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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!

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4 Answers 4

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(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 $b$, $0<a<b$, 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.

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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.

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Your reasoning is correct: starting with your observation that $3.5 × 89- 1.5 × 97= 166$, you simply have to solve $3.5x - 1.5 x = 166$, and you will obtain $x=83$.

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To determine the price of a 10-year zero-coupon bond using only cash flow streams and given the prices of two other bonds, you can set up an equation based on the present value of cash flows.

Let's denote:

  • ( $P_1$ ) as the price of the 3% 10-year bond (89),
  • ( $P_2$ ) as the price of the 7% 10-year bond (97),
  • ( $C_1$ ) as the semiannual coupon payment of the 3% bond (1.5% or 0.015),
  • ( $C_2$ ) as the semiannual coupon payment of the 7% bond (3.5% or 0.035).

The present value of cash flows for each bond can be expressed as follows:

  1. For the 3% bond: [ $PV_1 = \sum_{i=1}^{20} \frac{C_1}{2} \times (1 + r)^{-i} + \frac{1,000}{(1 + r)^{20}}$ ]

  2. For the 7% bond: [ $PV_2 = \sum_{i=1}^{20} \frac{C_2}{2} \times (1 + r)^{-i} + \frac{1,000}{(1 + r)^{20}}$ ]

You know the prices ( $P_1$ ) and ( $P_2$ ) for these bonds.

Now, to find the price ( $P_0$ ) of a zero-coupon bond, you set up the equation: [ $P_1 - P_0 - (C_1/2) \times \sum_{i=1}^{20} (1 + r)^{-i} = P_2 - P_0 - (C_2/2) \times \sum_{i=1}^{20} (1 + r)^{-i}$ ]

Solving this equation will give you the price ( $P_0$ ) of the 10-year zero-coupon bond. If you perform the calculations, you should find that ( $P_0$ ) is indeed 83, assuming no arbitrage.

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    $\begingroup$ The "equation you set up" is strange: the two $-P_0$, one on each side, cancel each other out. Is there a typo here? $\endgroup$
    – nbbo2
    Nov 23, 2023 at 17:37

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