Chapter 12: Valuation of investments (1) Flashcards
e. Fixed income analytics
Define the following terms:
- Clean price
- Dirty price
of a bond.
- 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.
- Dirty price - this is the price at which the bond is traded and includes accrued interest.
e. Fixed income analytics
Explain what is meant by:
1. Spot rate yield (or Zero rate)
2. Bond yield
3. par yield
4. Forward interest rate
- Spot rate yield (or Zero rate) - Rate of return on a zero-coupon bond - ie it is the interest rate that applies to a lump sum of money that is borrowed or lent over a specific period starting now.
- Bond yield - gross redemption yield - ie constant rate that makes discounted present value of capital and interest payments equal to market price of bond.
- Par yield - it is the constant coupon rate given current term structure of interest rates, would make price of bond equal to its par value.
- FRI - it is an interest rate that applies over a future period of time - implied by current pattern of spot yields
e. Fixed income analytics
Explain the process of bootstrapping.
It is the process of deriving the pattern of spot yields from observed market prices of coupon bearing bonds.
- Bonds first ordered in terms of outstanding maturity.
- Equation of value set up and solved for shortest bond, to find corresponding yield.
- Equation of value set up for the next shortest bond. First spot yield substituted in this equation, which is then solved for next spot yield.
- Process repeated for each successive bond to obtain spot yields at each bond term.
- Spot yield curve completed by interpolation between spot yields obtained and extrapolation at either end.
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
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
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.
- 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
e. Fixed income (Interest rate) analytics and valuation.
Explain what is meant by an interest rate future.
- Exchange traded and standardized equivalent of a FRA that a specified interest rate will apply over a specified period
- Available for a wide range of currencies, for terms up to 10 years
- Most often based on 3-month (90 day) interest and principal of 1 million currency units (for the US dollar product)
- Trade with specified delivery months (March, June, Sep and Dec)
- Purchaser/long party effectively agrees to lend $1m (for the US dollar product) for a period of 3 months at an agreed interest rate starting at an agreed future date
- Can be used to hedge against or speculate on changes in 3-month interest rates in future time periods.
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.
i. $10 000 (100 - 0.25 x (100 - Z))
ii. R^(4) = 100 - Z
Tick value = $25
e. Fixed income (IRF and FRA) analytics and valuation.
Explain how the futures price differs from the corresponding (un-margined) forward price when the asset is strongly negatively correlated with interest rates.
- if interest rates are constant (ie fixed) then value of CF are equal and hence price of FRA and IFR are equal
- This is not the case when interest rates vary unpredictably
- Suppose underlying asset price strongly negatively correlated with interest rates
-if asset prices increase (falls), investor with long futures position makes an immediate gain (loss) because of daily margining (making to market) - As such gains (losses) tend to happen when interest rates are low (high), this gain (loss) will tend to be invested (financed) at lower (higher) -than-average interest rate
- In contrast the investor with a long position (un-margined) forward position will not be affected in this way by interest rate movements.
- So, all else being equal, long futures position less attractive than equivalent long forward contract - hence futures prices lower than forward prices
e. Fixed income (IRF and FRA) analytics and valuation.
State the formula relating the (un-margined) forward interest rate and the futures interest rate.
Forward rate = Future rate - 0.5 x V x t_1 x t_2
where:
- t_1 is time (in years) of maturity of futures contract
- t_2 is time (in years) to maturity of rate underlying futures contract
- Squrt (V) is the standard deviation of the change in ST interest rate in one year (typically 1.2%)
Here forward and future rates are both expressed in continuously- compounded form. 0.5 x V x t_1 x t_2 is known as the ‘convexity adjustment’.
e. Fixed income (Interest Rate Swaps) analytics and valuation.
State two ways in which interest rate swap can be valued.
- Difference between the value of a fixed bond and floating rate bond
- As a total of series of FRA
e. Fixed income (Interest Rate Swaps) analytics and valuation.
Explain with an aid of a formula how to value a fixed rate bond.
B_fix = Sum (Kexp(-r_it_1) + Lexp(-r_n*t_n)
where:
- 1<= i <= n
- cashflows are k at t_i and L at t_n
- r_i is continnuosly compounded spot rate for maturity t_i
(r_i LIBOR - implicit assumption that risk associated with swap the same as risk associated with loans in interbank market)
e. Fixed income (Interest Rate Swaps) analytics and valuation.
Explain with an aid of a formula how to value a floating rate bond.
- At outset, value of any floating rate bond is equal to principal amount
- this is the also the case immediately after any 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, where k is floating rate payment that will be made on the next payment date due at time t_1
- hence, value today is its value before next payment date, discounted at appropriate spot rate r_1 for time t_1:
B_fl = (L + k)exp(-r_1t_1)
- floating rate payment based on forward rate at the beginning of the period
e. Arbitrage pricing and concept of hedging.
State the formula for forward price in terms of the spot price for an asset that pays:
i. no income
ii. Certain income with a present value of I during the lifetime of the forward
iii. income in form of a constant, continuous yield q.
i. F_0 = S_0 * exp(r*T)
ii. F_0 = (S_0 - I) * exp(r*T)
iii. F_0 = S_0 * exp((r - q)*T)
e. Fixed income (Interest Rate Swaps) analytics and valuation.
Explain how to value an interest swap as a series of FRA.
- Calculate forward rates for each LIBOR rates that will determine the swap cashflows
- calculate swap cashflows on the assumption that LIBOR rates will equal forward rates
- set swap value equal to net present value of these CF, discounted using appropriate LIBOR zero rates.
(In the absence of information better to assume spot yield curve is flat)
*do example on page 22 and if you get right delete this
e. Arbitrage pricing and concept of hedging.
Consider an asset that pays no income.
Explain how an investor can make a risk-free profit if F_0 < S_0 * exp(rT)
- sell the asset short at current spot price
- invest proceeds risk-free (to accumulate sum S_0* exp(rT)
- enter into long forward contract to buy asset at T at price F price F_0
This will generate risk free profit of S_0 * exp(r*T) - F_0 at T
