Question present value floating rate leg

[dla00]

Hi @David Harper CFA FRM

I'm quite confused on why when computing the present value of the floating leg only the notional amount and the LIBOR that corresponds to the last payment date is considered. I don't understand why we don't consider all the other LIBOR rates in the corresponding pay dates.

Can someone please explain why this is the case?

For instance, if we have a 1 million notional swap with a floating rate based on 6-month LIBOR and swap has remaining life of 15 months with pay dates at 3,9,15 months. Spot LIBOR rates are 5.4% at 3 months, 5.6% at 9 months and 5.8% at 15 months. LIBOR at last payment date was 5%.

Then it says that the present value of the floating leg is (1 million + 1 million * .05/2) / 1.054^0.25.

I don't understand the above logic. Why are we not considering the other dates? e.g. why there are no intermediate cash flows for the floating rate?

 

[David Harper CFA FRM]

Hi @dla00 Because it is such a common question over the years, I created this post (below my YT video on Interest Rate Swaps) at https://forum.bionicturtle.com/thre...n-vanilla-interest-rate-swap.22446/post-82735 i.e.,

FAQ: Why does a floater price to par on coupon settlement dates; aka, how are we able to value the floating-rate leg of the swap as a single cash flow when it's really a stream of future cash flows?

About the valuation of an interest rate swap "as if two bonds," a very popular question we get is: why is it that we can value the floating-rate leg by assuming only one cash flow? At first, it does seem strange given that the floating-rate leg of a swap will pay several coupons in the future (not just one)! The assumption, as I start to review in the video above (at about 15:00) is that a floating-rate note (FRN) prices exactly to par at each coupon settlement date. Put another way, you should be willing to pay $100.00 (or $1,000.00; i.e., face value) for a note/bond with a coupon that floats, if the floating rate used to determine the coupon (e.g., LIBOR) is the same rate that you will use as your discount rate (ie., to discount your expected future cash flows).

This is not immediately intuitive to many of us. Although for some people it probably is intuitive: first imagine you buy a 10-year fixed-rate annual-pay bond with a 4.0% per annum coupon rate while the (spot rate) term structure is conveniently flat at 4.0%; you will pay exactly $100.00; i.e., if coupon rate equals yield, the bond's price is par because the interest (the coupons) covers the yield, no more and no less, so you don't require any expected capital appreciation. If the interest rate increases to 5.0%, the price of your bond drops (per the inverse relationship between price and yield) because, if you want to sell the bond in the "new" 5.0% rate environment, nobody will pay $100.00 for it. Why would anybody pay "full price" for a bond yielding 4.0% if the new yield is 5.0%?! No, they will only pay, and you can only sell, at the price given by -PV(yield = 5%, 10, coupon = 4%, $100) = $92.28. The buyer now expects a 5.0% yield and they're getting $4.00/92.28 = 4.33% from the coupon plus the other ~ 0.67% from the expected capital appreciation as the bond will pull-to-par and redeem at $100.00 (it's actually not the only way to parse these components but that's not the topic here). In any case, in the case of a typical vanilla bond, the price goes down when the interest rate goes up because a lower price is required to make the same coupon stream competitive at the new interest rate. But if the coupons floats up or down in tandem with the discount rate, then the bond price does not need to adjust! If your bond is instead a floater and the interest rate increases from 4.0% to 5.0%, then the coupon increases also: the new coupon "handles" the entire yield need, no price adjustment is necessary. The price only needs to adjust during the time between coupons, when the coupon rate and the discount rate temporarily diverge.

Below I tried to illustrate why a floater must price to par at each coupon date. Very simple illustration. The XLS is here at https://www.dropbox.com/s/hwrv6m6svt5eqwu/070420-floater-prices-to-par.xlsx?dl=0

The only inputs are in yellow. The upper panel imagines a spot rate term structure {1.00% at 1.0 year, 1.50% at 2.0 years, ..., 4.00% at 5.0 years}. The spot rate term structure informs the forward rate term structure (in blue). The expected future cash flows are based on the forward rates; i.e., the forward rates, if unbiased, represent the expected future one-year spot rates. If we discount those cash flow, the bond's price is exactly $100.00. Put simply, if the coupon cash flows are paid at the same rate that we use to discount the cash flows, the bond prices to par. That's the idea.

In the lower panel, I simply added a 60 basis premium to the forward rate term structure. Then DCF valued the bond. First, again, if discounted at the revised forward rates, the price is (again) par. But second (bottom row), I discounted the revised cash flows at the original spot rates (which is also the same as discounting at the original forward rate); in this case, the bond becomes a premium-priced bond. That's because the discount rates are lower than the rates used to determine the cash flow. I hope that's interesting.

 

[HPASI2304]

Hi @David Harper CFA FRM, I would much appreciate if you could answer the below question:

Why do we use simple rates when it's about t-bills valuation, I mean, in the following exercise:

1. A US T-bill paying $10,000 after 13 weeks sells at a discount rate of 2.04%, how much does Tami pay for this T-bill?

10,000 (1-0.0204*(13/52)) = 9449

In this case, the discount rate is treated as a simple discount rate, not composed, could you please explain what is the rationale behind it?

Thank you beforehand,

Regards!