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 effecitvely 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 raes
• 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

Answer 5

Question 6

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

Answer 6

• As difference between values of fixed rate bond and floating rate bond
• As total value of series of forward rate agreement

Question 7

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

Answer 7

Discount each coupon and capital payment using appropriate spot rate. So

Bfix = Σ ke^-r₁t₁ + Le^-r_nt_n

Where:

• Cashflows are k at time t_i (1 < = i <=n) at time t_n
• r_i is continusously compounded spot rate for maturity t_i

Question 8

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

Answer 8

• 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 approproate spot rate r₁ for time t₁

B_fl = (L + K)e^-r_rt₁

Question 9

Explain how to value an interest rate swap as a series of forward rate agreements

Answer 9

• Calculate forward rates for each of LIBOR rates that will determine swap cashflows
• Calculate swap cashflows on assumption that LIBOR rates will equal forward rates
• Set swap value equal to net present value of these cashflows, discounted using appropriate LIBOR zero rates

Question 10

State the formula for the forward price in terms of the sport price for an asset that pays

• No income
• A certain income with a present value of I during the lifetie of the forward
• Income in the form of a consistent, continous yield q

Answer 10

• No income F₀ = S₀e^rT
• Certain income with present value I during lifetime of forward F₀ = (S₀ - I)e^rT
• Income in form of constant, continous yield q

F₀ = S₀e^{(r - q)T}

Question 11

Consider an asset that pays no income

Explain how an investor can make a risk free profit if F₀ < S₀e^rT

Answer 11

If F₀~~0e^rT~~

• Sell asset short at current spot price S₀
• invest sale proceeds risk free (to accumulate sum S₀e^rT)
• enter into long forward contract to buy asset at time T at price F₀

This will generate risk - free profit of S₀e^rT - F₀ for no initial outlay, at time T.

If F₀ > S₀, risk free profit by doing opposite of above

Question 12

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

Answer 12

• 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 requries futures contract to be closed out before matruirty date eg if hegder uncertain as to exact date when asset will be bought or sold
  • because basis of future cannot be predicted with certainty

Question 13

State the formula for the optinal hedge ratio h

Answer 13

h = pσ_s/σf

• σ_s 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 adn AF

Question 14

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 14

• Normality of increments in log asset prices
• independence of increments in log asset prices
• constancy of paramters for example, constant drift and voloatility paramters

Question 15

Describe how the empirial evidence contradicts the model assumption

Answer 15

1. Equity returns are more peaked and have fatter tails than normal distribution
   - days of no change or small change
   - and market crashes or jumps appear more often than suggest by ND
   - the longer the time horizon the more non-norma
2. 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
3. Financial market volatility varies in cerain systemic ways, eg tends to be higher during financial crises, recession and bear markets