Finding the 18 month spot rate - Fixed Income

CFA Level I
#1

Say we have the following data:

Maturity Coupon Price

0.5 0 96.78

1 0 94.36

1.5 6 97.55

What is the 18 month spot rate?

A) 4.2

B) 6.4

C) 7.8

These questions throw me, what’s the intuition and the quickest way to calculate this?

Thanks

#2

Having a hard time figuring out where to start with this… this is all the data they gave?

#3

I got 7.8%. Just calculated the YTM on the 1.5 year bond.

Not sure if thats correct though? Seems to simple

#4

I did that also, but I haven’t cracked this part of the curriculum yet. That just seemed to be the intuitive answer. I was under the impression that the spot rate equates the PV (spot price) to the value at a future date.

#5

YTM does not equal the 18 month spot rate, but they are darned close with the given data. I’m guessing your I/Y is 7.762%; I get a spot rate of roughly 7.81% using bootstrapping.

Look at the prices for the 0.5 and 1 year bonds: since both have zero coupons, the price is in effect the discount factor for the given maturity. Using these two values, project out the cash flows for times 0.5, 1.0, and 1.5, discount each one by the appropriate factor, and solve for the spot rate for the cash flow at time 1.5.

#6

I think you’re supposed to bootstrap… so pv=(pmt/(1+s1))+(pmt/((1+s2)^2))+(pmt/((1+X)^3))

solve for X, and maybe they’ve given the 1.5m bond to use as the PV??? I’ve only read through this… haven’t done a single mock question yet on any topic (AAAGH). So this is just my (prob wrong) opinion.

#7

Using the data provided, I got 7.79% entering 1.5 for N, -97.55 for PV, 6 for PMT, and 100 for FV. This made me think it was that straight forward. For the spot rate, should it only be calculated using zero coupons?

#8

I did something similar but the more I look at it I think the bootstrapping method is correct.

Regarding your calculation though, its a semi-annual pay bond i belive N should equal 3 (three semi-annual periods) and PMT should be 3 (6 is the annual coupon) - Then multiply I/Y by 2 to get annual YTM

#9

The 18-month YTM is not the spot rate; it’s the par rate.

To get from par rates to spot rates, you have to bootstrap. I wrote an article on this that may help: http://financialexamhelp123.com/par-curve-spot-curve-and-forward-curve/.

#10

yeah you can’t just use the 18 month spot rate and work out a normal TVM calc.

Sorry could someone please explain how to use the bootstrapping method for the question above? I tried following it in Schweser and got totally lost

Or for example a different version we have annual bonds :

Maturity Coupon Yield Price

1 3 3 1000

2 4 4 1000

3 5 5 1000

How would we caclualte the 2 year spot rate?

#11

Problem in original post done real fast:

Cash flows for 1.5 yr bond: $3 @ time 0.5, $3 @ time 1, $103 @ time 1.5

Price of 1.5 yr bond = PV (cash flows)

97.55 = 3 * 0.9678 + 3 *.9436 + 103 / (1 + spot /2)^3

97.55 - 2.9034 - 2.8308 = 103/ (1 + spot/2)^3

91.8158 = 103/(1 + spot/2)^3

(1+spot/2)^3 = 103 / 91.8158

1 +spot/2 = (103/91.8158) ^ (1/3)

1 + spot/2 = 1.03906

spot= 0.07812