Question by KT: How do I construct a synthetic zero coupon bond?
How do I create/construct a zero coupon bond using two coupon bonds with different prices but the same maturity date? I believe it is called a synthetic coupon bond or a STRIP. For example, if there are two coupon bonds maturing in 5 years. The price of bond #1 is $ 1,100 with a 8% coupon and the price of bond #2 is $ 1,250 with a 10% coupon rate. What would be the price and yield of a zero coupon bond also maturing in 5 years?
Answer by John W
Stripped bonds are when the coupons are detached and sold separately from the certificate which is then a zero coupon bond. This started in the 80’s where stripped bonds exploited a tax loop hole which has since been closed but has remained as there are other advantages to shifting the yield either to the coupons (purchasing the detached coupons) or to the bond maturity (purchasing the zero coupon bond).
Synthetics are when other financial instruments are arranged in combination to produce the same results, for example if you purchase only the detached coupons maturing on a specific date, you in essence have a synthetic zero coupon bond or strip.
In your examples, you did not mention if the coupons were annual, semi annual or quarterly, although it’s more common for them to be quarterly, I’ll assume them to be annual for simplicity sake. If we look at bond #1, the IRR of the cash flow is such that:
$ 1,100 = $ 80 * ( ( 1 – (1/R)^6 ) / ( 1 – 1/R ) – 1 ) + $ 1,000 / R^5
Hence bond #1 priced at $ 1,100 is giving you a return of 5.65% per annum as determined by it’s IRR so we can assume that the 5.65% per annum is the market rate and hence the investment products must at least return 5.65% in order to be competitive. If you stripped the coupons from this bond then the $ 1,000 face value certificate (the zero coupon or stripped bond) can only be sold a a price where the yield will also be the equivalent of a 5.65% per annum investment hence it’s price would be:
$ 1,000 / PV = 1.0565^5
PV = $ 1,000 / 1.0565^5
PV = $ 759.72
So the $ 1,000 face value bond certificate stripped of it’s coupons can be sold as a zero coupon bond for up to $ 759.72.
The strips ( the detached coupons ) could each be sold individually as synthetic zero coupon bonds of $ 80 face value maturing in 1, 2, 3, 4 and 5 years respectively and could be sold at prices up to $ 75.72, $ 71.67, $ 67.84, $ 64.21 and $ 60.78 respectively.
In theory, if bond #2 was available in the same market and was considered a comparable credit risk as bond #1, it should also have a IRR of 5.65% but it’s IRR is such that:
$ 1,250 = $ 100 * ( ( 1 – (1/R)^6 ) / ( 1 – 1/R ) – 1 ) + $ 1,000 / R^5
R = 1.0433 or 4.33%
We’ll have to assume that the difference between the IRR of bond #1 and bond #2 is due to the credit risk with the issuing company of bond #1 more likely to default than bond #2. One should actually take the risk of default into account with the pricing as the later payments are more likely to default than the earlier payments but as the probability of default is largely unknown and rarely is the risk premium representative of that risk, we can’t extrapolate from it for accurate calculations so for the time being we’ll ignore the default risk. However the stripped bonds and the strips from bond #2 can be assumed to have the same default risk as bond #2 from which they are derived. When the coupons are stripped, the bond certificate with a face value of $ 1,000 must be sold at a price that gives the investor the same IRR as bond #2 hence:
$ 1,000 / PV = 1.0433^5
PV = $ 1,000 / 1.0433^5
PV = $ 809.01
So the $ 1,000 face value bond certificate stripped of it’s coupons can be sold for up to $ 809.01 as a zero coupon bond.
And the detached coupons can be individually sold as $ 100 face value synthetic zero coupon bonds (strips) with maturities of 1, 2, 3, 4, and 5 years and can be sold a prices up to $ 95.85, $ 91.87, $ 88.06, $ 84.40, and $ 80.90 respectively.
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