# 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!

## 2 Answers

(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.