to be that the investor sells 12.4445 2-year bonds and purchases 1.1111 1-year bonds and deposits $1000 at 17% for 2 years. please help once again!

  7. #7
    Actuary.com - Level I Poster
    Join Date
    Oct 2007
    Location
    Colorado
    Posts
    21
    Typically, in the very limited experience I have working these arbitrage problems, an arbitrage opportunity can usually be taken advantage of by borrowing or lending now, and settling up later. So in this scenario, since you can invest $1000 at 17% for 2 years, in 2 years you will receive $1368.90 from that investment. You are not looking to make money on the bonds, or the 17% investment, you are looking for free money now. So you create two positions that offset each other, but gives you an advantage. This brings up the question, where did you get the money to invest at 17%. you borrowed it by selling bonds. So the $1368.90 you are going to receive in 2 years is going towards paying back the bonds, all of it. And now it is very similar to the previous problem. How many 2 year bonds will $1368.90 cover? Answer: $1368.90/$110 , which is 12.4445. Because you sold 12.4445 2 year bonds, you owe coupons worth $124.445 at time one. To get that cashflow at time one, use some of the proceeds from the 2 year bond sale to buy a 1 year bond to cover that liability. You need $124.445/$112 1 year bonds to cover it. Which is where you get 1.111.

    So, to summarize, you want free money. The 17% investment is nothing more than a way to borrow money and use the proceeds from the investment to cover the mony you borrowed. Start at the end, and work your way back to time 0. An inflow of $1368.90 at time 2 allows you to cover the bonds you sold (borrowing money at time 0). So you can sell 12.4445 2 year bonds. That sale gives you 12.4445*92.10 (the price) = 1,146.14 at time 0. But to cover the coupons owed in the two year bond, you spend 1.111*100(the price) to just cover the coupons. So how much does that leave you with, $1035.03. So you walk with $35 bucks, invest the $1000 to cover you position.

  8. #8
    Actuary.com - Level I Poster
    Join Date
    Jun 2007
    Location
    Texas
    Posts
    29
    thanks halfym,

    this definitely helps. I feel like this type is definitely a weakness for me. It just seems like i haven't found very many practice problems for this type. I think the part that didn't click to me earlier was the dedication/cash flow matching method and i was thinking too much arbitrage only.

Thread Information

Users Browsing this Thread

There are currently 1 users browsing this thread. (0 members and 1 guests)

Similar Threads

  1. August 2007 Question
    By crystal0 in forum SOA Exam P / CAS Exam 1 - Probability - with practice exam problems
    Replies: 18
    Last Post: July 1st 2008, 01:59 PM
  2. Question from exam
    By iamsammy84 in forum SOA Exam P / CAS Exam 1 - Probability - with practice exam problems
    Replies: 29
    Last Post: June 1st 2007, 10:03 AM
  3. Binomial tree hypothetical question (MFE) - possible exam question?
    By Hawgdriver in forum SOA Exam MFE - Actuarial Models, Financial Economics - with practice exam problems
    Replies: 2
    Last Post: April 23rd 2007, 03:28 PM
  4. Defective Question?
    By Junkmonkey in forum SOA Exam FM / CAS Exam 2 - Financial Mathematics - with practice exam problems
    Replies: 6
    Last Post: March 14th 2007, 10:25 AM
  5. Perpetuity due question
    By rheeh02 in forum SOA Exam FM / CAS Exam 2 - Financial Mathematics - with practice exam problems
    Replies: 3
    Last Post: May 5th 2006, 12:39 PM

Posting Permissions

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •