# Arbitrage-Free Approach: Need an Explanation

The question I’m confused on is:

“When the present values of the coupon payments are summed and added to the present value of the principal payment at maturity, the result should equal the current price of the bond. In order to calculate the arbitrage-free value of a bond, we determine the future coupon and principal payments for the bond. If the yield curve is positively sloped, the appropriate interest rate used to discount each cash flow will be different. However, in the event the yield curve is flat, all cash flows will be discounted in the same manner as in the traditional approach to valuing bonds.”

Is Zhang most likely correct in his description of calculating the arbitrage-free approach to fixed-income security valuation?

My Confusion: The rates used in the binomial interest rate tree will depend on the volatility assumption used, even if the yield curve is flat. So how does this mean that all cash flows are discounted in the same manner as the traditional approach?

It sounds to me as though they’re not talking about a binomial interest rate tree.

What makes you think that they are?

I’ve added the paragraph that proceeded the one I included above for more context:

I think what gets me is that it says ‘arbitrage-free approach to valuation’ which I thought always uses a binomial interest rate tree

"Zhang explains: “Our fixed-income team focuses on constructing a portfolio of fixed-income securities issued in local markets that trade in multiple markets around the world. We seek out securities we believe to be mispriced and identify these using an arbitrage-free approach to valuation, which can be described as follows. The cash flows of any fixed-income security can be thought of as a package of separate zero-coupon bonds. Each zero-coupon bond in such a package is valued separately at a discount rate corresponding to the interest rate on the yield curve at the time of the cash flow.

When the present values of the coupon payments are summed and added to the present value of the principal payment at maturity, the result should equal the current price of the bond. In order to calculate the arbitrage-free value of a bond, we determine the future coupon and principal payments for the bond. If the yield curve is positively sloped, the appropriate interest rate used to discount each cash flow will be different. However, in the event the yield curve is flat, all cash flows will be discounted in the same manner as in the traditional approach to valuing bonds."

Not in general.

All it’s saying is that the price you get by discounting each cash flow at its spot rate has to be the same as the price you get by discounting all of the cash flows at the bond’s YTM.

This makes perfect sense, I was over complicating the question.

Thank you Magician

My pleasure.