Chapter 12:Valuation of investments

Question 1

Explain what is meant by each of the following

Spot yield
Bond yield
Par yield
forward interest rate

Answer 1

Spot yield

Rate of return on zero coupon bond - interest rate at which can agree now to borrow or lend lump sum over time period beginning now
Bond yield

Gross redemption yield -

constant discount rate that makes discounted present value of capital and interest payments equal to maket price of bond

Par yield

Constant coupon that, given current term structure of interest rates, would make price of bond equal to its par value

Forward interest rate

Interest rate that applies over a future period of time - implied by current pattern of spot yields

Question 2

Explain the process of bootstrapping

Answer 2

Bootstrapping

​The process of deriving pattern of spot yields from observed market pices of coupon - bearing bonds

Bonds first odered in terms of outstanding maturity

Equation of value set up and solved for shortest bond, to find corresponding spot yield

Equation of value set up for next shortest bond.
First spot rate substituted into this equation, which is then solved for next spot yield

Process repeated for each successive bond to obtain spot yield for each bond term

Spot yield curve completed by interpolation between spot yields obtained and extrapolation at either end

Question 3

Explain what is meant by an interest rate future

Answer 3

Exchange traded and standardised equivalent of forward rate agreement

Available in wide range of currencies, for terms up to 10 years

Most often based on 3 months interest rate and principal of 1 million currency units

Purchaser/long party effectively agrees to lend 1 million over a 3 month period at agreed interest rate starting on agreed future time periods

Can be used to hedge against or speculate on changes in 3 month interest rates in future time periods

Question 4

Explain how the futures price differ from the corresponding (un - margined) forward price when the asset price is strongly negatively correlated with interest rates

Answer 4

Suppose underlying asset price strongly negatively correlated with interest rates

If asset price increase, then investor with long futures position makes immediate gain because of daily margining (marking to market)

As such gains tend to happen when interest rates low, this gain will tend to be invested at lower than average interest rates

Likewise, decreases in asset price, which lead to immediate loss to investor with long futures position, will tend to be financed at higher than average interest rates

In contrast the investor with long (un - margined) forward position wil not be affected in this way by interest rate movements

So all else being equal, long futures contract less attractive than equivalent long forward contract - hence futures prices lower than forward prices

Question 5

State two ways in which an interest rate swap can be valued

Answer 5

As difference between values of fixed rate bond and floating rate bond

As total value of series of forward rate agreement

Question 6

Explain with the aid of a formula how to value a floating rate bond

Answer 6

At the outset, value of any floating bond is equal to principal amount

This is also case immediately after coupon payment, as remaining payments can be thought of as brand new floating rate bond

So, immediately before payment date, its value will be L + K* is floating rate payment that will be made on next payment date due at time t1

Hence, value of bond is its value just before next payment date, discounted at appropriate spot rate r for time t1
Bfl = (L + K)e^-rxt1

Question 7

Give two reasons why it is generally not possible to hedge exactly using futures

Answer 7

Cross hedging

Asset whose price is to be hedged is not exactly same as asset underlying futures contract. So, spot price and futures price do not move in exactly same way

Basis risk arises

If hedge requires futures contract to be closed out before maturity date eg if hedger uncertain as to exact date when asset will be bought or sold
because basis of future cannot be predicted with certainty

Question 8

State the formula for the optimal hedge ratio

Answer 8

h = pσs/σf

σs is standard deviation of AS, change in spot prices over life of hedge

σf is the standard deviation in AF, change in futures price over the life hedge

p is the correlation coefficient between AS and AF

Question 9

State the three main assumptions regarding the distribution of investment returns often made in stochastic asset models, such as the lognormal model of security prices

Answer 9

Normality of increments in log asset prices

independence of increments in log asset prices

constancy of parameters for example, constant drift and volatility parameters

Question 10

Describe how the empirical evidence contradicts the model assumption

Answer 10

Equity returns are more peaked and have fatter tails than normal distribution

Variance of price changes grows at a slower than linear rate, suggesting that equity values exhibit long run mean reversion, contradicting assumption of independent increments

Financial market volatility varies in certain systemic ways, eg tends to be higher during financial crises, recession and bear markets