Part B i) Introduction Profiting from price differentials between related financial instruments through the simultaneous buying and selling of different securities is known as relative value arbitrage (RVA). We have chosen to buy a 1 X 1 payer swaption which will be hedged with interest rate options on the 1Y and 2Y rates to take advantage of a twist in the US LIBOR yield curve. A macroeconomic and PCA analysis will develop an insight consistent with those a trader might use to establish a position based on US LIBOR yield curves.

Investment Rationale After 30 odd years of a secular bond bull market (rising bond prices, declining bond yields), we think we are about to enter a secular bear market in bonds (rising yields). There are a few reasons for this: 1. The past decades have been marked by disinflationary pressures coming from Asia i.e. Asian manufacturing countries producing low cost goods. In order to support their manufacturing exports, they have bought US$ assets (predominantly US Treasuries) to keep the value of their own currencies low. The disinflationary pressures also mean declining bond yields (particularly long dated bonds, when expectations of inflation fall). Rising inflationary pressures (particularly food and energy inflation) in countries like China are forcing Asian countries to increase wages and also allow their currencies to appreciate (through increased interest rates). Rising wages increase the price of exports. Rising Asian currencies mean Asian countries no longer need to buy US$/US Treasuries. The imbalances of the last few years (Asia selling to US and lending to the US (through buying Treasuries)) have seen US policy rates fall to zero. The US allowed the Asia/US imbalance to continue by continually lowering rates to fund excess consumption by US consumers. Policy rates have got nowhere to go now but up. There is also a chance that we are about to enter a secular inflationary cycle driven by booming commodity prices. The emergence of the middle class in Asia and their demand for commodities will drive this. Developed countries are heavily indebted. Rising yields will reflect high debt to GDP ratios and problems with fiscal deficits. U.S.A GDP is about US$ 14trn. Debts including unfunded liabilities are about US$ 70trn. Investors will become more worried about this

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problem over the coming year and will demand a higher risk premium for investing in Treasuries as seen recently with S&P putting the AAA status of the U.S.A under negative watch [1].

Interest Rate Curve Movements PCA is commonly used in analysis of fixed income portfolios which have a large number of correlated risk factors. The process allows us to reduce the dimensionality of the risk factor space to three components (eigenvalues selected which will be discussed below) simplifying the analytical process. This is done by establishing a linear combination between variables to extract the variance, removing the variance and identifying a second linear combination (eigenvectors). The process provides easily interpreted values with large explanatory powers. By undertaking principal component analysis on the US LIBOR curves we can analyse the historical curve movement and plan to hedge against some more common occurrences such as parallel shifts. Analysis will be performed on the covariance matrix which captures every movement in the variables including their individual volatilities (this is ignored by the correlation matrix). Taking two years of data, we have analysed movement in eight different US LIBOR curves ranging from one to ten years to get an overall picture of the type of movement our investment will be exposed to over its horizon. To establish what factors we will be analysing, we must assess the explanatory power of each factor. Data (below) describes the Eigenvalues which were observed from the dataset. We can see the power of each factor reflected in the variability explained. The first three components explain 99% of the variance giving strong assurance that the risk of movement to be noted in our curve will be from one of these factors.

The factors are: 1st Risk Factor: a parallel shift in the yield curve. This accounts for c. 85% of the variance. 2nd Risk Factor: a twist or steepening of the yield curve. This accounts for c. 13% of the variance. 3rd Risk Factor: a bowing of the yield curve i.e. rates at the long and short end of the curve move in one direction. This accounts for c. 1% of the variance. Having performed testing we have arrived at the following factor sensitivities:

A graphical representation of these first three factor sensitivities will offer further detail:

PC 1 is mildly sloping, similar to an upward sloping yield curve, showing us it is almost constant as a function of maturity. Should an increase in the first principal component be noted then amounts at the shorter end will shift by a larger amount than towards the long end (if PC 1 was a straight line then all would move the same amount). This analysis supports our assessment that when there is a shift it is not always parallel, which is part of our potential profit source. PC 2 is an almost linear decreasing function of maturity moving from positive to negative telling us that if PC 2 increases then the yield curve moves up at the short end and down at the long end. Reflection on the period in which sample taken (past two years) would support this as large variability has been noted. PC 3 is humped moving from negative to positive and back to negative again. An increase in PC 3 will shift the yield curve down at the ends and up in the middle. The small chance of this happening will be address in the selection of swaption length []. Interpretation of PCA and macroeconomic analysis performed above will allow us arrive at an appropriate RVA trade. Clearly the parallel shift Is the most common curve movement with 85% of the variance coming from this factor. Accordingly this will be hedged out through interest rate swaps.

Trade Summary Our trade consists of buying a 1X1 payer swaption along with a IY payer swap and a 2Y receiver IR swap. If the strike rate of the swaption is less than the prevailing market swap rate at expiry then the swaption will be exercised. From the macro analysis we conclude that we expect interest rates to move higher at shorter ends of the yield curve in a faster manner than the longer end. These rates are more sensitive to Fed policy change with longer rates a factor of inflation expectations and growth. The reason for this is that the longer dated the bond, the more time risks that are involved. At 2Y+, the years economic growth and inflation are more susceptible to change than over 1Y. Bondholders are paid more of a premium for the risks of changing inflation and/or economic growth. A sharp rise in inflation or signals the economy is growing too quickly means governments will increase policy rates to cool the economy or dampen inflation. Impact can vary depending on interest rate sensitive positions held in a portfolio i.e. the more volatile movement noted at the short end of the curve where we have fixed interest rate payments in exchange for floating can profit if interest rates rise more at 1Y rate then 2Y rate. While the 2Y horizon is quite near the 1Y, we believe the sensitivity of the short end of the curve over the short investment horizon will see this move more than may be seen by the longer end of the curve. We expect these rates to rise in the short term as the overall health of the American economy is called into question and doubts as to the ability of the government to raise the debt ceiling continue. This will be seen in shorter rates at the 1Y with 2Y+ affected to a lesser extent. This is key to our choice of buying a 1X1 payer swaption. Factors from our PCA analysis have also played a part. Focusing on the 1Y and 2Y as the key rates we will use to hedge our swaption position we can see that when there is one unit of a parallel shift the 1Y rate will decrease by .64189 bps while the 2Y rate will decrease by .47883 bps. Considering that c. 85% of the variance is made up of parallel shifts in the curve we have concluded that the most likely movement in the coming days is a parallel shift up but also consider the second risk factor (twist) a distinct possibility i.e. 145 of the time this has occurred. We can conclude from this that if preceding trends follow our position will be delta hedged c. 85% of the time as a parallel shift will be noted on these occasions. The curve movement we are hoping for is an increase at the immediate short end with no corresponding increase in the longer end of the curve i.e. the 1Y (and shorter maturities) to increase by more than the 2Y and longer maturities.

Interest Rate Sensitivities

This hedging strategy is implemented on the assumption that the curve moves in parallel shifts i.e. the movement of one rate such as the 1Y rate will note a commensurate movement in other periods i.e. 2Y rate thus making the delta hedging possible. Taking the first key rate (1Y), and holding all things equal, an increase in the zero rate will decrease the value of P(t,) resulting in a subsequent fall in the value of the swaption. The par swap offsetting hedge will note a commensurate increase in value. As the maturity payer swap will rise in value at the 1Y interest rate, the par swap hedge will be long the maturity payer swap. Similarly, an increase in the zero rate for the maturity will result in a decline in the value of P(t,) and accordingly a rise in the value of the swaption. An offsetting change in value in the par swap hedge will see the position balanced. The negative duration inherent in a long position in the maturity payer swap will increase the value of the position for an interest rate risk. Accordingly, the appropriate position to be taken is to be long the receiver swap or short the payer swap. This ensures the position is hedged and exposure to zero rate movement addressed (for parallel shifts). Profit and loss on this position will be derived from movement in two US LIBOR rates, the 1Y and 2Y, as illustrated below:

We can see from this diagram that the 1Y payer swaption will have a positive NPV for a rate increase and negative for a decrease. The opposite is the case for the 2Y position which will note a negative NPV for rate increase and positive for a decrease. Essentially, this is how the hedge works. A parallel movement in either direction will see a positive and negative NPV generated i.e. the loss on the 1Y payer will be equal to the profit on the swaption plus the profit noted on the 2Y receiver leaving the portfolio profit neutral. A steepening or flattening will note a change in the slope of the curve and potential for profit as long as the increase in the value of the swaption is greater than loss on the hedging swap positions. This also brings the potential for losses as a tilt in the opposing direction will result in a movement out of the money. Being able to accurately predict a curve

movement presents an opportunity for the skilled trader to use tools such as this for profit.

Results Over the course of the five day period we observed the following curve movement. While not immediately obvious from the diagram we observed a decline in the 1Y period from .4243 to .3873 (8.7% decline) while the 2Y rate noted a movement from .9978 to .8691 (12.9% decline). Overall there was a slight tilt downwards. We have observed sensitivities on the curve to be the greatest between the 1.5Y and 3.5Y with % changes in the yield ranging from 10.2% to 13.5% which quickly

dissipated to c.5% levels at other points in the curve. These movements are further analysed in the P&L section.

P&L Analysis While intraday movements on positions are noted, total profit and losses during the period sum to a loss of 2% or $1,973 as illustrated below:

Refer Appendix 4 for screenshots of same. We can observe the distance between the two key rates described above (IY & 2Y) widening from the day 0 gap of .

037% to the day 4 gap of .1287% which indicates the absence of a parallel shift in the yield curve which has huge implications for our portfolio. In general, a fall in yields will see a steepening of the yield curve and yield increases will note a flattening as the short end of the curve tends to be more susceptible to central bank policy. In light of this, analyses of the NPV of the hedge positions over the period will allow us describe the changes in the yield curve. Beginning on Day 0 both swaps will have a zero value as the cashflows generated from the fixed and floating legs will equal zero. The 1Y payer swap will have a positive NPV if the yield curve moves up, whereas the 2Y receiver will have a positive NPV if the yield curve moves down. Over the five days we can see we can see the value of the swaption has dropped by 34%, most significantly after day 1 trading which noted a 19% decrease. Day 1 (12-April-2011) noted a mild pivotal shift in the curve as the short end decreased with an increase in the long end. The decrease in the long end was significantly more than the increase in the short end causing sharp swaption and payer swap declines in value. Intuitively this makes sense as the swaption has locked in an option to pay fixed rates; currently rates at this time are lower than agreed fixed rate causing losses on swaption. The payer swap has declined in value for the same reason while the receiver swap which has fixed in payments at a lower rate than current floating rates has noted a substantial gain. The receiver increase is furthered into day 2 helping bring the portfolio into profit. The last two days of the trade saw the portfolio again return to losses as yield continued to decline in a nonparallel fashion. We can see from the P&L that the losses suffered on the swaption and 1Y payer are greater than profits on the 2Y receiver highlighting the greater movement down in the 1Y rate compared to the 2Y rate (for day 3 & 4). Considering that PVBP at inception of the trade is 3.6748 and having analysed the curve movement, losses seem reasonable. These movements are further investigated by analysing the trades Greeks.

Hedge Effectiveness Analysis Consideration of the overall effectiveness of the hedged position can be done through observation of actual results. Overall we can see declines in swaption value largely offset by increases in the receiver swap. Bearing in mind that the delta hedged position only takes into account small moves, the large movement noted on day 1 has given rise to a 4.8% loss as you would expect. These losses were retraced over the coming days with rates returning towards day 0 levels. For small intraday movements we can conclude the hedge was robust enough to offer protection from large losses. While the larger movement created losses, the $38,050 decrease in swaption value was matched by a $34,096 increase in the receiver swap which went a long way to offsetting the swaption loss. Losses at this date were compounded by payer swap losses.

The Greeks Vega In arriving at the amount of price movement to be attributed to volatility we have calculated the price of the option as of Day 0 noting the ATM volatilities previously obtained from Reuters. End of trading volatilities were obtained which noted increases across the majority of timeframes. As with all options, an increase in volatility will increase the value of the option (holding all things equally). Holding all Black model variables constant and replacing the Day 0 volatilities with Day 4 (fifth day of trading) volatilities we ran the model again which noted an increase in volatility and accordingly a profit. There was a 2.2647 bps increase in the swaption value which translates into an $11,324 profit from volatility. Theta Taking the model inputs and again holding all things constant with the exception of a change in todays date from Day 0 to Day 4, the price of the option was again calculated. A decrease in value of 1.9381 bps was noted which translates into a loss of $9,691. Swaption time decay has increased importance as option maturity approaches i.e. at 30 days to maturity the negative value has more weight than at 180 days. With one year to maturity we will not see a huge impact of time decay on this position.

Reflection on Trade In hindsight, the purchase of a longer dated swaption (such as 1X5) to take advantage of the declining yields further along the yield curve would have produced higher returns for the portfolio.

Appendices [1] Pimco, 2011. Investment Outlook. [investment report] April 2011, ed. Allianz: Pimco. (report attached) [2] Alexander, C. Practical Financial Econometrics: Market risk Analysis Vol.2.p64.John Wiley and Sons, 2009. [3] PCA Codeclc clear

Price_Data=xlsread('PCA_Data.ods'); X=Price_Data(:,1:8); % X=price2ret(X(:,1:6)); % X=flipud(X(:,1:8)); [Factor_Sensitivities,Principal_Components,Eigenvalues]=princomp(X); plot(Factor_Sensitivities(:,1:3)); title('Factors Driving Yield Curve'); xlabel('Date'); ylabel('Level'); legend('PC 1','PC 2','PC 3');

[4] Trade Screenshots Day 0 1x1 Payer

Day 0 2Y Receiver

Day 0 1Y Payer

Day 0 - Portfolio

Day 1 - Portfolio

Day 1 1X1 Payer

Day 1 -2Y Receiver

Day 1 1Y Payer

Day 2 1X1 Payer

Day 1 - Portfolio

Day 1 2Y Receiver

Day 2 1Y Payer

Day 3 1X1 Swaption

Day 3 2Y Receiver

Day 3 2Y Payer

Day 4 - Portfolio

Day 4 2Y Receiver

Day 4 1Y Payer

Day 4 1X1 Payer

Day 4 1X1 Payer

[5] Code for Greeks% Today's Date : 24th March 2010 % First, input initial market data (rates and volatilities) .... market_deposit_rates=[.15475 .28525 .44750 .60675 .77725]/100.*365/360; MDR=market_deposit_rates; % BID Par Swap Rates : [ 2Y, ..., 10Y ]. Semi-annual compounded. swap_rates=[.9490 1.554 2.088 2.4750 2.83 3.107 3.327 3.486 3.652]'./100; % ATM Vols Array % 3mx1Y 3mx2Y 3mx3Y 3mx4Y 3mx5Y % 6mx1Y 6mx2Y 6mx3Y 6mx4Y 6mx5Y % 1Yx1Y 1Yx2Y 1Yx3Y 1Yx4Y 1Yx5Y % 2Yx1Y 2Yx2Y 2Yx3Y 2Yx4Y 2Yx5Y % 3Yx1Y 3Yx2Y 3Yx3Y 3Yx4Y 3Yx5Y % 4Yx1Y 4Yx2Y 4Yx3Y 4Yx4Y 4Yx5Y % 5Yx1Y 5Yx2Y 5Yx3Y 5Yx4Y 5Yx5Y % Day Zero Vols Swaption_Vols_Matrix=... [79.5 64.6 52.5 44.8 39.7; 81.9 62.9 51.3 43.2 38.1; 68 52.2 43.7 37.5 33.7; 44.9 37.8 33.6 30.3 28.2; 33 29.7 27.6 26 24.9;

26.9 25.3 24.3 23.5 22.9; 23.8 23.2 22.6 21.9 21.4;]./100; % % Day 4 Vols % Swaption_Vols_Matrix=... % [76.3 64.2 52.3 44.6 40.1; % 83.3 64 51.7 43.4 38.2; % 72.1 54.3 44.4 38 34.2; % 46.1 35.8 32.1 30.6 28.8; % 33.8 29.7 27.3 26.2 24.9; % 27.5 25.6 24.3 23.4 23; % 23.9 23.2 22.8 22.3 21.4;]./100; % These inputs will have been generated as the "market rates" output from the HJM_Sim.m % forward curve simulation engine function file i.e. [MDR, par_swap_rates]. % Need to compute P so as to compute the FPSR and PVBP values [quarterly_zero_curve,quarterly_discount_curve, quarterly_forward_curve, swap_rates_full] = ... zero_curve_generator(market_deposit_rates, swap_rates); P=quarterly_discount_curve; delta=0.5; initial_date = '04/11/11';% Date when forward-starting swap was FIRST issued at PAR ! start_date = '04/13/13';% Start Date of fwd-starting swap expiry_date = '04/13/13'; swap_maturity_date = '04/13/17';% Forward Swap Maturity Date today_date = '04/15/11';% This is the valuation date. fixed_basis = '30/360'; Indicator='Payr'; % Or can be 'Rcvr' Fk=.036528 % 'ATM'; %Strike level of forward par swap rate. delta=0.5;% Approximation for fixed-leg day-count accrual fraction. NP=50*10^6; [Black_Swaption_Price, Fwd_Swap_Rate, PVBP] = ... Generic_Black_Swaption_Price_Calculator(Indicator, Swaption_Vols_Matrix, today_date, expiry_date, Fk, ... P, initial_date, start_date, swap_maturity_date, delta, NP, fixed_basis) disp [Note : Swaption price in basis points per unit notional principal]