Actuarial Outpost Spot Rates and Forward Rates
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#1
06-16-2017, 03:17 PM
 CrunchinNumbers Member CAS SOA Join Date: Jul 2016 Studying for MFE? IDK Yet College: University of Missouri - St. Louis Favorite beer: Urban Chestnut: Zwickel Posts: 30
Spot Rates and Forward Rates

I thought I had a decent grasp of spot rates, and how they go together with forward rates... until this problem. I don't know if it is the wording, if it is the fact that the spot rate at the end of year 2 is for a zero coupon bond ending in year 4, or what. But somehow the answer is simply:

(1+j)*(1.04)*(1.07)^2 = (1.1)^4

Can someone break this down for me? I got 10.09% the first time around:

(1.07)^2/(1+j) = 1.04

But that is wrong, and the answer comes out to be 22.96%
#2
06-16-2017, 04:05 PM
 Breadmaker Member SOA Join Date: May 2009 Studying for CPD - and nuttin' else! College: Swigmore U Favorite beer: Guinness Posts: 4,053

1+j= 1.4641/1.19070, so j= 22.96%

For this question, you have 2 possible paths: invest for 4 years (RHS), or keep reinvesting at time 1 and time 2 (LHS). To avoid arbitrage, both sides have to equal.
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#3
06-16-2017, 04:34 PM
 CrunchinNumbers Member CAS SOA Join Date: Jul 2016 Studying for MFE? IDK Yet College: University of Missouri - St. Louis Favorite beer: Urban Chestnut: Zwickel Posts: 30

Quote:
 Originally Posted by Breadmaker For this question, you have 2 possible paths: invest for 4 years (RHS), or keep reinvesting at time 1 and time 2 (LHS). To avoid arbitrage, both sides have to equal.
Ah ok, I see now. So I guess the trick is to know that the spot rate for the four year zero coupon is the same every year? And does "at the end of year two" imply that it is for the 3rd spot rate?

Because I am seeing:

(1+j) is 1st spot rate
(1.04) is 2nd spot rate (says forward rate in question?)
(1.07) is the 3rd and 4th

Is that correct? This stuff is puzzling to me if you can't tell.
#4
06-16-2017, 11:57 PM
 Breadmaker Member SOA Join Date: May 2009 Studying for CPD - and nuttin' else! College: Swigmore U Favorite beer: Guinness Posts: 4,053

Yes, n year spot means that you invest for n years and the interest rate payable stays constant for each year. Spot rates are typically quoted at time 0. The only spot rate we know at time 0 is the 4 year rate: we have to derive the 1 year spot rate based on the implied forward rates given.

The phrase "at the end of year two" does not refer to a spot rate: rather, this is an implied forward rate. All it means is that if I invest for 2 years from time 0 (either by a 2 year spot, or 1 year spot, followed by the 1 year forward rate at time 1), then at time 2, I could reinvest my accumulated balance at an implied forward rate of 7%/year for 2 years.

Another possible path is to invest at 1 year spot, then reinvest at the implied 3 year forward rate at time 1. The math is very similar to your original question above - and you could even calculate this implied forward rate!
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#5
06-19-2017, 09:29 AM
 CrunchinNumbers Member CAS SOA Join Date: Jul 2016 Studying for MFE? IDK Yet College: University of Missouri - St. Louis Favorite beer: Urban Chestnut: Zwickel Posts: 30

Quote:
 Originally Posted by Breadmaker Another possible path is to invest at 1 year spot, then reinvest at the implied 3 year forward rate at time 1. The math is very similar to your original question above - and you could even calculate this implied forward rate!
Could you show how to get the implied 3 year forward at time 1? I understand your first part, and appreciate the insight. It helped a lot, honestly.

To get the implied 3 year forward I took:

(1.07^3)/(1.04^2) - 1 = .1326

and

(1.07^4)/(1.07^3) - 1 = .07

because I am not sure which one to use, but using this equation after that:

(1+j)*(1+3 year implied forward) = (1.1^4)

solving for j doesn't give me the answer with either forward.
#6
06-19-2017, 01:52 PM
 bigb Member Non-Actuary Join Date: Jul 2010 Posts: 9,175

If you wanted to calculate f(1,3) you would need to know the spot rates for years 2,3,4 which were given. This wasn't asked in the problem so not sure why you need to do that here (unless its for your edification).

(1+f(1,3))^3 = 1.04*1.07*1.07

(also note that 1.2296*(1+f(1,3))^3 = 1.10^4)

Last edited by bigb; 06-19-2017 at 01:57 PM..
#7
06-19-2017, 03:52 PM
 CrunchinNumbers Member CAS SOA Join Date: Jul 2016 Studying for MFE? IDK Yet College: University of Missouri - St. Louis Favorite beer: Urban Chestnut: Zwickel Posts: 30

Quote:
 Originally Posted by bigb unless its for your edification.
Yeah, I just wanted to know how it would work, but like Breadmaker said it all needs to be equal to avoid arbitrage, so that makes sense. I am just trying to get a full grasp of these spot, forward, and swap concepts. I feel this is my weakest area for this test, and since they removed the options portion I fear there will be a couple of questions in August.

Thank you!
#8
06-20-2017, 10:11 AM
 flee2thecleve14 CAS SOA Join Date: Jan 2017 College: University of Notre Dame Posts: 7

This is the approach I took given the information provided in the problem. Solving this is a matter of interpreting the variables given correctly and applying the correct equations. It may also help to draw out a timeline to visualize the different spot rates and forward rates and how they are related.

Given:
s4 = 0.10 = four year spot rate
f[2,4] = 0.07 = two-year forward rate from end of year two to end of year four
f[1,2] = 0.04 = one-year forward rate from end of year one to end of year two

Solve 2 equations, 2 unknowns:
(1) (1+s2)^2*(1+f[2,4])^2=(1+s4)^4
s2 = 0.1308
(2) (1+s1)*(1+f[1,2])=(1+s2)^2
s1 = 0.2296
#9
06-20-2017, 02:07 PM
 Breadmaker Member SOA Join Date: May 2009 Studying for CPD - and nuttin' else! College: Swigmore U Favorite beer: Guinness Posts: 4,053

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#10
06-20-2017, 04:53 PM
 NattyMo Member Non-Actuary Join Date: Nov 2016 Posts: 606

To get the hang of what is going on with this topic, I deliberately think of spot rates as forward rates starting at time zero.

When this problem mentioned "the expected spot rate at the end of year two" I thought that was lame. As described it's a forward rate.
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