# fixed income

I am having problem understanding the logic behind EOC # 23: Rd 10.

Explanation given:- Inter-market carry trades do not, in general, break even if each yield curve goes to its forward rates. Intra-market trades will break even if the curve goes to the forward rates because, by construction of the forward rates, all points on the curve will earn the “first-period” rate (that is, the rate for the holding period being considered). Inter-market trades need not break even unless the “first-period” rate is the same in the two markets. If the currency exposure is not hedged, then breaking even also requires that there be no change in the currency exchange rate.

Can anyone help?

Rd 20 I apologize.

I get a feeling the CFAI is making Fixed Income increasingly difficult with every passing year for the Level III candidates. Though the underlying concept is really simple, I believe the language is overcomplicated.

1. In an inter market carry trade, the yield is on account of 2 sources : the interest rate differential and the basic presumption of IRP not holding in the short run. However, it becomes simpler for me if the currency pair is hedged. I only need to focus on the relative difference of the forward rates. Now, if the forward rates are not same, I have an almost arbitrage opportunity in the first period. Borrow @ Lower rate and imvest @ higher rate. The only way this will not happen is when the first period forward rates are same across both countries. It becomes a break even point for me. Guaranteed ( in the backdrop of currency pair being hedged )

2. The intra market carry trade- should be much simpler to understand. First the currency pair problem is non existent here. Now regardless of whichever maturity pair we pick up( say for example 2 yr. And 5 yr . i.e. we borrow @ 2 yr. Rate and imvest in the 5 yr. Rate) , so long the actual yield realised is that of the forward rate predicted, both the rates have moved in the same trajectory. It is a simple case of pure expectation theory.Obviouslybyou don’t win just by placing your borrowing and investing bets on the same trajectory regardless of the maturity pair. Hence this becomes break even.

HerbsDelite,

Thanks for your response. I still am not clear.

1. When we invest , taking into account the first period forward rates, how does it become the same?

2. I got lost after - both the rates move in same trajectory…

Care to elaborate please?

Thanks

Suppose that the yield curve evolves as predicted by the forward curve. To make things simple, I’ll consider only two bonds, and they’ll both be zeros: a 1-year and a 5-year. Everything here will hold for coupon bonds as well, and bonds of any maturities.

The original discount factor for the 1-year bond is:

1 + s1

while the original discount factor for the 5-year bond is:

1 + s5 = (1 + s1)(1 + 1f1)(1 + 1f2)(1 + 1f3)(1 + 1f4)

One year later, the 1-year bond is paid off at par, so the yield is obviously s1. The yield curve has evolved as predicted by the forward curve, which means that:

• The original 1-year forward rate starting in 1 year is now the 1-year spot rate:
• s1-new = 1f1-old = 1f1
• The original 1-year forward rate starting in 2 years is now the 1-year forward rate starting in 1 year:
• 1f1-new = 1f2-old = 1f2
• 1f2-new = 1f3-old = 1f3
• 1f3-new = 1f4-old = 1f4

The discount rate on the (formerly 5-year, now) 4-year bond is:

1 + s4-new = (1 + s1-new)(1 + 1f1-new)(1 + 1f2-new)(1 + 1f3-new)

= (1 + 1f1)(1 + 1f2)(1 + 1f3)(1 + 1f4)

If you compare this discount rate to the original discount rate for the 5-year bond, you’ll see that what’s missing is the factor of (1 + s1), which means that this bond, too, has earned a yield of s1. The same holds for all cash flows, so it holds for coupon-paying bonds, and it holds for bonds of any maturity.

Remember the key stipulation: the yield curve has to evolve exactly as predicted by the forward curve.

I don’t think in the whole wide world anybody can explain any better than the Magician. You should get it now.

@ Thanks Bill

You’re too kind, and you’re quite welcome.

Magician,

Thanks for your help. Can you give me a numerical example please?

Thanks and I appreciate your help.

If you followed what I wrote, you could do a numerical example yourself.

Sigh.

Suppose that (a portion of) the par curve is:

• p1 = 1.00%
• p2 = 1.80%
• p3 = 2.20%
• p4 = 2.40%
• p5 = 2.50%

The (first 5 years of the) spot curve is:

• s1 = 1.0000%
• s2 = 1.8073%
• s3 = 2.2150%
• s4 = 2.4200%
• s5 = 2.5223%

You could work these out yourself; it’s Level I stuff.

The (first 5 years of the) forward curve is:

• 0f1 = 1.0000%
• 1f1 = 2.6210%
• 2f1 = 3.0354%
• 3f1 = 3.0374%
• 4f1 = 2.9325%

Again, you could work these out yourself; it’s still Level I stuff.

A 1-year, \$1,000 par zero sells today for:

\$1,000 / (1 + s1) = \$1,000 / 1.01 = \$990.10

A 5-year, \$1,000 par zero sells today for:

\$1,000 / (1 + s5)5 = \$1,000 / 1.0252235 = \$882.89

One year from today, the formerly 1-year, \$1,000 par zero is redeemed for \$1,000, so the yield is:

\$1,000 / \$990.10 – 1 = 0.0100 = 1.00%, the original s0.

One year from today, the new 4-year spot rate will satisfy:

(1 + s4-new)4 = (1 + 0f1-new)(1 + 1f1-new)(1 + 2f1-new)(1 + 3f1-new)

= (1 + 1f1)(1 + 2f1)(1 + 3f1)(1 + 4f1)

= (1.026210)(1.030354)(1.030374)(1.029325)

= 1.12142

The price of the (formerly 5-year, now) 4-year, \$1,000 par zero is:

\$1,000 / (1 + s4-new)4 = \$1,000 / 1.12142 = \$891.72

The yield on the (formerly 5-year, now) 4-year, \$1,000 par zero is:

\$891.72 / \$882.89 – 1 = 0.0100 = 1.00%, the original s0.

Brilliant as usual.

Just to append to it for the OP to elucidate further

"

One year from today, the new 4-year spot rate will satisfy:

(1 + s4-new)4 = (1 + 0f1-new)(1 + 1f1-new)(1 + 2f1-new)(1 + 3f1-new)

= (1 + 1f1)(1 + 2f1)(1 + 3f1)(1 + 4f1[AB1] )

_ - This is the key assumption. _

In case the above equality is not satisfied, then the yield curve has not shaped up as was predicted at time=0. Theoretically, the new forward rate starting from the end of 01st yr. or the begining of the 2nd NEED NOT NECESSARILY evolve as per the OLD forward rates predicted at time=0. In that case the TRAJECTORY is not maintained.

Thanks both Magician and Herbsdelight for taking time to write an excellent explanation.

It is very clear now.