# How does spot rate give a no-arbitrage price of the bond?

Can anyone help me explain why using a spot rate give a no-arbitrage price of the bond? Having an example will be much appreciated! Thanks.

You can think of a coupon paying bond as a bunch of smaller zero coupon bonds.

Example: A 3 year, 5% coupon bond could be four different bonds (stripping the coupons and selling them). Call this Bond XYZ

Bond 1: 1 year, \$5 face value
Bond 2: 2 year, \$5 face value
Bond 3: 3 year, \$5 face value
Bond 4: 3 year, \$100 face value

If you had the following yearly spot rates: 2%, 3%, 4% then which do you discount them by?

If you discount them all by 4% then an investor could buy Bond XYZ at \$102.78

If you use the spot rates the price is \$105.76

An investor could make risk free profits by buying the \$102.78 bonds, selling the \$105.76. They keep the \$2.98

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I see. Thank you

I am struggling to calculate the no arbitrage price of the bond at \$105.76 using those spot rates. Maybe I am looking at your example wrong but I am reading it as the 1 Year spot rate= 2%, 2 year= 3%, and 3 year=4%.

Using those spot rates to find the non arbitrage price of a bond, wouldnâ€™t you use the formula:
5/(1.02) + 5/(1.03)^2 + 105/(1.04)^3 = \$102.96

What formulas are you using to find \$105.76?

Thanks for the help

Oh no. Youâ€™re right! I was treating the 3% and 4% as forward rates. Ex. f(1,1) and f(2,1). Thatâ€™s my fault.

Zroubal, youâ€™re right. The cash flows are

5/(1.02) + 5/1.03^2 + 105/1.04^3

For PV of 102.96.

Good catch!

The YTM equivalent to the 2/3/4 spot rates is 3.935%.
At any other YTM underlying the bond such as the 4% of MGreg99 there is a mispricing as the spot rates per nganh96 are indeed the no-arb rates.

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