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# Law of one price

#### atandon

##### Member
Hi David,

Could you pls elaborate on the definition for law of price which says -
"two securities with exactly the same cash flows should sell for the same price". Do we mean by cash flow -> the coupon payment for bond A & B should be same if payment made semi annually (for e.g) to be considered under the bracket of law of 1 price.

Regards,
atandon

#### Aleksander Hansen

##### Well-Known Member
Not necessarily, a sufficient condition is that you can synthetically replicate bond A in any way shape or form such that their price should be the same. In the world of finance that would make the two assets identical. That is, for any superset of assets X, that spans states of the world such that A can be replicated, the law of one price must hold. Indeed, the subset of X that replicates A need only span those states also spanned by A; that is they are of full rank. As a corollary, one can thus replicate the security by means of the canonical basis of linearly independent vectors formed by the Dirac delta.

#### atandon

##### Member
Sorry Hansen, but the explanation you provided was not very clear to me. I would like to understand the context with a very simple example of 2 bonds.

#### David Harper CFA FRM

Staff member
Subscriber
Hi atandon

I'd elaborate by offering how Tuckman applies the law of one price: he says, if it is true, there must be a single, one and only one, consensus discount function (set of discount factors) for risk-free cash flows. Although you and I may assign different present values to $1.0 received in six months (maybe you think a certain dollar in six months is worth$0.95 and I think its worth $0.97), that is a-okay, but, if the risks are identical (i.e., riskless), the market can't find disagreement simultaneously among various securities pricing that dollar-in-six-months differently (whether it's riskless coupon or riskless return of par), the market's securities need to agree on a single d(0.5), or there should be an arbitrage. Simplest application is bootstrap: say 6-month T-note pays 4% semi-annual coupon and prices at par ($100). Price at par implies 6-month spot/zero rate = 4%. Now say also trading is a 1-year T-note that pays a 4% semi-annual coupon, and happens to price at par; i.e., $100 price =$2 coupon/[1+z(0.5)] + $102/[1+z(1.0)]^2. We can infer the one year rate, z(1.0), only because we assume the z(0.5) = 4% per the six-month bond; without that assumption, there are an infinite number of solutions to z(1.0) as we'd have two unknowns and one equation. Tuckman's law of one price "in action" is the assumption that we can retrieve spot price information from the six-month bond and re-use the z(0.5) discount factor (which is just a re-packaged spot rate) as a "public good." Here is how GARP has tested it (cleverly, IMO): "The following table gives the prices of two out for three US Treasury notes for settlement on August 30, 2008. All three notes will mature exactly one year later on August 30, 2009. Assume annual coupon payments and that all three bonds have the same coupon payments date. Coupon ----Price 2 - 7/8 ---- 98.40 4 - 1/2 ---- ???? 6 - 1/4 ---- 101.30 Approximately what would be the price of the 4-1/2 US Treasury note? a. 99.20 b. 99.40 c. 99.80 d. 100.20 Answer to GARP's 2010 here: http://www.bionicturtle.com/forum/threads/2010practice-exam-page12-question18.4468/ I wrote my own variation on this "law of one price" question here at T4.10.1, see http://www.bionicturtle.com/forum/threads/p1-t4-10-time-value-of-money.4879 Thanks, #### Aleksander Hansen ##### Well-Known Member That is [an easy, yet] very clever question as it tests ones understanding of two important concepts. I also like the fact that one can solve it in two ways, so it's quite forgiving: through YTM or prices. Your example easily beats mine in terms of both pedagogy and application. #### atandon ##### Member Many Thanks David, that was very useful. #### noalv4 ##### Member Hi David, Please note that in the study notes you gave this example with a wrong data: it the third bond: 5.25 instead of 6.25... #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber Hi noalv4 - Yes, agreed, according to our project manager, that error was fixed on 3/15 and saved up to a revision in the PDF, thanks! #### sohinichowdhury ##### Member Subscriber @David Harper CFA FRM I have a question on the Law of one Price. Since the discount factors incorporate the compounding frequency information, a d(1) with annual compounding is NOT equal to a d(1) with semiannual compounding, correct? Mathematically, the first one is 1/(1+r), the second one is 1/(1+r/2)^2. Put differently, can I use the d(1) backed out from the price of a 1yr zero coupon bond and use it to price a 1 yr Treasury bond that pays semiannual coupon? I think no since the d(1) for the Treasury bond will need to use a different compounding frequency. Can you please confirm? If that is the case, is the only way to solve for the Treasury bond price to back out the 1yr spot rates? (I also need to be provided the 0.5 yr spot rate). Thank you! #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber Hi @sohinichowdhury I really like your question (because I love financial philosophy)! The law of one price says that absent confounding factors (e.g., credit risk is a big one), there is only a single discount factor at each maturity. This is my expression but it is equivalent to Tuckman's definition: "This reasoning is an application of the law of one price: absent confounding factors (e.g., liquidity, financing, taxes, credit risk), identical sets of cash flows should sell for the same price." -- Tuckman, Bruce; Serrat, Angel. Fixed Income Securities: Tools for Today's Markets (Wiley Finance) (p. 55). Wiley. Kindle Edition. Let me briefly illustrate. But we will assume we are referring to the riskfree term structure (i.e., no credit risk) and my interest rate is too high but that's only to make comparisons easier. Let's say that one-year riskfree rate is 5.00% per annum with semi-annual compounding. If that is the case, then the one-year discount factor is, d(1.0) = 1/(1+0.050/2)^2 = 0.9518144. Consequently, the only correct theoretical price for the one-year (riskfree) bond is$95.18144.

It is a violation of the law of one price to assume that, additionally, the one-year riskfree rate is 5.00% per annum with quarterly compounding. That interest rate implies a different discount factor; 0.9515243. This demonstrates that 5.00% per annum is insufficient to specify an interest rate!

Instead, if the one-year discount factor is 0.9518144, then we can infer per [($100.00/$95.18144)^(1/4)-1]*4 = 4.96913% that the correct one year rate is also 4.96913% per annum with quarterly compounding. Or for that matter, the correct one year rate is ln(100/95.18144) = 4.9385% per annum with continuous compounding.

The law of one price says each maturity has only one discount factor, which is the same as saying that (in this case) a one-year zero-coupon bond can only have one price (!), but this implies a different stated (aka, nominal) rate for each different compound frequency. I'm not sure quite how to untangle you application, but hopefully that's enough guidance to apply. Thanks,

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#### sohinichowdhury

##### Member
Subscriber
@David Harper CFA FRM Thank you, David! I understand now why the discount rates are so much more important than the spot rates. The discount rates are unique by maturity, the interest rates depend on the compounding freq. Its a bit discomforting to realize that I picked up this key concept this late, with only 5 days to go before the test. But thank you so much.