Valuing Cash flows...

Valuing cash flows using zeros

Zeros are zero coupon bonds, which pay off at the maturity at a specified price.

Any portfolio of Bonds can be represented as a collection of ZCBs some of which will pay after one year, some will pay after 2 some will pay after 3 and so on.

Such a portfolio is called a replicating portfolio. The cost of replicating portfolio represents the price of the BOND portfolio. If the price of the replicating portfolio is more than its market price, then buy it and if it is less, then sell it ! The difference will be called as the arbitrage.

In general, arbitrage is the opportunity to make money by just buying and selling. Whether buying at lower cost and selling at a higher first or selling at higher first and buying at lower later doesn’t change the meaning of Arbitrage.

Wikipedia definition : In economics, arbitrage is the practice of taking advantage of a state of imbalance between two or more markets: a combination of matching deals are struck that capitalize upon the imbalance, the profit being the difference between the market prices. When used by academics, an arbitrage is a transaction that involves no negative cash flow at any probabilistic or temporal state and a positive cash flow in at least one state. A person who engages in arbitrage is called an arbitrageur. The term is mainly applied to trading in financial instruments, such as bonds, stocks, derivatives and currencies. Entrepreneurs seek out arbitrage markets (i.e. insurance) to earn a profit. For example, consumers benefit greatly from entities who locate arbitrage markets within the insurance industry. Insurance arbitrage markets are an immensely profitable trend that Wall Street is eyeing. Those insurance firms in a non-arbitrage environment will likely have difficulty competing against an arbitrage business model.

Law of one price: Portfolios with equal payoffs have equal prices.

Forward interest rate = _Tf_T+1 = (M_T+1 /M _T – 1 ) = ( d_T / d_T+1 – 1 )

MT is what 1 dollar will become in T time.

dT is price of a zero coupon Bond.

Buying a bond for one year and forwarding it to three years, or buying it for 4 years, and similarly all types of combinations will give same payoff.

_Tf_T+n = ((M_T+1 /M _T)ⁿ – 1 ) = ( (d_T / d_T+1)ⁿ – 1 )

From T year to T + n years.

Discount function is given by

dT = 1 / (1 + ₀f₁ ) *( 1+ ₁f₂ ) ….(1 + _T-1 f _T)

y_T = ((1 + ₀f₁ ) *( 1+ ₁f₂ ) ….(1 + _T-1 f _T)) ^T – 1

y_T = (1 / dT )^T – 1

_nf_n+T = [(1 + Y _{n + T} )^n+T / (1 + Y _n )ⁿ]^1/T – 1

= [d(n) / d(n + T )]^1/T – 1

d(t) = 1 / ( 1 + y_T) ^T

General Bond Valuation

C = stated annual coupon rate of bonds

m = # of coupon payments per year.

c = periodic coupon actually paid

R = Stated rate

i = effective period rate.

k = years to maturity

n = number of coupon payments.

An i Value of annuity for n years at i interest.

P_z1= price of a zero coupon bond maturing in 1 year.

P b = c * Ani + m*Zⁿ

	BOND A	BOND B
R = Current stated interest rate	12.500%	12.500%
C = stated annual coupon rate of bonds	8.750%	12.625%
m = # of coupon payments per year.	2.000	2.000
c = periodic coupon actually paid	0.044	0.063
i = effective period rate.	0.063	0.063
k = years to maturity	12.0	12.0
n = number of coupon payments.	24.0	24.0
An i Value of annuity for n years at i interest.	12.266	12.266
Pz1= price of a zero coupon bond maturing in 1 year.	0.941	0.941
Bond Price should be	77.002	100.767

PERIOD	SPOT RATE	FORWARD RATES	DISCOUNT FUNCTION	STATED COUPON PAYMENT	STATED MATURITY PAYMENT	STATED BOND CASH FLOW	BOND PRICE	YIELD TO RETURN
1	8.15%	8.15%	0.92464	100		100	92.46	8.39%	92.26	0
2	8.20%	8.25%	0.85417	100		100	85.42	8.39%	85.12	0
3	8.35%	8.65%	0.78616	100		100	78.62	8.39%	78.53	0
4	8.40%	8.55%	0.72424	100		100	72.42	8.39%	72.45	0
5	8.40%	8.40%	0.66812	100		100	66.81	8.39%	66.84	0
6	8.40%	8.50%	0.61635	100		100	61.63	8.39%	61.67	0
7	8.40%	8.40%	0.56858	100		100	56.86	8.39%	56.9	0
8	8.40%	8.30%	0.52452	100	1000	1100	576.98	8.39%	52.49	524.9
9	8.40%	8.50%	0.48388			0	0.00	8.39%	0	0
10	8.40%	8.60%	0.44638			0	0.00	8.39%	0	0
					1000
							1091.2	1091

Estimating forward rates

Observe Price for a large number of coupon paying bonds

Denote Cash Flow as CFit

Dt be discount coefficients

P1 = d1 CF11 + d2 CF12 + ….+ e1

P2 = d1 CF21 + d2 CF22 + … + e2

…

…

Pn = d1 CFn1 + …

Yield to maturity

Yield to maturity

Calculate yield to return on the any call date. The lowest yield to call is known as yield to worst.

Bond Price Dynamics

Any bond can be brought to its Present value.

Vo = FV * ( 1 + R ) ^–T

dV/ dR = - T * ( 1 + R )^-T-1* FV

Derivatives are negative – Price of zero coupon bond decreases with interest rates.

T affects the rate of decrease.

Measuring Bond Performance

Holding period return

HPR = ( C_t,t+1 +(P_t+1 - P_t)) / P_t

Time shape of cash flows

Price of a bond is dependent on the cash flows. Thus if the payoffs are concentrated in the beginning, then it will not be sensitive to changes in interest rates.

Graph of bond prices for different bonds show different curves

Term structure of interest rates and yield curves

Relationship between interest rates and time to maturity

These curves can be flat, monotonically increasing or decreasing or humped.

Term structure Theories

There are four term structure theories

1. Market Segmentation Theory

Separate forces of supply and demand for funds in the short term Vs long term markets, with interest rates determined independently in each segment.

Investors have a strong preference for the certain maturity and are sufficiently risk averse such that they only trade within their desired maturity segment no matter how large of yield differentials exist at other maturities.

2. Preferred Habitat theory

In this case, the investors are not that risk averse as in the M S T. In this case, they might depart from the preferred maturity if there is sufficient return to compensate them for the additional risk.

Example : (1 + y2 ) ² = ( 1 + y1 )(1 + 1f1 + P )

P > 0

If there is relatively small demand for 2 year bonds, then 1f2 can be increased to boost that.

Thus there is a bias in the forward rate.

Preferred habitats :

Lenders borrowers

Short term Banks Corporations for inventory

Long term Insurance companies Corporations for long term assets

3. Expectations theory

Investors’ expectation of future rates determines the shape of yield curve. Expectations of future increases in the short term rates leads to higher yields on long term bonds. In this theory, the forward rates are unbiased estimates of expected future spot future returns.

For a given time zone, the expected return from investment strategy of identical risk is identical If yield curve is upwards sloping investing in long term bonds does not mean a person will expect to earn higher returns. The difference in short term and long term bonds is not important to them.

4. Liquidity preference theory

This theory predicts an upward bias in the yield curve.

It predicts that the liquidity premium is positive and increasing with maturity. In other words, observing an upward sloping curve does not mean spot rates will increase in the future.