Chapter 12: Valuation of investments (1)

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

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 formula how to value a fixed rate bond

Answer 6

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 7

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

Answer 7

• 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 8

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

Answer 8

• 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 9

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 9

• 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 10

Consider an asset that pays no income

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

Answer 10

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 11

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

Answer 11

• 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 12

State the formula for the optinal hedge ratio h

Answer 12

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 13

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 13

• 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 14

Describe how the empirial evidence contradicts the model assumption

Answer 14

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

Question 15

e. Fixed income analytics

Define the following terms:

1. Clean price
2. Dirty price
   of a bond.

Answer 15

1. Clean price - Price of bond that appears on the dealers screens and is the quoted price and excludes accrued interest. The interest is that deemed to have accrued to the bondholder since the date of the last coupon date.
2. Dirty price - this is the price at which the bond is traded and includes accrued interest.

Question 16

e. Fixed income analytics and valuation

(i) the forward rate R_F from time T_1 and T_2

(ii) the instantaneous forward rate R_F and time T

Answer 16

Forward rate:

R_F = (R_2T_2 - R_1T1)/(T_2 - T_1)

Instantaneous forward:

R_F = R_1 + T*change in R/change in T

Here all interest rates are compounded continuously.

(if an expression for the spot rate is not given as a continuous and differential function time then estimate R_F as (R_F1 + R_F2)/2

Question 17

e. Fixed income (FRA) analytics and valuation

Explain what is meant by a forward rate agreement.

State how it may be used and how it may be valued.

Answer 17

• OTC agreement that specified interest rate will apply to specified principal over specified future time period
• Used to speculate or hedge against future interest rate changes
• valuing by discounting notional cashflows that will be exchanged at the start and end of the forward period
• e.g., effective interest rate of R_K (T_2 - T_1) applies to principal of L over period from T_1 and T_2, then value of FRA to purchaser (lender) is:

V = -L(1+ R_1)^(-T_1) + L(1 + R_K(T_2 - T_1))(1 + R_2)(-T_2) —> spot rates compounded T_2 - T_1

or

V= L(R_K - R_F)(T_2 - T_1)exp(-R_2T_2) —> spot rates compounded continuously

Question 18

e. Fixed income (IRF) analytics and valuation.

State a formula for each of the following in terms of the quoted price (Z) of an interest rate future:

i. the contract price of an interest rate future

ii. the implied interest rate.

Answer 18

i. $10 000 (100 - 0.25 x (100 - Z))

ii. R^(4) = 100 - Z

Tick value = $25

Question 19

Answer 19

Question 20

Define a hedge

Answer 20

A hedge is defined as a trade to reduce risk

Question 21

Basis risk may arise if: (3)

Answer 21

1. The asset whose price is to be hedged is not exactly the same as the asset underlying the futures contract.
2. The hedger is uncertain as to the exact date when the asset will be bought or sold.
3. The hedge requires the futures contract to be closed out well before its expiration date.

Question 22

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

Answer 22

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