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Text Box: How to Solve the Buying and Selling Bonds Problems Note from Joe: This tip covers what used to be the hardest problems on the exam (10 or 15 years ago). These are problems that essentially involve buying bonds at a discount or premium to the par value, holding them just a portion of the time until maturity, then reselling them at a profit or loss. The complex part is that usually the interest rates underlying all the prices are given, but the actual prices are not. Rather, you must first figure out the sales prices, then use the various sales prices and the coupon values to derive the interest you earn on your investment. There are some variants on this we often cover in our seminar, but we want to get you started here. These problems all tend to sound something like this “If I purchase a $1000 n-year bond with a c% semiannual coupon rate at a price where I expect to actually earn x% interest (it is implied that this is over the full life of the bond), but in k years I sell it to someone else who expects to earn y% on his investment (it is assumed that this is over the remaining n-k years), what actual interest rate (I’m going to call this ‘i’) will I earn over the life of my (presumably k year investment)? These can be fairly confusing, especially with the three interest rates c, x, and y floating around that are actually only peripheral to the answer. The trick is not to think of them as the true interest rate you’re looking for, but just as inputs you use to compute a final stream of payments. You start by purchasing the bond (for present unknown $P) as the amount invested. You view the coupons (of unknown $C) you get as interim payments, and the monies you receive when you sell the bond ($SP) as the final larger ‘balloon payment you get at the end of what turns into a k year investment. You then use the interest rate solver function on the calculator , plugging in P , C, SP and (remember this) 2k, to get the semiannual rate. [Why the semiannual rate? As I understand it, the calculator’s interest rate solver can only handle interim payments (the coupons) that are paid as often as the one period of the interest rate you’re looking for.] Then you take the rate the calculator computes, call it i’ and take (1+i’)²-1=i to annualize it. In a nutshell, I know there’s a lot of symbols here, but let me illustrate this with a chart.  So, you can see, just by following the basic formulas for pricing a bond [which, by the way, can be done pretty simply on the calculator using the bond pricing functions, especially if x and y are quoted as semiannual rates], you can set up the values to plug into the calculator to give you the half-year equivalent of the interest rate you seek. [Also, to respond to the likely quibbling, I do realize that there are special symbols for the semiannual interest rate, etc. instead of my primed interest rates. Click here for the details. I just wanted to illustrate the concept and not let the notation (which you are advised to know) get in the way.] [Another trick to see if you’re in the ballpark. There’s so much interest and final balloon principal payment to go around. If you buy at a price valued using x% and sell at a price reflecting y%, if y<x then the buyer pays a high price for the last years, so i>x. Similarly, if y>x, then i<x.] Example Let me finish with an example. Suppose you purchase a $5,000 10 year bond with a 4% annual coupon rate, paid semiannually to yield 6% (convertible semiannually). After 6 years you sell it to another investor who expects to only realize 5%, also convertible semiannually. What yield, convertible semiannually, do you earn on your investment. What effective annual rate do you earn? We take this step by step: Step 1. Figure out what the coupons are Easily , since the principal is $5,000, not $1,000, C = $5,000×.04/2=$100. Step 2. Figure out what the purchase price is. Since the interest rate stated is in semiannual terms, this is simplified to $100×a20¬.03’ + $5,000×(1/(1.03))20 = $4.256.13. Step 3. Figure out what the secondary purchaser paid. Again, using the semiannual rate, this becomes $100×a8¬.025’ + $5,000×(1/(1.025))8 = $4,820.75. Step 4. Figure the half-year interest rate Plugging in an investment of $4,256.13, generating Payments of $100 over 12 periods, coupled with an addition balloon payment of $4820.75, we get a half-year rate of 3.27%. Step 5. Figure the full year rate This would be 2×.3.27% = 6.54% compounded semiannually, or an annual effective rate of 1.0327²-1=6.65%.