FEC Financial Engineering Club

FECFinancial Engineering Club

Welcome • Website coming soon • Committee • applications due • at midnight! • Email uiuc.fec@gmail.com to get on mailing list

Agenda • Interest rates and returns • Bonds • Bond risk • Other fixed income instruments

Interest Rates

Interest Rates • Compensation to the owner of an asset (generally cash) for loss of the asset’s use • Ex) You deposit $1000 into your savings account. They pay you a small interest rate since they may use your deposit for various reasons and you cannot readily use it without a withdrawal. For principal amount invested/borrowed rate (per arbitrary period) number of periods invested/borrowed • Simple interest: • Ex) $100 invested now at 6% per month for 10 months: interest earned = $100 x .06 x 10 = $60.

Interest Rates • Compound Interest: Interest earned each period is reinvested at the same rate • Compound interest earned = • Ex) $100 invested now at 6% per month , compounded monthly for 10 months: • Amount earned = $100 x .106^ 10 = $179.08 • Interest earned = $179.08 – 100 = 79.08 • Interest at a rate r per period, compounded N1 times per period, but invested over N2 compounding periods earns the interest: • When is per year this is known as the effective annual interest rate

Interest Rates • Ex) What is the value of $100, invested at a rate of 5% annually for two years, compounded monthly? • 100 x (1+.05/12)^24 = 110.4941 • Continuously compounded interest rate: = • What is the value of $100, invested at a rate of 5% annually for two years, continuously compounded? • 100 x = 100 x = 110.5171

More Scenarios • Ex) Suppose you invest $1000 at 5% per year today and in every subsequent year until 2020 (7 investments). What is your investment worth in 2030? • 7 investments of $1000. • First one for 16 years • Second for 15 years • Etc $1000 $1000 $1000 $1000 $1000 $1000 $1000 ???? February 2018 February 2020 February 2030 February 2017 February 2019 February 2015 February 2014 February 2016

More Scenarios • Alternative approach—consider the value of the investment in 2020, once all investments have been pooled, then accrue interest from 2020 to 2030. • Investment in February 2020 • This is known as an annuity • Valueannuity = P x • $1000 x = 8142.01 • Invest this for 10 years: • 8142.01*(1.05)^10 = $13262.36 $1000 $1000 $1000 $1000 $1000 $1000 $1000 ???? February 2018 February 2020 February 2030 February 2017 February 2019 February 2015 February 2014 February 2016

Time Value of Money • Up until now, we have been considering how much money is worth in the future, after being invested at different rates • One can always invest their free cash at some interest rate • Opportunity cost: the cost of a choice; the amount of economic value forgone by doing A instead of B • There is an opportunity cost to holding cash—it can always be invested. Ideally it would earn interest in the future and be worth more. • Money today is worth more in the future • It can be invested • Money in the future is worth less today • You can invest current cash to grow into a future sum

Time Value of Money • Future value: the value of an asset at a specific date in the future • Effectively what we have been calculating FV = Present Value * • Present value: the value today of an asset in the future, if it exists • The reverse of what we have been calculating PV = This method of reducing a future cash flow to its value today is known as discounting it back to today

Present Value • Ex) How much money would you need to invest at 5% per year to earn $1000 in 3 years? • Alternatively, how much is $1000 in 3 years worth now at a rate of 5%? • Ex) At what rate would you need to invest $100 annually to earn $250 in three years? • PV = = = $863.84 100 = = r =

Which Interest Rate? • There are many different types of interest rates—which one do I use in my calculations? BTMM <GO> In Bloomberg

Term Structure of Interest Rates • Interest rates do not remain constant over time and over borrowing tenures • Default risk—the chance that the borrower may default, or be unable to pay interest to the lender. A longer borrowing time increases the default risk. • Opportunity costs • Interest rates reflect economic conditions • The shape of an interest rate curve over time is known as the yield curve

Term Structure of Interest Rates • The quantitative values of the interest rate at different term lengths is known as the term-structure of interest rates • This refers more to bond yields than regular interest rates • More on this in Bonds • Who determines how interest rates change? • The Federal Reserve—the central banking system of the united states—uses monetary policy to force the Fed Funds rate to a target set by them (the fed funds target rate) • This helps determine several other rates • For interbank lending, rates are based on LIBOR (London Interbank Offer rate), which is determined by the British Banker’s Association

More on Interest Rates • Interest rates are pivotal in valuing future cash flows and therefore the majority of financial products. • Here we have calculated the present and future value of streams of cash flows with certain interest rate environments • There are sophisticated models for interest rates which are used heavily on interest rate/fixed income derivatives such as floors, caps, floorlets, caplets, swaps, swaptions, etc • LIBOR Market Model (LMM): • Hull-White

Bonds

Bond Basics • A debt instrument in which an investor loans money to the issuer (by buying the bond) and the issuer agrees to repay the principal with interest over the life of the bond until it matures. • A bond has several key features • PAR value (also known as face value) is the notional amount that is borrowed by the issuer and hence the amount on which is paid interest • You may by the bond for cheaper /more than (discount/premium to) PAR • Maturity is the date at which the issuer has agreed to repay the principal • Coupon the interest rate specifying the regular interest payments • Usually a fixed amount at regular intervals over the life of a bond (like an annuity) • In this regard bonds are referred to as fixed-income instruments • Market Value—if the bond is traded in a secondary market, you may buy it after it is issued at this price

Bond Basics • Ex) In January 2014, Goldman Sachs issues a bond with a PAR value of $10,000 with a semiannual coupon of 5%. It matures in January 2019. • PAR = $10,000 • Coupon = 5%, semiannually • Maturity is January 2019 $10,500 -$10,000 $500 $500 $500 $500 $500 $500 $500 $500 $500 Jan 2016 July 2018 July 2015 July 2016 Jan 2018 Jan 2019 July 2014 Jan 2017 Jan 2014 Jan 2015 July 2017

Where do Bonds Come From? • Bonds are a major way institutions finance their operations • Interest payments on bonds are tax-deductible, making them cheap financing option • However, they are risky—too many short term obligations to creditors can cause default • Governments also issue bonds • (US) Treasury bills are short term (< 2 years until maturity) bonds that are generally zero-coupon • (US) Treasury bonds are longer term instruments • Bonds issued by governments are generally referred to as sovereign debt

Where Do Bonds Come From?

Zero-Coupon Bonds • A zero-coupon bond is a bond with no coupon. • However it is bought at a steep discount to PAR • Ex) Goldman Sachs sells a zero-coupon bond with a PAR-value of $10,000 for $8,500 that matures in 5 years. What is the effective interest rate (yield) to the borrower? $10,000 -$8,500 PV =  8,500 =  r = .033038 t = 4 t = 3 t = 5 t = 1 t = 0 t = 2

Zero-Coupon Bonds • Note that buying a zero-coupon bond is equivalent to lending money • You lend the value at which you buy it to the issuer and you earn the yield • Conversely, short-selling a zero-coupon bond is equivalent to borrowing money • This concept is very important for future financial engineering applications

Valuing a Bond • For valuation purposes a bond is simply a stream of future cash flows • Must know your discounting rate r—the rate at which you can borrow cash • Ex) In January 2014, Goldman Sachs issues a bond with a PAR value of $10,000 with a semiannual coupon of 5%. It matures in January 2019. What is this bond’s value? Suppose we can borrow cash at 4% annually $10,500 -$10,000 $500 $500 $500 $500 $500 $500 $500 $500 $500 Price = PV(all the cashflows) = + + … + = 12,715.18 Jan 2016 July 2018 July 2015 July 2016 Jan 2018 Jan 2019 July 2014 Jan 2017 Jan 2014 Jan 2015 July 2017

Valuing a Bond • Note that the value of a bond is given by PV(Bond) = ti is the time at which cash flow i is realized CFi is the ith cash flow at time ti , which may not be equal for all i ri is the interest rate at time ti

Yield To maturity • Note that the market price of the bond may not be equal to the present value of the bond? • What discount factor will equate the present value of the bond to the market value? Market Value = PV(Bond) = • This is known as the yield to maturity • If you buy the bond and hold it, what is your equivalent yield or interest rate on the bond

Yield To Maturity • Ex) In January 2014, Goldman Sachs issues a bond with a PAR value of $10,000 with a semiannual coupon of 5%. It matures in January 2019. You can buy this bond in the secondary market for $11,000. What is its yield to maturity? Recall that Price = PV(all the cashflows) = PV(CF1) + PV(CF2) + … + PV(CF10) = + + … + = 11,000 Easy way: using Excel’s solver  y = .0774 How do I arrive at this?

Calculating Yield to Maturity • More rigorous ways: Root-finding methods like Newton-Raphson, bisection method • Newton-Raphson: Given a function f(x) and an initial point iterate via

Calculating Yield to Maturity • Bisection Method—Guaranteed to converge on an interval where and have opposite signs • Binary search on the interval , evaluate at midpoint and update interval appropriately to keep the signs of and opposite • Terminate when within a reasonable range of 0 or a and b are very close • Here • is the market price

Yield To Maturity

Yield To Maturity • Fundamental property of bond prices: they are inversely related to interest rates

Bond Risk

Bond Risk • How can we measure the risk of the price of a bond? • If you need to sell your bond today, you may have lost money • Some risks of bonds (Qualitative) • Default risk—probability that issuer will be unable to repay (default) principal and interest rates • Interest rate risk—implicit assumption of bond pricing/discounting that we will be able to reinvest at the rate we discount at. This may not be true. • What is the primary factor that directly affects a bonds price: • Cash flows—these do not change after bond is issued • Interest rates—subject to change

Duration • Formal definition: The average time until maturity of a bond, weighted by cash flow • This version of duration is known as Macaulay Duration, named after Frederick Macaulay • Ex) Goldman Sachs sells a zero-coupon bond with a PAR-value of $10,000 for $8,500 that matures in 5 years. $10,000 -$8,500 Duration = For a zero-coupon bond, the Macaulay duration is equal to its maturity t = 4 t = 3 t = 5 t = 1 t = 0 t = 2

Duration • In general Macaulay Duration = • Ex) In January 2014, Goldman Sachs issues a bond with a PAR value of $10,000 with a semiannual coupon of 5%. It matures in January 2019. What is this bond’s Macaulay Duration? $10,500 -$10,000 $500 $500 $500 $500 $500 $500 $500 $500 $500 r = .04 Price = 12,715.18 Macaulay Duration = = 4.175847 Jan 2016 July 2018 July 2015 July 2016 Jan 2018 Jan 2019 July 2014 Jan 2017 Jan 2014 Jan 2015 July 2017 Sum = 53,096.64081

Duration • Why would average time until maturity be related to how sensitive a bond’s price is? • Re-examine the pricing formula: PV(Bond) = • Duration is the sensitivity of a bond’s price to interest rates. • A more useful metric—Modified duration: Modified Duration(r0) = , where P is the present value of the bond, r is the variable interest rate, and r0 is a numerical rate

Duration • Note that Macaulay Duration = • However, if rates are continuously compounded Macaulay Duration =

Duration • Example) Suppose your bond has a (modified) duration of 5. If the interest rate rises by 1%, how does the price of your bond change? • Increases by 1% • Decreases by 1% • Increases by 5% • Decreases by 5%

Convexity • Note that changes in bonds’prices with respect to interest rates are not linear • How many continuous derivatives would you say has? • Convexity is the second derivative of a bonds price wrt interest rates, normalized by price: Convexity(r0) = PV(Bond) =

Convexity • Example) Suppose your bond has a (modified) duration of 5 and a convexity of 15. It is valued at $100. If the interest rate (currently at 5%) rises by 1%, How much does duration change? by 1*15 = 15% how does the price of your bond change?

Duration and Convexity Calculate how bond prices change given changes in yield: P2 =P1 + -duration * (Δr) + .5*convexity* (Δr)2 (Taylor’s expansion) • In the previous example: • P1 = 100 • Δr = .01*5 = .05 • duration = 15 • convexity = 5 • If interest rates change by Δr =.01, the bonds price changes to P2 = • 100 + 5*(.05) + .5*15*(.05)2= 100.0313 P1 ΔP P2 P2-duration * (Δr) Convexity correction r2 r1 Δr

Immunization • Suppose you, as a borrower, had several (floating-rate) interest expenses. • Since interest rates may change, you wish to hedge your risk here. • What is your risk? • Hedging—Buying and selling of assets so as to use some of their features to ‘cancel’ out risks of another. • If I can match the duration of my liability (the owed interest rate expense) with that of an asset (specifically a bond), I can buy the bond and have a net duration of zero.

Immunization • Ex) You owe $1000 in 2 years. You would like to invest in bonds now to meet that obligation in the future. Your borrowing rate is 9%/year and you can invest in the following two bonds: How much do you invest in each? First note that PV(1000) = You want to invest in Bond 1 and in Bond 2 to meet this obligation: Want to match the duration of your obligation (2 years): Thus x = 6.88 ‘units’ of bond 1 and y = 3.24 ‘units’ of bond 2

Immunization • What was the point of immunization? • Hedging higher-order derivatives is simple • It follows the same process • Hedging n derivatives will lead to a system of n linear equations—therefore we need n bonds is the present value of bond i is the amount of ‘units’ of bond i is the duration of bond i is the jth price-normalized derivative of bond i refer to the obligation

Immunization This is summarized conveniently by

Questions?

Next Lecture (2/26/2014) • The world of trading • Trading ecosystem • Market microstructure • Roles of brokers, traders, exchanges

Upcoming External Events • IMC Financial Markets Tech Talk on Hadoop –Thursday, Feb 13th 6pm – 2240 DCL • UIUC MSFE Information Session – Wednesday, Feb 28th 4pm – 106B Engineering Hall

Thank you! • Facebook: http://www.facebook.com/UIUCFEC • LinkedIn: http://www.linkedin.com/financialengineeringclub • Email: uiuc.fec@gmail.com • Next Meeting: • Statistics Primer • Wed. 19thFeb. 6-7pm • 1310 DCL • Committee • applications due • at midnight!

FEC Financial Engineering Club