# Construct a zero coupon bond

Suppose a 3% 10-year bond is trading at 89 and a 7% 10-year bond is trading at 97. Then (assuming no arbitrage) the price of a 10-year zero-coupon bond would be:

The answer should be 83. How using only cash- flows (no excel formulas) I would be able to get 83?

When I multiply the price of the first cashflow by semiannual coupons of second CF and subtract the price of the second CF multiplied by semiannual coupons of first: 3.5 * 89- 1.5 * 97= 166

How to get the 83 using only cash flow streams?

thanks!

(Assuming that the two coupon bonds have exactly the same schedules, and that you're settling when the accrueds are 0.)

Consider a portfolio consisting of \$7 long 3% bond and$3 short 7% bond.

This portfolio costs 7 * 89 - 3 * 97 = 332.

Every time you receive a 7 * 3% coupon from the 3% bond position, you pay out the same 3 * 7% amount for the 7% bond position. They offset each other exactly. The only time when the net cash flows are non-zero is at maturity, when you receive \$7 principal, pay out \$3 principal, and are left with net \$4, the difference in coupon rates. So the portfolio is equivalent to \$4 of zero-coupon bond.

So the fair price of just 1 zero-coupon bond is 332 / 4 = 83.

Generally, if the bond coupons are $$a$$ and $$b$$, $$0, and if the bond dirty prices are $$p_a$$ and $$p_b$$ respectively, then the portfolio consisting of long $$\\\\frac{b}{b-a}$$ of the $$a\%$$ bond and short $$\\\\frac{a}{b-a}$$ of the $$b\%$$ bond has coupon cash flows $$a\frac{b}{b-a}-b\frac{a}{b-a}=\frac{ab-ba}{b-a}=0$$ and final principal cash flow $$\frac{b}{b-a}-\frac{a}{b-a}=\frac{b-a}{b-a}=1$$ and is equivalent to \\$1 of zero-coupon bond. (In other words, in the numeric example above, long $$\\\b$$ of the $$a\%$$ bond and short $$\\\a$$ of the $$b\%$$ bond is equivalent to $$\\\(b-a)$$ zero-coupon bond.)

This replicating portfolio costs $$\frac{b}{b-a} p_a - \frac{a}{b-a} p_b=\frac{b \times p_a - a \times p_b}{b-a}$$. But this is a kind of formula that you should not memorize, but rather should be able to derive on the fly in real life situations.

If you buy one 7% coupon bond and sell one 3% coupon bond, you pay $$97-89=8$$ dollars. The money you pay give you in return a series of 4 dollar incomes in the next 10 years.

By no arbitrage, a series of 1 dollar income in the next 10 years shall cost 2 dollars (a quarter of the above cash flow). And a series of 3 dollar income shall cost 6 dollars (3 quarters of the above cash flow).

Now decompose the cash flow of the 3% coupon bond, you can see that is a series of 3 dollar income (value 6 dollars) and a ten year zero-coupon bond. The total value is 89 dollars, so the 10-year zero coupon shall cost you 83 dollars.