NEW YORK UNIVERSITY STERN SCHOOL OF BUSINESS



Professor Richardson

International Fixed Income

Sample Exam

Solutions will be posted on my website by Thursday, April 27, at 5pm

TRUE / FALSE

1. Consider a 10-year default-free fixed-rate bond. Even though an inverse floater (from this bond, benchmarked against a 6-month floating rate note) has a higher duration than the underlying fixed rate bond, it's convexity is not necessarily larger.

False. The inverse floater is just a levered version of the underlying fixed-rate bond. Because this bond has positive duration and convexity, the inverse floater does also. In fact, if the split of the principal was 50-50 between the inverse floater and floater, it would be almost twice the convexity (with a small adjustment for the floater).

2. It is possible that the duration of a foreign 10-year zero bond in terms of its sensitivity to US interest rates is greater than the duration of a domestic US 10-year zero bond (even assuming the same interest rate level in the two countries).

True. If foreign interest rates change more when US rates change, its beta will be greater than one, and its duration to US rates will exceed that of the US bond. For many international bond markets though, the beta tends to be less than one.

3. In order to get rid of currency risk in holding a foreign bond for a given period, one can sell a forward contract over that same period. This way, one necessarily gets rid of the currency risk, and is left with a domestic US bond.

False. When you sell a currency forward, you only get rid of the amount sold forward. Because the price of the bond is uncertain over a given period, the investor will still face some residual interest rate risk. That said, a substantial portion of the exchange rate risk can be hedged.

4. One stylized fact about bond returns is that they are not that highly correlated across countries. To the extent that bond returns are predictable (and let's assume they are), this means that the predictable component is also not highly correlated. In other words, there is more than one factor that drives global expected returns.

False. Bond returns can have a predictable and an unpredictable component, so that, if the unpredictable component dominates the bond's variation and it is uncorrelated across markets, you can still get highly correlated predictable components even if bond returns don't move together that closely. This in fact characterizes the G7 countries we studied in class.

5. The duration of say a 30-year floating-rate Brady bond will always be less than a corresponding 30-year fixed-rate Brady bond on the same country even if the fixed-rate bond has some of its coupons guaranteed. (For the purpose of this question, you can assume the same probability of default and the principal amounts are the same and guaranteed as well.)

False. If the default probability is close to one, then the floating rate bond will have a duration equal to a 30-year zero due to its guaranteed principal. In contrast, the 30-year fixed rate bond's duration will be somewhat less because it has some guaranteed coupons.

6. During the Mexican Peso crisis of 1994, the interest rate on Tesebonos shot up. Since these bonds are dollar denominated, this can only mean there was an increase in the probability of default. (Assume devaluation risk has no impact on US rates and that convertibility of the currency was not an issue).

True. Tesebonos include an interest rate risk and a default risk component. As we know US rates did not move that much during this period; thus, the only explanation is that there was increase in the probability the bonds would not pay off.

7. In one of LTCM's trades, they took a long position in Italian government bonds and a short position in corresponding German bonds. If monetary union was to take place, then these bonds should eventually converge in value to avoid arbitrage. (Assume no tax issues exist.)

True. As we've seen in class, bonds denominated in the same currency (e.g., Euro) should offer the same rates if they have the same level of default. In fact, for the most part, all these bonds have converged quite closely in value.

8. Even if the default risk of a swap is zero, a fixed-against-Libor swap will have a swap rate greater than the prevailing par yield rate on Treasuries.

True. A swap is a derivative contract which finds the fixed rate that sets a fixed-rate bond equal to a floating rate bond based on Libor. Since Libor represents a spread over treasuries, and the swap's underlying floating rate-note has no default risk, the fixed rate should offer a spread over Treasuries.

PROBLEMS

1. Consider the following information:

σ(US)=10 basis points weekly, σ(euro)=20 basis points weekly, S(euro/$)=1.05,

σ(S)=100 basis points weekly.

Assume zero correlation between euro rates and the exchange rate.

a.What is the volatility of the $-adjusted return on euro-denominated bonds with durations equal to 1 and to 10.

The volatility can be expressed as

Thus, for a duration of 1, we get 102 basis points; and, for a duration of 102, we get 224 basis points.

b. What % of this volatility is due to exchange rate volatility?

For durations of 1 and 10, we respectively get 98.04% and 44.64%, i.e.

c. Comment on why these %'s differ?

Exchange rates are 5 times more volatile than Euro interest rates, which means, given the zero correlation, the volatility of these bond returns can be described almost completely by exchange rates for low duration bonds; as the duration increases, and the interest rate sensitivity gets increased, exchange rates become relatively less important.

d. Explain what happens to these bonds if US rates go up by 100 basis points. Give an estimate of the % change in value. (Ignore convexity.)

We don't have enough information to answer this question. We would have to have either (I) Euro rates Beta with US rates and US rates correlation with exchange rates, or (II) forward currency rates to convert foreign bond's cash flows to US cash flows.

2. Consider the two following term structures in the US and UK respectively:

6-month 1-year

US 6% 6.5%

UK 5% 4.5%

Assume semi-annual compounding and the $/Pd exchange rate is 1.5

a. What is the value of a 5% $100 par US bond?

US rates correspond to .9709 and .9380 discount factors, so that the bond's value is

2.5(d0.5 + 102.5(d1 = 98.58

b. What is the forward currency curve for $/Pds.

The appropriate formula is:

Thus, using the current exchange rate of 1.5 $/pds, we get respectively 1.515 and 1.530

c. If you were to convert the cash flows of the US bond into pounds using the forward market, describe what those cash flows would be?

Using these forward rates, you could convert the $2.50 and $102.50 in 6 months and 1 year into 1.65 pounds and 66.99 pounds, respectively.

d. What is the value of this US bond in pounds?

This is easy - you can just immediately convert the $98.58 at the current exchange rate into 65.72 pounds. That is, you don't need to discount all the pound cash flows. It holds by no arbitrage.

3. Consider the following characteristics of a 1-year semi-annual 8% Brady bond with the consequences of default meaning that the bond never gets paid off: (I) guaranteed principal, and (II) 10% probability of default each period. Assume the discount factors for 6-months and 1-year are respectively .95 and .90.

a. What is the value of the Brady bond?

4(1-p)(d0.5 + 4(1-p)(1-p)(d1 +100(d1 = non-guaranteed + guaranteed part

4(.90)(.95 + 4(.81)(.90 +100(.9= $96.34

b. Is the strip spread greater than or less than 25%. (Show your work).?

4(.90)(.95 + 4(.81)(.90 = $6.34 - the nonguaranteed part

The question is we need to find the spread that satisfies the following:

A discount factor .95 and .90 implies respectively 10.53% and 10.82%. Adding 25% to these discount rates gives us the following value:

This means that the strip spread needs to be less than 25% to explain the $6.34.

c. If the bond had no guarantee, would its duration be higher or lower? Explain your answer in detail.

If the bond had no guarantee, then its duration would be lower. This is because the guarantee ensures that the principal represents a greater fraction of the market value of the bond. Since the principal has longer maturity, it also pushes up the interest rate sensitivity of the bond. (Note that the probability of default here is fixed at 10% - all we are doing is varying the guarantees).

4. Today, time 0, a newly issued (zero cost) 1-year semi-annual pay plain vanilla interest rate swap has a swap rate of 8%. The time 0.5 cash flow to the counterparty who is long $100 notional amount of the swap is $1.00. Assume there are no swap spreads.

a) What is the 1-year par rate?

par rate = current swap rate = 8%

b) What is the 0.5-year zero rate?

100(r0.5-0.08)/2 = 1 ( r0.5 = 10%

c) What is the price of $1 par of a 1-year zero?

4(d0.5 + 104(d1 = 100 ( d1 = (100 - 4(d0.5)/104 = (100 - 4/1.05)/104 = 0.924908

d) What is the value of $100 par of a 1-year semi-annual pay inverse floating rate note that pays 16% minus floating?

This inverse floater is equivalent to two 1-year 8%-coupon bonds minus one floater. Each 8% bond is priced at par and the floater is priced at par, so the inverse floater is also worth par: $100.

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