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# GARP 2010 Practice Exam

#### intuit2k2

##### New Member
Q 18 - GARP 2010 Practice Exam

Here are the prices for 2 out of 3 US Treasury Notes for settlement on August 30, 2008. All 3 bonds will mature exactly 1 year later on August 30, 2009. Assume annual coupon payments and that all three bonds have the same coupon payment date.

Coupon Price
2 7/8 98.4
4 1/2 ?
6 1/4 101.3

What is the price of the 4.5 US treasury Note?

Solution: 2.875x + 6.25(1-x) = 4.5; x = 52%

implying that the 4.5 is 52% 2 7/8 and 42% 6 1/4

ie P = 52% * 98.4 + 48% * 101.3 = 99.8
Could anyone provide intuition for this?

#### kthwow

##### New Member
I just think of this question very simply...
First, assume the no arbitrage situation... Value of coupon 2 7/8 with price 98.40 equals that of coupon 4 1/2 with price ? or that of coupon 6 1/4 with price 101.30
second, in this situation we dont care of buying between #0.6(weighed quantity) 98.40 plus #0.4(weighed quantity) 101.30 and #1.0(whole buying) price ?(which is 4 1/2 coupon).
third, in the context of present value, price depends on yield and future cash-flow...(function of these)
In the question, all is same except for coupon(which is varible), we can set no arbitragy strategy, 2.875% * x% + 6.25%*(1-x%) = 4.5% * X (which is the whole percentage we would like to buy) so we can solve for X% (which is 52%) and another is 48% (1-52%).
Adding weighting Value of two US treasury notes equals unknown one US treasury notes price...
So what we want to know is 99.80 (=52%*98.40+42%*101.30)

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
…. And just to piggyback on kthwow's no arbitrage explanation, this means to apply (Tuckman's) law of one price: the idea that equivalent cash flows must cost you the same (have the same price to you). If you have $99.80, then one "portfolio" buys the 4 1/2 coupon bond and earns$4.50. The second portfolio, consisting of the other two bonds, if it costs the same 99.80 must produce the same 4.50 cash flow (coupon). The mix solved for is the necessary combination (52% and 48%) that produces an identical cash flow (4.50) so it should cost the same (99.80). The solution breaks this into two steps:

1. If i have 100% to invest, what mix of bonds #1 and #3 give me the same cash flow (coupon) as 100% of bond #2
2. Knowing the mix (52%/48%), there is only one bond #2 price that satisfies the "law of one price."

David

#### faryal

##### New Member
…. And just to piggyback on kthwow's no arbitrage explanation, this means to apply (Tuckman's) law of one price: the idea that equivalent cash flows must cost you the same (have the same price to you). If you have $99.80, then one "portfolio" buys the 4 1/2 coupon bond and earns$4.50. The second portfolio, consisting of the other two bonds, if it costs the same 99.80 must produce the same 4.50 cash flow (coupon). The mix solved for is the necessary combination (52% and 48%) that produces an identical cash flow (4.50) so it should cost the same (99.80). The solution breaks this into two steps:

1. If i have 100% to invest, what mix of bonds #1 and #3 give me the same cash flow (coupon) as 100% of bond #2
2. Knowing the mix (52%/48%), there is only one bond #2 price that satisfies the "law of one price."

David
hi.... how to get 52%? like how can we find that bond 2 has 52% weight od bond 1?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @faryal please see previous discussions at either

#### Evelyn.K

##### New Member
Hi,

I am trying to solve a past year FRM question but couldn't quite understand how to get the answer.

Following are the settlement prices of 2 out of 3 US Treasury notes for settlement on Aug 30 2008. All 3 notes will mature exactly one year later on 30 Aug 2009. Assume coupon pmts and all 3 bonds have the same couple payment date.

Couple Price
2 7/8 98.40
4 1/2 ?
6 1/4 101.30

Approximately what would be the price of the 4 1/2 bond? (Answer: 99.80)

Cheers.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Evelyn.K I moved your question to this preexisting thread. The links go to paid forums I think but here is a highlight:
David Harper (Mar 19, 2013): Hi Sunny,

It is really just a linear interpolation, but the intuition is something like:
we can pay 100% of unknown price (P) in order to receive the $4.5 coupon cash flow ... or, if law of one price prevails, we should be able to get the same cash (no arbitrage) by splitting up the 100%*P into two bonds: x% to the 98.40 bond which will give us$2.785*x, plus
(1-x%) to the 101.30 bond which will give us $6.25*(1-x) our assumption is that$4.5 = 2.875*x + 6.25*(1-x)
solving for x,
4.5 = 2.875x + 6.25 - 6.25x = 6.25 +2.875x - 6.25x;
-1.75 = -3.375x, and
x = -1.75/-3.375 ~= 52%;

i.e., we get can "reconstitute" the single \$4.5 coupon with two partial coupons if we split up unknown (P) into 52% of the 2.875 bond and the remaining 48% into the 6.25 bond.
According to this, P = 52%*98.40 + 48%*101.30 = 99.80. It's ultimately "merely" a linear interpolation, but i hope this helps,

..... and it looks like a member noticed a glitch with this question (I do know that GARP often gets these setups incoherently), Mad_Mac wrote:
Mad_Mac (Feb 16th, 2014): What makes this question particularly confusing is that the implied interest rate curve has such a bizarre shape.

We're told in the question that we're dealing with treasury notes maturing exactly one year from now, and that there is one (annual) coupon remaining. At first glance it would appear that this is a straight-forward question, with the coupon being paid together with the principal, and that there is only one zero-coupon interest rate that is of relevance.

However, when you back out what this zero-coupon rate is, you find different values for the two bonds. We can therefore infer that there are several cash flows, and that the bonds are not simply the present value of a combined principal & coupon payment in one year from now.

So in other words, it has to be the case that were dealing with bonds that will pay a coupon at some point before maturity. We're told that the coupons are paid annually, so we can therefore conclude that there are only two cash flows remaining: the payment of the coupon, and the payment of the principal.

We're not given the exact payment date for when the coupon is paid. All we know is that it is not paid together with the principal.

Let:
DFCP = DiscountFactorCouponPayment ; and
DFPP = DiscountFactorPrincipalPayment

2.875 x DFCP + 100 x DFPP = 98.4
6.25 x DFCP + 100 x DFPP = 101.3

This solves for DFCP = 0.859259 and DFPP = 0.959296

Now we can find the price of the 4 1/2 Treasury Note: 4.5 x 0.859259 + 100 x 0.959296 = 99.796

Other Notes:

DFPP is 0.959296 , which translates into an annual interest rate of 4.2%. Nothing strange with that value.

However, lets now turn to DFCP. With a factor of 0.859259, the rate, before annualising, is 16.4%.

Again, we don't know when the coupon will be paid, but no matter what assumptions you make, the shape of the interest rate curve will be very strange looking.

Say for example that the 16.4% rate is for a cash payment 11 months from now. The annualised rate would then be almost 18% vs the 4.2% 12-month rate.

If the 16.4% rate is for a cash payment 6 months from now, then the annualised rate would be almost 33%.

I can appreciate that the point of the question is to illustrate how the law of one price, and interpolation work. The point of this reply is rather to show that the traditional approach that you'd take when solving fixed income questions still works. However, linear interpolation is clearly a quicker way to solve the problem.