1 Flashcards
Bonds
Bonds have a FACE VALUE which is the final payment paid on the MATURITY DATE along with the last of the coupon payments paid every so often.
Correct price
Equilibrium price: non arbitrage pricing (for which arbitrage strategies don’t occur)
If too low or hard go then profits can be made and unstable as the demand for the low price causes it to rise so a price that’s stable is one that doesn’t introduce arbitrage opportunities
Eg selling price can be worked out from prices for others
Amount invested for n years with an interest rate compounded a number of times a year
An amount A invested for n years at a yearly interest rate r compounded m times a year yields at the end of this a period
A( 1 + (r/m)) ^mn
Present value
The present value, of a payment occurring in the future, is the number resent price of entitlement to that payment
•deposit £A yields £B withdrawn after t years so we pay A now to receive B in t years- the present value of B paid in t years is A
PRESENT VALUE —————-multiply by (1 + (r/m))^mn to FUTURE PAYMENT —————-multiply by (1 + (r/m))^-mn to PRESENT VALUE
Continuously compounded interest rates
An amount A invested for t-years accruing a yearly interest rate of r compounded continuously, yields at the end of this period £Ae^rt
present value of an amount X with continuously compounded interest is Xe^-rt
Example: borrow £1000 for 3 years at an 5% annual interest rate compounded every 6 months
Example: borrow £1000 for 3 years at an 5% annual interest rate compounded every 6 months
A=1000
r=0.05
m= 2 (times a year)
n= 3 (years)
Loan = 1000(1+ 0.05/2)^ 6 = £1158.69
Example: pay in X for ten years paying an interest rate of 3% per year compounded yearly
Example: pay in X for ten years paying an interest rate of 3% per year compounded yearly
After ten years we have
X ( 1 + (0.03/1))^10
= 1.03^10• X
So if we wAnt £1 in ten years must deposit 1/1.03^10 = 0.74
(Present value of £1 is 74p and the right price to pay for this is this, otherwise we could could perform an arbitrage strategy by borrowing and repaying loans )
Example: £100,000 mortgage with a fixed annual interest rate of 6% compounded monthly, what are the repayments per month if we are repaying in 20 years
Example: £100,000 mortgage with a fixed annual interest rate of 6% compounded monthly, what are the repayments per month if we are repaying in 20 years
The bank is willing to pay for all future payments a certain amount: based on the total of the present values
Let the repayments be P
The LAST (future payment) repayment P has present value P(1+ (0.06/12))^ -(240)
The second to last (future payment) has present value
P(1+(0.06/12))^-239
…
The second has present value P(1+0.06/12)^-2
The first payment has present value P(1+0.06/12)^-1
The bank wants the sum of all the PRESENT values of the repayments to equal 100,000:
100,000= (from k=1 to 240)Σ P(1+0.06/12)^-k
Hence. P=716.43
(Note: the future value of 100,000 /240 is 1379 which is not equal to this, nor is 100000/240 = 416)
Zero coupon bond
A bond in which only one payment (final) is paid on the maturity of the bond with no other payments
The price p of a £B face value zero coupon with maturity in t years is therefore the PRESENT VALUE p of £B paid t years into the future
Proposition 1: zero coupon bond and continuously compounded interest
PROPOSITION 1: let P be the price of a zero coupon bond with face value £1 and maturing in t years. Let r be the continuously compounded interest rate for t year deposits and loans. Then
P = exp ( -rt)
PRESENT VALUE FOUND FROM FUTURE VALUE SO NEGATIVE EXPONENT
Proof: arbitrage strategies otherwise
P> e^-rt
Issue a zero coupon bond with face value £1 and maturing in t years, collect price P and deposit this for t years, withdrawing e^rt after t years then pay face value £1 of bond and pocket pe^rt -1 >0
P< e^-rt
Similarly borrow P for t years and buy the bond.. finally pocketing 1-Pe^rt >0
Discount curve
P(t)
Function of time P(t) whose value at any t ≥ 0 is the present value of one unit of currency paid in t years. Equivalently P(t) is the price of a 0 coupon bond with face value 1 maturing in t-years.
PRESENT VALUE OF FUTURE PAYMENT
So if we are FINDING this from a future payment we use a negative power
Yield curve
Y(t)
or r(t)
Function Y(t) whose value at any time t>0 is the interest paid on t year deposits
Y(5) is the value of the interest paid on a 5 year deposit
Values Y(t) SPOT INTEREST RATES OR YIELDS
• yield curves aren’t directly observable finding interest rate which fit observes prices of assets, by linear interpolation
Converting between Spot interest rates/ yields and present values
Finding present value from “future payment” (yield) so we use negative power in exponent
P(t) = exp(-Y(t)*t)
Y(t) = - log (P(t)) /t
P(t) is Finding the present value of zero coupon of ONE UNIT of currency so we can use to multiply by units for more than £1.
Constructing yield curves
- use the equation to convert between P(t) and Y(t)
- P(t) can be found: when it’s a zero coupon bond ….by dividing present price by the face value ( as defined for one unit only)
•P(t) can be found when not a zero coupon bond:
By calculating the total payments made worth ( ie summing the present value for each payment, which changes as payments are made at different times)
•Y(t) is found from p(t)
REMEMBER CALCULATIONS ONLY APPLY TO ZERO COUPIN BONDS
Example finding P(t) and Y(t) Semi annual coupon bonds Face value: 100 |100|100|100|100 Maturity yrs: 0.25| 0.5| 1. |1.5. |2 Annual interest: 0. |. 0. | 0. |. 4. |6 Price: 98.3|96.5|93.7|95.5|97.2
•P(0.25) = 98.3/100 , Y(0.25) = 6.86%
•P(0.5) = 0.965 Y(0.5) =7.13%
•P(1)= 0.937, Y(1) =6.51%
Following aren’t zero coupon bonds:
• P(1.5)=
Annual interest is 4, paid 2 a year cost is 95.5 which is the present value of the total of these FUTURE payments
= £2 each 0.5 years
£2 £2 £2 £102 |——|——-|——|—--|——| 0 0.25 0.5 1 1.5 2
2P(0.5) \+ 2P(1) \+ 2P(1.5) \+ 102 P(2) = 95.5
And using others P(2) =0.826
• Y(2)) =7.42%
spot interest yields
(n/ year payments found from = annual interest*FACE VALUE)/ payments per year, not price but face value)
