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# YouTubeT4-21: Fixed income: Law of One Price

#### Nicole Seaman

##### Chief Admin Officer
Staff member
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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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#### RajivBoolell

##### New Member
Subscriber
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
Rajiv

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
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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!