VII. ARBITRAGE AND HEDGING WITH FIXED INCOME INSTRUMENTS AND CURRENCIES

7.1. Arbitrage with Riskless Bonds Riskless bonds can be replicated with portfolios of other riskless bonds if their payments are and made on the same dates. Consider Bond D, a 3-year, 20% coupon bond selling for $1360. bA = 0 bB = 10/3 bC = -7/3

In Matrix Format =

B. Fixed Income Hedging Fixed income instruments provide for fixed interest payments at fixed intervals and principal repayments. In the absence of default and liquidity risk (and hybrid or adjustable features), uncertainties in interest rate shifts are the primary source of pricing risk for many fixed income instruments.

7.2. Fixed Income Hedging A bond maturing in n periods with a face value of F pays interest annually at a rate of c with yield y.

Bond Sources of Risk In general, bond risks might be categorized as follows: Default or credit risk: the bond issuer may not fulfill all of its obligations Liquidity risk: there may not exist an efficient market for investors to resell their bonds Interest rate risk: market interest rate fluctuations affect values of existing bonds.

Fixed Income Portfolio Dedication Assume that a fund needs to make payments of $12,000,000 in one year, $14,000,000 in two years, and $15,000,000 in three years.

7.3. Fixed Income Portfolio Immunization Bonds, particularly those with longer terms to maturity are subject to market value fluctuations after they are issued, primarily due to changes in interest rates offered on new issues. Generally, interest rate increases on new bond issues decrease values of bonds that are already outstanding; interest rate decreases on new bond issues increase values of bonds that are already outstanding. Immunization models such as the duration model are intended to describe the proportional change in the value of a bond induced by a change in interest rates or yields of new issues.

Bond Duration Bond duration measures the proportional sensitivity of a bond to changes in the market rate of interest.

Deriving Bond Duration

Deriving Bond Duration

Calculating Bond Duration 2-year, 10% $1,000 Bond selling for $966.20

Portfolio Immunization Immunization strategies are concerned with matching present values of asset portfolios with present values of cash flows associated with future liabilities. The simple duration immunization strategy assumes: Changes in (1 + y) are infinitesimal. The yield curve is flat (yields do not vary over terms to maturity). Yield curve shifts are parallel; that is, short- and long-term interest rates change by the same amount. Only interest rate risk is significant.

Immunization Illustration Assume a flat yield curve, such that all yields to maturity equal 4%. The fund manager has anticipated cash payouts of $12,000,000, $14,000,000 and $15,000,000. The flat yield curve of 4% implies a value for the liability stream is $37,816,120.

Immunization Illustration Yields to maturity equal 4%. Cash payouts of $12,000,000, $14,000,000 and $15,000,000. Value for the liability stream is $37,817,193.90. We calculate bond and liability durations:

Duration and Immunization Portfolio immunization is accomplished when the duration of the portfolio of bonds equals the duration (-2.048): DurA ∙ wA + DurB ∙ wB + DurC ∙ wC = DurL wA + wB + wC = 1 -1.962 ∙ wA – 2.8437 ∙ wB – 3 ∙ wC = -2.048 wA + wB + wC = 1 There are an infinity of solutions to this two-equation, three-variable system.

Duration and Immunization, Continued = = Solving the system: = = = =

Convexity The first two derivatives can be used in a second order Taylor series expansion to approximate new bond prices induced by changes in interest rates as follows:

Convexity and the Taylor Series The first two derivatives can be used in a second order Taylor series expansion to approximate bond prices after changes in interest rates:

Convexity Illustration

Duration, Convexity and Immunization Portfolio immunization is accomplished when the duration and the convexity of the portfolio of bonds equals the duration and convexity (6.38) of the liability stream:   DurA ∙ wA + DurB ∙ wB + DurC ∙ wC = Duro ConvA ∙ wA + ConvB ∙ wB + ConvC ∙ wC = Convo wA + wB + wC = 1 -1.962 ∙ wA – 2.837 ∙ wB – 3 ∙ wC = -1.975 5.41 ∙ wA + 10.30 ∙ wB + 11.09 ∙ wC = 6.38 wA + wB + wC = 1 The single solution to this 3 X 3 system of equations is wA = 0.106, wB = 5.166 and wC = -4.272. This system provides an improved immunization strategy when interest rate changes are finite.

5.4. Term Structure, Interest Rate Contracts and Hedging The Pure Expectations Theory: The Yield Curve can be bootstrapped

Simultaneous Estimation of Discount Functions Three coupon bonds are trading at known prices. Bond yields or spot rates must be determined simultaneously to avoid associating contradictory rates for the annual coupons on each of the three bills.

Spot and Forward Rates Spot rates are as follows: Forward rates are as follows:

The Evolution of Short-Term Rates Merton [1973] Rendleman and Bartter [1980] Vasicek Cox, Ingersoll and Ross [1985]

Vasicek Process

7.5. Arbitrage with Currencies Triangular arbitrage exploits the relative price difference between one currency and two other currencies. Suppose the buying and selling prices of EUR 1 is USD 1.2816. However, the South African Rand (SFR) has a price (buying or selling) equal to USD 0.2000 or EUR 0.1600. Since USD 0.20 = EUR 0.16, dividing both figures by 0.16 implies that USD1.25 = EUR1. But, this is inconsistent with the currency price information given above, which states that USD1.2816 = EUR1.0. In terms of the SFR, it appears that the USD is too strong relative to the EUR, so we will start by selling USD0.20 for SFR1 as per the price given above. We will cover the short position in USD by selling EUR0.16, which actually nets us .16 ∙ USD 1.2816 = USD0.2051. We will cover our short position in EUR by selling SFR at the price listed above. USD SFR EUR Sell USD0.20 for SFR1 -0.2000 +1.0000 Sell EUR0.16 for USD0.2051 +0.2051 - 0.1600 Sell SFR1.0 for EUR0.16 -1.0000 +0.1600 Totals +0.0051 0 0

Parity and Arbitrage in FX Markets Purchase Power Parity (PPP) Interest Rate Parity (IRP) 3. Forward rates equal expected spot rates 4. The Fisher Effect 5. The International Fisher effect. Collectively, these conditions are often referred to as the International Equilibrium Model.

Purchase Power Parity in Spot Markets PUS0 = S0 ∙ PEU0 For example, if the spot price of gold in U.S. markets is PUS0 = USD 1400 per ounce and EUR 1000 in German markets, the spot exchange rate must be USD 1.400/EUR. Any single deviation from these rates will lead to an arbitrage opportunity.

Purchase Power Parity in Forward Markets PUS1 = F1 ∙ PUK1 PUS0 = $20.00; PUK0 = £12.50; S0 = 1.6 πus = 10%; πUK = 8% PUS0 = $22.00; PUK0 = £13.50; F1 = 1.6296 The dollar will weaken against the pound based on the projection in forward markets.

Interest Rate Parity PUS0 = $20.00; PUK0 = £12.50; S0 = 1.6 πUS = 10%; πUK = 8% PUS0 = $22.00; PUK0 = £13.50; F1 = 1.6296 The forward rate suggests that the dollar will weaken against the pound.

Exchange Rate Expectations F1 = E[ S1 ]

The Fisher Effect The Fisher Effect within a given country states that the relationships among the nominal interest rate (i), the real interest rate (i') and the expected inflation rate E(π) is:   (1 + i) = (1 + i')(1 + E(π))

The International Fisher Effect The international counterpart to the Fisher Effect is the final parity condition - the International Fisher Effect. The International Fisher Effect draws from the Fisher Effect and Purchase and Interest Rate Parity. For example: (1 + ius) (1 + E[πus]) × (1 + i'us) (1 + iuk) = (1 + E[πuk]) (1 + i'uk)