present value of the floating-rate bond

CFA Level II
#1

Hi all, i am trying to figure out the pricing of floating rate bond, and i am having problem understanding the 1, for example

on Financial exam 123, it says “The present value of the floating-rate bond is easy: PVfloat=1.0. The reason that it’s 1.0 – par – is that at every coupon date the coupon resets to the current market rate; you’ll recall from Level I that when the coupon equals the market rate (YTM), the bond sells at par.”

Can some one explain why PV float =1? thanks

#2

The coupon rate is exactly the same as the discount rate, so it is a par bond.

Since floating bonds set the coupon rate in arrears, the coupon to be paid in 6 months, for example, is set today as the 6-month LIBOR and discounted at the 6-month LIBOR. Hence, the present value of the future cash flows is 1.

Numeric example:

6-month libor = 5%

Face value = 100

Price today = (100 + 5) / 1.05 = 100

The price of a floating rate bond is always equal to the face value at the payment dates.

Hope this helps.

#3

Remember your Level I Fixed Income:

  • If coupon > YTM, price > par
  • If coupon = YTM, price = par
  • if coupon < YTM, price < par

So, if the floating rate (coupon) = LIBOR = YTM, then the price = par.

#4

Fixed that for you.

Today’s rate change applies to the next (future) coupon payment: the rate is set in advance.

#5

You are totally right, thanks.

The coupon is set in advance and paid in arrear. Had the words confused.

Apologies.

#6

I’m confused (almost certainly because I’m misinterpreting your post) by this. I thought a bond traded at a discount when coupon rate < YTM?

#7

Correct.

I fixed it.

Good eye.

#8

so we have two period, period one is the period immediately after the currentt time point and period two is the one after, we are setting 1*2, and we are setting the interest rate at 2 at discount rate at period one level?

#9

I think i am confused the YTM is set at a later period and the discount rate is the interest rate of the next period so the YTM in the later period is set equal to the rate at the next period? say when you set FRA 1*2, when at period 0, you are seting floating rate between 1 and 2 the same as discount rate between 0 and 1?

#10

Suppose that you have a 5-year, floating-rate, USD1,000 bond with annual coupon payments and annual reset; the coupon rate is 1-year USD LIBOR. The 1-year USD LIBOR rates are:

  • Today: 5.0%
  • One year from today: 6.0%
  • Two years from today: 4.0%
  • Three years from today: 4.5%
  • Four years from today: 5.5%

This means that the coupon payments are:

  • One year from today: USD50 (= 5.0% × USD1,000)
  • Two years from today: USD60 (= 6.0% × USD1,000)
  • Three years from today: USD40 (= 4.0% × USD1,000)
  • Four years from today: USD45 (= 4.5% × USD1,000)
  • Five years from today: USD55 (= 5.5% × USD1,000)

Starting 5 years from today and working back, we have that:

  • The PV of the bond 4 years from now is USD1,000 (= (USD1,000 + USD55) / 1.055)
  • The PV of the bond 3 years from now is USD1,000 (= (USD1,000 + USD45) / 1.045)
  • The PV of the bond 2 years from now is USD1,000 (= (USD1,000 + USD40) / 1.040)
  • The PV of the bond 1 year from now is USD1,000 (= (USD1,000 + USD60) / 1.060)
  • The PV of the bond today is USD1,000 (= (USD1,000 + USD50) / 1.050)

Draw a timeline, put in all of the cash flows, and discount them back each year at the prevailing 1-year USD LIBOR rate; it’s straightforwaard.

#11

“One year from today: 6.0%”

“One year from today: USD50 (= 5.0% × USD1,000)”

are you doing the right calculations?

#12

Interest Rates and Price of Bond are inversely related and so if Coupon is higher than YTm(mark rate should have fallen) then you would expect that the price of a bond should be higher than par or quoted at a premium.

In other words, the Bond is paying higher than market rates prevailing (YTM) therefore price is higher. We should be careful to see what are the 2 variables which move inversely. Here in this case it is the YTM (Market rates) and Bond Price.

#13

In a floating rate bond, the next coupon to be paid is the LIBOR today.

So if the 1-year LIBOR today (t=0) is 5%. Then, at the end of year 1, the floating rate bond will pay 5% coupon. So it would be $1000 x 5% = $50 coupon.

Look at the S2000 calculations and note that the floating rate bond always has a value of 1000 at each coupon payment date. It is not much complicated at all.

#14

Yes: the rate is set _ in advance _ and paid _ in arrears _.

Today’s 1-year USD LIBOR rate of 5% establishes the coupon payment one year from today : USD50.