# Fixed Income in a Financial Crisis

In the first part of our report, we investigate if a 35 basis points yield spread represents mispricing of two bonds, both with the same maturity but one with a coupon rate of 10. 625% and the other 4. 25%. Our investigation also determines if the yield spread represents an arbitrage opportunity. In our investigation, we calculate the theoretical yield spread between the two bonds and compare the figure with the observed yield spread. It is cited in the case that the observed yield spread could be due to different liquidity premium for each bond or simply due to different durations.

Through our calculations, we discover that the difference in duration between the two bonds does result in different theoretical yields, 2. 899% for Bond4. 25 and 2. 639% for Bond10. 625. However, the theoretical yield spread should be -26 basis points instead of the observed 35 basis points. Hence, we conclude that while difference in duration does result in a difference in yield, the observed yield spread of 35 basis points cannot be explained by duration. Further, the 61 basis points difference between the observed and theoretical yield spreads indicates that Bond10.

## Fixed Income in a Financial Crisis Essay Example

625 is underpriced relative to Bond4. 25 and the proposed strategy of long Bond10. 625 and short Bond4. 25 can work. In the second part of our report, we investigate if Franey’s duration-neutral strategy is immune to shifts in yield curves. We perform illustrative examples that show that the value of the arbitrage portfolio increases in the event of a steepening yield curve and decreases in the event of a flattening yield curve. Additionally, we highlight that even with a duration-neutral strategy, parallel shifts in the yield curve will still affect the portfolio’s value due to the effect of convexity.

Further, as the overall portfolio’s convexity is negative, any parallel shift in the yield curve will lead to a decrease in portfolio value. We support our points with detailed calculations found in the Appendix. Lastly, we take into account the more practical aspects of Franey’s proposed strategy by considering the relevant market risk and institutional risk factors. The market risks involved include yield curve risk, inflation risk and credit risk. Institutional risks include changes to financing costs that Franey might incur, such as a higher hair cut and margin call requirements should the yield spread widen.

Our risks evaluation shows that the institutional risks of margin calls and higher hair cut represent the most significant risks to Franey’s strategy. Inability to meet additional capital requirements could lead force selling of Franey’s position, resulting in unplanned losses. We take into further consideration the unpredictability of liquidity premiums during the time of the case study and determined that Franey’s proposed high leverage arbitrage strategy was highly risky.

Through our analysis, it becomes clear to us that while the observed 35 basis points yield spread represents an arbitrage opportunity, the unpredictability of liquidity premiums means that to execute this arbitrage strategy safely, an ability to hold the portfolio for a long period or through to maturity is required. Given the high degree of leverage that Franey is intending to take, we think that the proposed trade is too risky and should not be made. 1) Verifying an Arbitrage Opportunity Price Table 1 – Theoretical Bond Prices and Yields.

The 3. 62% calculated actual yield for Bond10. 625 differs from the 3. 61% provided in the case study. This is likely due to the difference in precision of Periods used by Bloomberg and our excel calculations (ie. Bloomberg may not have used periods of 0. 5667). We calculated the theoretical yield and theoretical price of each bond by summing the individual present value of the cash flows, which were discounted using the respective zero-coupon rates provided for each period up to August 2015.

Since the zero-coupon rates given are at an annual rate, we obtained the semi-annual rate by taking half of the respective annual zero-coupon rate of each period. Through this method, we obtained theoretical yields of the 4. 25% coupon bond and 10. 625% coupon bond to be 2. 899% and 2. 639% respectively. The corresponding theoretical prices of the bonds are $108. 27 for the 4. 25% coupon bond and $149. 31 for the 10. 625% coupon bond (see Table 1 above). Comparing each bond’s theoretical yield and price to its actual yield and price, we find that both bonds are underpriced.

However, this alone is insufficient to conclude that an arbitrage opportunity exists, since our calculated theoretical prices ignore the effects of liquidity premium. As, in this exercise, we are unable to calculate what the true liquidity premium for each bond should be, we consequently do not have a true price for each bond to compare with and ascertain whether each bond is underpriced (ie. both the theoretical and actual prices may not be the true price). Nonetheless, we have calculated the implied liquidity premium for each bond as the difference between the theoretical yield and actual yield (see Table 1 above).

Yield Spread To better determine if an arbitrage opportunity exists, the theoretical and actual yield spreads of both bonds can be compared. This determines whether both bonds are properly priced relative to each other, eliminating the need to have true prices for both bonds to determine if each bond is rightly priced. Based on our calculations, the theoretical yield spread (Bond10. 625 – Bond4. 25) should be -26 basis points (see section below on “Duration” for more explanation) instead of the 35 basis points observed, representing a difference of 61 basis points.

Therefore, as price distortions stemming from liquidity premiums subside, we expect the yield spread to tend towards the theoretical -26 basis points. In the unlikely event that the effects of liquidity premium persist for a long time, the yield spread should still converge to zero near to or at maturity of both bonds. As such, we conclude that the 10. 625% coupon bond is underpriced relative to the 4. 25% coupon bond and the 4. 25% coupon bond overpriced relative to the 10. 625% coupon bond. Hence, we believe that an arbitrage opportunity exists and Franey’s strategy of long 10.

625% coupon bond and short 4. 25% coupon bond will work. Duration We find that duration is unable to explain the apparent mispricing. Our calculated theoretical yields for both bonds indicate that the bond with higher duration has a higher yield than the bond with lower duration. This is in line with the slope of the interpolated yield curve, which is upward sloping. When the yield curve is upward sloping, the bond with higher duration should command higher yield since a greater proportion of its present value will be reduced by higher future interest rates, as compared to the bond with lower duration.

Conceptually, the scenario observed in the case study where the bond with higher duration has a lower yield than the bond with lower duration should exist only when the yield curve is downward sloping. Therefore, duration does not explain the apparent mispricing. The price difference observed may be attributable to other factors such as the premiums on a bond. Examples are premiums paid on interest rate risk, default risk and liquidity preference.

However, the price difference seems to be primarily driven by the liquidity premium the “on-the-run” bond commands. This is because default risk would be similar as the bonds are issued by the same organization and since the yield curve is expected to increase, the premium paid on interest rate risk would be minimal. Also, the -26 basis points yield spread as a result of difference in duration indicates that there is significant mispricing in the observed yields, since the yield spread before taking into account effects of liquidity premium should be -26 basis points and not 35 basis points.

Testing Impact of Shifts in Yield Curve on Arbitrage Portfolio Value To find out the effects of non-parallel shifts in the yield curve, we simulated two scenarios: 1) steepening and 2) flattening. Steepening Table 2 – Illustrative Example of Steepening Yield Curve In the case of steepening, we assumed that the interest rate will increase cumulatively by 0. 1% for every period (one period is six-months) and decrease in the same manner in the case of flattening. After including these effects to the given spot rates, we used the new interest rates to discount the cash flows of the two bonds.

In the case of steepening, the gain on the portfolio will be 28. 64% or $49. 93 (=$224. 29 – $174. 36) based on the new portfolio value of $224. 29 and an initial actual investment of $174. 36 (=10. 625% bond value ($1,418. 28 + $23. 68 accrual) – 4. 25% bond value ($1,256. 37 + $11. 23 accrual)). If we calculated returns based on the theoretical bond prices and consequent theoretical initial investment of $221. 91, the gain will be 1. 07% or $2. 38. These figures are calculated based on each $1,000 face value of Bond10. 625 longed and $1,185.

60 Bond4. 25 shorted. Flattening Table 3 – Illustrative Example of Flattening Yield Curve In the case of flattening, the gain in the portfolio will be25. 56% or $44. 57 (=$218. 93 – $174. 36) based on the new portfolio value of $218. 93 and an initial investment of $174. 36. If we used the theoretical initial investment of $221. 91, there will be a loss of 1. 34% or $2. 98. Similarly, these figures are calculated based on each $1,000 face value of Bond10. 625 longed and $1,185. 60 Bond4. 25 shorted. Discussion of Results

In conclusion, in both cases of the non-parallel shifts in the yield curve, the value of the arbitrage portfolio will increase since the current prices of the two bonds are mispriced relative to each other. However, if the two bonds are trading at their theoretical prices, steepening of the yield curve will increase the value of the arbitrage portfolio but in the case of flattening, it will decrease. We would like to point out that even though Franey’s strategy is duration neutral, the value of the arbitrage portfolio will change even if there is a parallel shift.

This is due to the effects of convexity. Based on our calculations, the 4. 25% bond has a higher dollar convexity than the 10. 625% bond. Therefore, when interest rates increase, the 4. 25% bond will experience lower price depreciation due to convexity and when interest rates decrease, the 4. 25% bond will experience higher price appreciation due to convexity. Since the portfolio convexity is negative (see appendix 1 for detailed calculations and breakdown), the value of the arbitrage portfolio will decrease if there are any changes in interest rates. 3) Market risks

Market risk which in this case refers to the exposure to movements in the interest rates and shifts in the yield curve is best measured by duration as it is a summary measure of maturity, coupon and yield. Bonds with lower duration are less sensitive to change in yield while bonds with higher duration are more sensitive to changes in yield. Further, it is imperative to point out that in volatile interest rate environment, duration is an inadequate measure of interest rate risk as it does not serve as a good approximation during large interest rate shifts. a) Yield curve risk

As discussed previously, even when a portfolio is duration-neutral, it would still be exposed to yield curve risk. More specifically, the portfolio would no longer be duration-neutral in the event of large parallel shifts or non-parallel shifts in the yield curve. While this primarily stems from the differences in maturities of bonds that comprise a bond portfolio, other factors like the yield and cash flows of bonds could also cause yield curve risk to arise in a portfolio. Bonds with longer maturities tend to be more sensitive to shifts in yield curve and this translates into greater price volatility.

However, what we find most unique about this portfolio is that it comprises of merely two bonds which have the same maturity. In addition, when we simulated a steepening and flattening of the yield curve, we saw a gain on the arbitrage portfolio even though it was no longer duration-neutral. The yield curve risk would only arise if the bonds were trading at their theoretical prices, in which case, would have already been sold to realize the arbitrage profit. Hence yield curve risk is not very relevant in this case. b) Inflation risk

This is the risk that will diminish the value of coupon cash flows received from the bond due to inflation. This risk is of moderate nature c) Reinvestment risk This is the risk that coupons payments received from the bond would be reinvested at lower than expected rates due to fall in interest rates. Reinvestment risk is typically expected to be offset by capital gains on the bond’s market value when interest rate falls. 4) Institutional Risks a) Liquidity risk / Risk of margin call As this is an arbitrage strategy, some form of financing would be needed from brokers and in return, Franey would be required to post margin.

Any extreme market volatility could lead to widening of yield spreads due to higher perceived illiquidity of the Treasury which expires earlier. This would in turn require Franey to post higher margins. In the event of insufficient funds, this could ultimately lead to a margin call. Franey would be forced to realize the losses on his position even though he would have otherwise made a profit if he had held both bonds to maturity according to his arbitrage strategy. b) Credit risk Credit risk is the risk that the issuer fails to pay coupons or ultimately defaults on the bond.

This is less relevant as compared to portfolios comprising institutional bonds as US Treasuries as generally considered to be free of credit risk. 5) Conclusion We have shown in our report that there exists an arbitrage opportunity given the difference between the theoretical yield spread (-26 basis points) and actual yield spread (35 basis points). Despite so, a key uncertainty lies in the timing of realisation of the arbitrage profit. As stated in the case study, if held to maturity, the arbitrage strategy would yield a riskless profit with present value of $26 for each $1,000 face value of Bond10.

625 longed and $1,185. 60 Bond4. 25 shorted, before taking into consideration financing costs. However, the case suggests that Franey would like to realise his profit and close out his position as soon as the yield spread falls back down to zero. Hence how soon Franey is able to realise the arbitrage profit will depend on the direction and magnitude of change in the yield spread of the 2 bonds. We further observe that interest rate environment at the point in time of the case study was volatile.

This is evident from the fact that Treasury yield spreads that have historically been near to zero have widened significantly and more frequently during the period of the case study. Such volatility contributes to the risk of this arbitrage strategy since Franey is in a way speculating that the yield spread will narrow significantly soon. Should the yield spread narrow quickly and greatly, the strategy will yield high returns with little financing costs incurred. However, should the yield spread widen or fail to narrow greatly, more financing costs will be incurred.

Institutional risk factors, as elaborated in the earlier section, will also come into play. In particular, additional capital requirements resulting from yield spread widening or increase in hair cut represents a significant risk. Should Franey be compelled to force sell part of his portfolio at unfavourable prices, he could incur substantial losses that would render the arbitrage strategy unprofitable. It is clear to us that the ability to hold onto this arbitrage portfolio for a long period or through to maturity is important in rendering this arbitrage strategy a safe one.

Considering the high degree of leverage that Franey is planning to use for this strategy and the possible significant additional capital requirements that Franey may have to contribute should conditions change unfavourably, Franey’s holding power is questionable. This is thus a high-risk strategy. Keeping in mind the unpredictability of yield spreads during the time of the case study, we are of the opinion that Franey should adopt a more prudent approach and not make the trade. Appendix 1