1. VII. ARBITRAGE AND HEDGING WITH FIXED INCOME INSTRUMENTS AND CURRENCIES

2. 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

3. In Matrix Format =

4. 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.

5. 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.

6. 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.

7. 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.

8. 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.

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

10. Deriving Bond Duration

11. Deriving Bond Duration

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

13. 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.

14. 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.

15. 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,
We calculate bond and liability durations:

16. 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
∙ wA – ∙ wB – ∙ wC =
wA wB wC = 1
There are an infinity of solutions to this two-equation, three-variable system.

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

18. 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:

19. 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:

20. Convexity Illustration

21. 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
∙ wA – ∙ wB – ∙ wC =
5.41 ∙ wA ∙ wB ∙ wC = 6.38
wA wB wC = 1
The single solution to this 3 X 3 system of equations is wA = 0.106, wB = and wC = This system provides an improved immunization strategy when interest rate changes are finite.

22. 5.4. Term Structure, Interest Rate Contracts and Hedging
The Pure Expectations Theory:
The Yield Curve can be bootstrapped

23. 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.

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

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

26. Vasicek Process

27. 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 However, the South African Rand (SFR) has a price (buying or selling) equal to USD or EUR
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 USD = 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 = USD We will cover our short position in EUR by selling SFR at the price listed above.
USD SFR EUR
Sell USD0.20 for SFR
Sell EUR0.16 for USD
Sell SFR1.0 for EUR
Totals

28. 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.

29. 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.

30. 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 =
The dollar will weaken against the pound based on the projection in forward markets.

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

32. Exchange Rate Expectations
F1 = E[ S1 ]

33. 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))

34. 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)