Valuation of a Capped Floater
A floater floats.
If you have a floating-rate bond, and the rate adjusts to whatever the market rate is, what do you expect the price of the bond to be?
That's right!
Premiums and discounts are offered when coupon rates are "too low" or "too high." A floater is always just right.
Not quite.
That would be the case if the investor were getting a better deal than the market suggests. But that's not the case with a floater.
No.
Discount prices are needed to induce investors to accept coupons that are too low; that's not the case with a floater.
Now imagine that the floater has a maximum coupon rate that will be paid. In other words, it's a __capped floater__. Starting from a par value, where do you think the price of this capped floater will move?
Exactly!
Actually, it will be below 100.
This is a bad deal for the investor, so now a discount price will have to be offered. It's possible that the rates will never reach the cap, but you never know. What you can do, though, is model it.
Take this interest rate tree for a three-year, annual coupon, capped floater, which is capped at a 6% coupon.
The coupon is __set in arrears__, meaning that the coupon rate is decided at the start of the period and paid at the end. So the Time 1 payment, no matter if it's up or down, will be 4%.
What's the highest coupon that could be paid in Time 3?
Yes!
The potential rates are shown in the tree for Time 2, since coupons are set in arrears. But the 8.49% coupon is above the 6.00% cap, so that's the maximum amount that can be paid at the end, with the face value.
No. Recall that this is a capped floater. The coupon won't ever be this high.
No, that's not true.
The coupon payments in Time 3 are set in arrears, so you just need those Time 2 rates.
Then, with traditional backward induction, the potential rates of 6.00%, 6.00%, and 4.66% are discounted one period using the Time 2 rates. That leaves you with Time 2 values of
$$\displaystyle \frac{106}{1.084936} = 97.702$$
$$\displaystyle \frac{106}{1.062922} = 99.725$$
and
$$\displaystyle \frac{104.6614}{1.046614} = 100 $$.
How would you best explain that third value of exactly 100 from the lowest branch?
Nope.
There's no luck in an interest rate tree. Just beautiful, lognormal design.
That's right.
Without the cap, all values would be 100, leading back to 100 in Time 0. It's just when you run into the cap that the 6% coupon will be discounted by something larger, leading to an inevitable discount price in Time 0.
No, it's not just coincidence.
Taking these values back to Time 1, you'll notice that the rates are both below 6%. So no problem there, and no adjustment to be made. Add those coupons, do the typical discounting and averaging, and you end up with a Time 0 discount price of 99.36 (small rounding differences may occur, depending on how you do the math).
Since the "uncapped" floater is priced at par, what can you say about the value of the cap?
No, it's positive for the issuer. But the value is 0.64.
Yes! The value is 0.64 (and it's positive, not negative, for the issuer).
It's just the difference between par (the value of an "uncapped" floater) and the capped floater. That difference here is
$$\displaystyle 100 - 99.36 = 0.64 $$.
That's value for the issuer, and a negative for the investor, which is why it's subtracted from par in the bond's value. Remember, a floater floats; a capped floater also floats, but it may hit a ceiling.
In summary:
[[summary]]
Par
A premium
A discount
Below 100
Above 100
6.00%
8.49%
It's unknown
Luck
It's a floater
Coincidence
It's 0.64
It's negative for the issuer
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