Arbitrage-Free Valuation
Bond A here should be the same price as Bond A there.
If it's cheaper here than there, what would you do?
Poetic, but no.
No. You would lose money doing that.
Absolutely.
If you think of doing this at the same time, you're not really even using cash to do it. So simultaneously, you can sell it there and use the money to buy it here. You sell it for more there than you pay here, and a riskless profit is born. That's __arbitrage__.
You're not supposed to be able to do that. And you usually can't.
The __law of one price__ really just says that "Bond A here should be the same price as Bond A there." Two assets should have the same price. Also, two assets that provide the same benefits (like the same amount of cash at the same time) should sell at the same price. If that's true, then those assets demonstrate __arbitrage-free valuation__. How do you think the law of one price could hold, and yet Bond A and Bond B with the same cash flows and timing sell at different prices?
No. If the bonds have the same cash flows, the IRR of those cash flows would be the same, and that's what the yield is. But risk is a good answer. A riskier bond would be cheaper.
Sure. A riskier bond would be cheaper.
Think now about a yield curve. You have the typical simple example of Bond A with 10 years of coupons and a face value. If you discount it at some discount rate _z_, like 6%, what are you assuming about the yield curve?
No. A slope wouldn't be 6% in this case. The slope would be zero.
That's right.
The yield curve would be flat with each maturity having a 6% spot rate. Good luck finding that in the real world. Usually, each maturity is a different rate, and usually longer maturities have higher rates. Each cash flow must be discounted at the appropriate spot rate to get the value of the bond.
No. If it were zero, then the cash flows wouldn't need discounting at all.
Now consider that you do that for Bond A, and the market price is too low. You use spot rates of zero-coupon bond yields that investors are getting right now, and the market price of Bond A is below what you calculate. If you think about Bond A as a series of little zero-coupon bonds all expiring at different times, what could you do to arbitrage?
No. The bond price is too low. Selling it and going from there will lose you money.
Exactly!
Buy it, rip it apart, and sell off the pieces. This is called __stripping__, since you strip each coupon off of the bond and sell it off. If Bond A's market price is too low, then the discount rate used to value it is "too high," relative to what others demand for the same cash flows. So buy it here, and sell it there.
Also, if Bond A's market price were too high, then you could do the opposite: gather up the little zero-coupon bonds you need to build Bond A, and sell it at the "too high" price for a nice arbitrage profit. That's called __reconstitution__.
Probably not. They won't want your coupon bond—they just want zeros.
These are two main labels for arbitrage opportunities, and they are pretty much the same thing. If you add up 100 units of Asset C to make one unit of Asset D, and they have different returns, there's an arbitrage opportunity called __value additivity__, since you're adding up a lot of one thing to make another of different value. If it's just one or two of Asset C to make Asset D, then it's called __dominance__.
For example, here are four assets and their prices today and payoff in a year:
| Asset | Price Today | Payoff in One Year |
|---|---|---|
| E | 2.50 | 2.62 |
| F | 100 | 105 |
| G | 200 | 210 |
| H | 300 | 310 |
If you know that E and F have the same risk, and G and H have the same risk, then which pair looks more like dominance?
No. That's value additivity.
You got it! E and F together can be exploited by value additivity.
Since F pays a 5% return, and E's return is a little less, then just use 40 Es to make F'. From there, F' costs 100 today and will pay 104.80 in a year, so sell F' to buy F. Arbitrage.
Then exploit G and H as well. G is definitely a better deal, so sell two H and buy three G. There's no up-front cost. You'll owe 620 in a year while you're receiving 630. Arbitrage.
But again, it's rare to find deals like this, and exploiting arbitrage opportunities makes them go away quickly. But enjoy looking.
To summarize:
[[summary]]
For any bond, the price is really just a matter of doing a few things: estimate the cash flows (without options, those are given), determine the discount rate, and then discount the cash flows to the present. You've done that before many times.
Buy it here and sell it there
Buy it there and sell it here
Buy and sell it everywhere
Different risk
Different yields
It's zero
It's flat at 0.06
It's upward sloping at a slope of 0.06
Buy Bond A, and sell it in the zero-coupon market
Buy Bond A, and sell it in pieces as zero-coupon bonds
Sell Bond A, using the proceeds to purchase zero-coupon bonds that match the cash flows
E and F
G and H
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