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YouTube T4-21: Fixed income: Law of One Price

Nicole Seaman

Director of FRM Operations
Staff member
The Law of One Price says that only one discount factor exists at each maturity, absent confounding factors. On the first sheet, I demonstrate why "spot rate of 4.0%" is imprecise, yet "discount factors do not lie." On the second sheet, given observed bond prices, depending on the Law of One Price, I show how we can bootstrap the discount function (i.e., set of discount factors)

David's XLS is here: https://trtl.bz/2TZicOd

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Hi @David Harper CFA FRM , I hope you are safe and well.

How can the discount factor from of 2% coupon be "carried over " to a 5% coupon. I do not know if i'm expressing it right but I have a bit of difficulty in understanding why the discount factor of the first row can be used on the second row.

My reasoning is : because the 6m period coupons are not the same, we cannot make use of the same discount factor. I thought the law of 1 price applied in the case where the FV were the same.

I know i'm misunderstanding something but I cannot place it.
Many thanks

David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi @RajivBoolell Thank you, I am doing fine and I hope you are well also.

The example in the video includes (on the first row) a six-month bond that pays a 2.0% s.a. coupon with an observed (i.e., given as input) price of $100.40. As this first bond matures in only six months, there is only one cash flow and it is easy to retrieve the six-month discount factor, d(0.50), because PV = FV(0.5)*d(0.5), or specifically $100.40 = $101.00*d(0.5) = $101.40 so that d(0.5) = 100.40/101.00 = 0.99406. $101.00 is the face plus the final $1.00 coupon. This is equivalent to a spot rate of [(1/0.99406)-1]*2 = 1.1951%; I do like to remind folks that spot/zero rates are interchangeable with discount factors.

The one-year bond that pays a 5.0% s.a. coupon has an observed (given as input) price of $103.20. If the bond trades fairly (i.e., neither trades rich nor trades cheap) then its market price should equal its theoretical price as given by discounting the cash flows:

$103.20 = $2.50*d(0.5) + $102.50*d(1.0)

You are correct that in six months each bond pays a different coupon: the first pays $1.00 (½ of 2.0% of $100.00) and the second pays $2.50 (½ of 5.0% of $100.00). But the law of one price asserts that the six-month discount factor, d(0.5), is the same for both of them, if they are risk-free bonds. More generally, Tuckman says "absent confounding factors (e.g., liquidity, financing, taxes, credit risk), identical sets of cash flows should sell for the same price." (Tuckman Chapter 1). I prefer to restate this law, as I did in the video, as follows: absent confounding factors, the law says that only one discount factor exists at each maturity. This is equivalent to: absent confounding factors, the law says there is only one spot rate (aka, zero rate) at each maturity. Because a discount function (i.e., a vector of discount factors) describes a term structure, we can elaborate further: the law of one price says there is only one theoretical risk-free spot rate term structure! Continuing with the example, we can assume there is only one value for the six-month discount factor and we already inferred it from the six-month bond, d(0.5) = 0.99406. That suggests that we can bootstrap/solve for d(1.0) by assuming we know d(0.5):

$103.20 = $2.50*0.99406 + $102.50*d(1.0)

If the second bond can assume a different d(0.5), then Tuckman shows how an exploitive arbitrage is possible. Please note the confounding factors are quite material. If the second bond is not riskfree (e.g., if it's a corporate bond), then a credit spread applies (its discount factors would be lower because its spot rates would be higher). I hope that's helpful!