Lecture 3 Fixed-income securities: Valuation
Lecture 3 1 / 27 There are many different types of fixed-income securities. See the book for many examples.
In the lectures we will focus on Annuities Perpetual Finite-life Bonds Coupon bonds Zero coupon bonds
Introduction
A fixed-income security is, essentially, a financial asset that gives the holder a given (i.e. fixed) stream of cash flows.
Lecture 3 2 / 27 In the lectures we will focus on Annuities Perpetual Finite-life Bonds Coupon bonds Zero coupon bonds
Introduction
A fixed-income security is, essentially, a financial asset that gives the holder a given (i.e. fixed) stream of cash flows.
There are many different types of fixed-income securities. See the book for many examples.
Lecture 3 2 / 27 Bonds Coupon bonds Zero coupon bonds
Introduction
A fixed-income security is, essentially, a financial asset that gives the holder a given (i.e. fixed) stream of cash flows.
There are many different types of fixed-income securities. See the book for many examples.
In the lectures we will focus on Annuities Perpetual Finite-life
Lecture 3 2 / 27 Introduction
A fixed-income security is, essentially, a financial asset that gives the holder a given (i.e. fixed) stream of cash flows.
There are many different types of fixed-income securities. See the book for many examples.
In the lectures we will focus on Annuities Perpetual Finite-life Bonds Coupon bonds Zero coupon bonds
Lecture 3 2 / 27 Assume that we stand at the beginning of a period, and that the amount A at that time has been paid out.
If we let r > 0 denote the per-period interest rate, then the present value PV of the perpetuity is given by
∞ X A PV = . (1 + r)k k=1
Annuities (1)
A perpetual annuity or perpetuity is a financial asset that gives the holder a given amount A at the end of every period from now on and forever.
Lecture 3 3 / 27 If we let r > 0 denote the per-period interest rate, then the present value PV of the perpetuity is given by
∞ X A PV = . (1 + r)k k=1
Annuities (1)
A perpetual annuity or perpetuity is a financial asset that gives the holder a given amount A at the end of every period from now on and forever.
Assume that we stand at the beginning of a period, and that the amount A at that time has been paid out.
Lecture 3 3 / 27 Annuities (1)
A perpetual annuity or perpetuity is a financial asset that gives the holder a given amount A at the end of every period from now on and forever.
Assume that we stand at the beginning of a period, and that the amount A at that time has been paid out.
If we let r > 0 denote the per-period interest rate, then the present value PV of the perpetuity is given by
∞ X A PV = . (1 + r)k k=1
Lecture 3 3 / 27 Alternative 1: Use the formula for a geometric series.
Alternative 2: Note that ∞ ∞ X A A X A PV = = + (1 + r)k 1 + r (1 + r)k k=1 k=2 ∞ A 1 X A = + 1 + r 1 + r (1 + r)k k=1 A PV A = + ⇒ PV = . 1 + r 1 + r r
Annuities (2)
How can we calculate this sum?
Lecture 3 4 / 27 Alternative 2: Note that ∞ ∞ X A A X A PV = = + (1 + r)k 1 + r (1 + r)k k=1 k=2 ∞ A 1 X A = + 1 + r 1 + r (1 + r)k k=1 A PV A = + ⇒ PV = . 1 + r 1 + r r
Annuities (2)
How can we calculate this sum?
Alternative 1: Use the formula for a geometric series.
Lecture 3 4 / 27 ∞ A X A = + 1 + r (1 + r)k k=2 ∞ A 1 X A = + 1 + r 1 + r (1 + r)k k=1 A PV A = + ⇒ PV = . 1 + r 1 + r r
Annuities (2)
How can we calculate this sum?
Alternative 1: Use the formula for a geometric series.
Alternative 2: Note that ∞ X A PV = (1 + r)k k=1
Lecture 3 4 / 27 ∞ A 1 X A = + 1 + r 1 + r (1 + r)k k=1 A PV A = + ⇒ PV = . 1 + r 1 + r r
Annuities (2)
How can we calculate this sum?
Alternative 1: Use the formula for a geometric series.
Alternative 2: Note that ∞ ∞ X A A X A PV = = + (1 + r)k 1 + r (1 + r)k k=1 k=2
Lecture 3 4 / 27 A PV A = + ⇒ PV = . 1 + r 1 + r r
Annuities (2)
How can we calculate this sum?
Alternative 1: Use the formula for a geometric series.
Alternative 2: Note that ∞ ∞ X A A X A PV = = + (1 + r)k 1 + r (1 + r)k k=1 k=2 ∞ A 1 X A = + 1 + r 1 + r (1 + r)k k=1
Lecture 3 4 / 27 A ⇒ PV = . r
Annuities (2)
How can we calculate this sum?
Alternative 1: Use the formula for a geometric series.
Alternative 2: Note that ∞ ∞ X A A X A PV = = + (1 + r)k 1 + r (1 + r)k k=1 k=2 ∞ A 1 X A = + 1 + r 1 + r (1 + r)k k=1 A PV = + 1 + r 1 + r
Lecture 3 4 / 27 Annuities (2)
How can we calculate this sum?
Alternative 1: Use the formula for a geometric series.
Alternative 2: Note that ∞ ∞ X A A X A PV = = + (1 + r)k 1 + r (1 + r)k k=1 k=2 ∞ A 1 X A = + 1 + r 1 + r (1 + r)k k=1 A PV A = + ⇒ PV = . 1 + r 1 + r r
Lecture 3 4 / 27 Annuities (3)
Theorem The present value of a perpetual annuity giving the amount A every period and where the constant per-period rate is r > 0, is given by A PV = . r
Lecture 3 5 / 27 We use the previous formula to get 25 000 PV = = 625 000 USD. 0.04
Annuities (4)
Example A perpetual annuity gives the owner 25 000 USD per year. What is the present value of this annuity if the yearly interest rate is 4%?
Lecture 3 6 / 27 Annuities (4)
Example A perpetual annuity gives the owner 25 000 USD per year. What is the present value of this annuity if the yearly interest rate is 4%?
We use the previous formula to get 25 000 PV = = 625 000 USD. 0.04
Lecture 3 6 / 27 If we again let r denote the per-period interest rate, then the present value of this type of annuity is given by
n X A PV = . (1 + r)k k=1 A quick way of valuing this is to observe that a finite-life annuity can be seen as the difference between a standard perpetual annuity starting now and one where the first payment is at time n + 1.
Annuities (5)
A finite-life annuity, or annuity, is a financial asset that pays the same amount A for a finite number of periods n.
Lecture 3 7 / 27 A quick way of valuing this is to observe that a finite-life annuity can be seen as the difference between a standard perpetual annuity starting now and one where the first payment is at time n + 1.
Annuities (5)
A finite-life annuity, or annuity, is a financial asset that pays the same amount A for a finite number of periods n.
If we again let r denote the per-period interest rate, then the present value of this type of annuity is given by
n X A PV = . (1 + r)k k=1
Lecture 3 7 / 27 Annuities (5)
A finite-life annuity, or annuity, is a financial asset that pays the same amount A for a finite number of periods n.
If we again let r denote the per-period interest rate, then the present value of this type of annuity is given by
n X A PV = . (1 + r)k k=1 A quick way of valuing this is to observe that a finite-life annuity can be seen as the difference between a standard perpetual annuity starting now and one where the first payment is at time n + 1.
Lecture 3 7 / 27 The value at time n of a perpetual with first paymenat at time n + 1 is A/r.
A/r The present value of this perpetual is (1+r)n .
The value of the annuity is the difference between these two values.
Annuities (6)
We have The present value of a perpatual is A/r.
Lecture 3 8 / 27 A/r The present value of this perpetual is (1+r)n .
The value of the annuity is the difference between these two values.
Annuities (6)
We have The present value of a perpatual is A/r. The value at time n of a perpetual with first paymenat at time n + 1 is A/r.
Lecture 3 8 / 27 The value of the annuity is the difference between these two values.
Annuities (6)
We have The present value of a perpatual is A/r. The value at time n of a perpetual with first paymenat at time n + 1 is A/r.
A/r The present value of this perpetual is (1+r)n .
Lecture 3 8 / 27 Annuities (6)
We have The present value of a perpatual is A/r. The value at time n of a perpetual with first paymenat at time n + 1 is A/r.
A/r The present value of this perpetual is (1+r)n .
The value of the annuity is the difference between these two values.
Lecture 3 8 / 27 Annuities (7)
Theorem The present value of an annuity giving the amount A every period for n periods and where the constant per-period rate is r > 0, is given by
A A 1 A 1 PV = − · = 1 − . r r (1 + r)n r (1 + r)n
Lecture 3 9 / 27 We use the formula above to get
25 000 1 PV = 1 − = 202 770 USD. 0.04 1.0410
Annuities are typical payment streams in life insurance (pensions), either as a finite-life or annuity or as a version of a perpetuity in which you get the amount A until you die.
Annuities (8)
Example A finite-life annuity gives the owner 25 000 USD per year for 10 years. Whis it the present value of this annuity if the yearlt interest is 4%?
Lecture 3 10 / 27 Annuities are typical payment streams in life insurance (pensions), either as a finite-life or annuity or as a version of a perpetuity in which you get the amount A until you die.
Annuities (8)
Example A finite-life annuity gives the owner 25 000 USD per year for 10 years. Whis it the present value of this annuity if the yearlt interest is 4%?
We use the formula above to get
25 000 1 PV = 1 − = 202 770 USD. 0.04 1.0410
Lecture 3 10 / 27 Annuities (8)
Example A finite-life annuity gives the owner 25 000 USD per year for 10 years. Whis it the present value of this annuity if the yearlt interest is 4%?
We use the formula above to get
25 000 1 PV = 1 − = 202 770 USD. 0.04 1.0410
Annuities are typical payment streams in life insurance (pensions), either as a finite-life or annuity or as a version of a perpetuity in which you get the amount A until you die.
Lecture 3 10 / 27 We know from the Main theorem of present values that if we use the same constant interest rate to discount all these cashflows, then having this cash flow is equivalent of having its present value at time 0:
x is equivalent to (PV, 0, 0,..., 0).
Annual worth (1)
Given is a stream of cash flows
x = (x0, x1,..., xn).
Lecture 3 11 / 27 Annual worth (1)
Given is a stream of cash flows
x = (x0, x1,..., xn).
We know from the Main theorem of present values that if we use the same constant interest rate to discount all these cashflows, then having this cash flow is equivalent of having its present value at time 0:
x is equivalent to (PV, 0, 0,..., 0).
Lecture 3 11 / 27 This means that we want to find a stream of cash flows
(0, A, A,..., A)
that is equivalent to x.
The value of A such that this is satisfied is called the annual worth over n years of x.
Annual worth (2)
Alternatively, we could ask which finite-life annuity with the same life length as the stream of cash flows, i.e. n in this case, is equivalent to x.
Lecture 3 12 / 27 The value of A such that this is satisfied is called the annual worth over n years of x.
Annual worth (2)
Alternatively, we could ask which finite-life annuity with the same life length as the stream of cash flows, i.e. n in this case, is equivalent to x.
This means that we want to find a stream of cash flows
(0, A, A,..., A)
that is equivalent to x.
Lecture 3 12 / 27 Annual worth (2)
Alternatively, we could ask which finite-life annuity with the same life length as the stream of cash flows, i.e. n in this case, is equivalent to x.
This means that we want to find a stream of cash flows
(0, A, A,..., A)
that is equivalent to x.
The value of A such that this is satisfied is called the annual worth over n years of x.
Lecture 3 12 / 27 (2) Calculate A by using the fact that
A 1 r PV = 1 − ⇔ A = PV · . r (1 + r)n 1 − 1/(1 + r)n
Annual worth (3)
To get the annual worth of a stream of cash flows x do as follows.
(1) Calculate the PV of x:
n X xk PV = . (1 + r)k k=0
Lecture 3 13 / 27 Annual worth (3)
To get the annual worth of a stream of cash flows x do as follows.
(1) Calculate the PV of x:
n X xk PV = . (1 + r)k k=0
(2) Calculate A by using the fact that
A 1 r PV = 1 − ⇔ A = PV · . r (1 + r)n 1 − 1/(1 + r)n
Lecture 3 13 / 27 The present value of a machine which costs x per period and if the period is N years is
∞ ∞ X x X x PV = = x + k k k=0 (1 + r)N k=1 1 + [(1 + r)N − 1] x (1 + r)N = x + = x · (1 + r)N − 1 (1 + r)N − 1 1 = x · . 1 − 1/(1 + r)N
Annual worth (4)
We can use the annual worth concept to get a value per cycle in the cycle problem.
Lecture 3 14 / 27 ∞ X x = x + k k=1 1 + [(1 + r)N − 1] x (1 + r)N = x + = x · (1 + r)N − 1 (1 + r)N − 1 1 = x · . 1 − 1/(1 + r)N
Annual worth (4)
We can use the annual worth concept to get a value per cycle in the cycle problem.
The present value of a machine which costs x per period and if the period is N years is
∞ X x PV = k k=0 (1 + r)N
Lecture 3 14 / 27 x (1 + r)N = x + = x · (1 + r)N − 1 (1 + r)N − 1 1 = x · . 1 − 1/(1 + r)N
Annual worth (4)
We can use the annual worth concept to get a value per cycle in the cycle problem.
The present value of a machine which costs x per period and if the period is N years is
∞ ∞ X x X x PV = = x + k k k=0 (1 + r)N k=1 1 + [(1 + r)N − 1]
Lecture 3 14 / 27 (1 + r)N = x · (1 + r)N − 1 1 = x · . 1 − 1/(1 + r)N
Annual worth (4)
We can use the annual worth concept to get a value per cycle in the cycle problem.
The present value of a machine which costs x per period and if the period is N years is
∞ ∞ X x X x PV = = x + k k k=0 (1 + r)N k=1 1 + [(1 + r)N − 1] x = x + (1 + r)N − 1
Lecture 3 14 / 27 1 = x · . 1 − 1/(1 + r)N
Annual worth (4)
We can use the annual worth concept to get a value per cycle in the cycle problem.
The present value of a machine which costs x per period and if the period is N years is
∞ ∞ X x X x PV = = x + k k k=0 (1 + r)N k=1 1 + [(1 + r)N − 1] x (1 + r)N = x + = x · (1 + r)N − 1 (1 + r)N − 1
Lecture 3 14 / 27 Annual worth (4)
We can use the annual worth concept to get a value per cycle in the cycle problem.
The present value of a machine which costs x per period and if the period is N years is
∞ ∞ X x X x PV = = x + k k k=0 (1 + r)N k=1 1 + [(1 + r)N − 1] x (1 + r)N = x + = x · (1 + r)N − 1 (1 + r)N − 1 1 = x · . 1 − 1/(1 + r)N
Lecture 3 14 / 27 r = x · 1 − 1/(1 + r)N
Annual worth (5)
To get the annual worth we use the formula derived earlier with n = ∞ (since the stream of cash flows goes on for ever):
A = PV · r
Lecture 3 15 / 27 Annual worth (5)
To get the annual worth we use the formula derived earlier with n = ∞ (since the stream of cash flows goes on for ever):
A = PV · r r = x · 1 − 1/(1 + r)N
Lecture 3 15 / 27 If the interest rate r = 4%, then 0.04 A = 10 000 · = 2 246 USD 1 1 − 1/1.045 0.04 A = 12 000 · = 1 782 USD 2 1 − 1/1.048 If on the other hand r = 32%, then 0.32 A = 10 000 · = 4 264 USD 1 1 − 1/1.325 0.32 A = 12 000 · = 4 307 USD 2 1 − 1/1.328
Annual worth (6)
Recall our example: Machine 1: 10 000 USD and replacement after 5 years. Machine 2: 12 000 USD and replacement after 8 years.
Lecture 3 16 / 27 0.04 A = 12 000 · = 1 782 USD 2 1 − 1/1.048 If on the other hand r = 32%, then 0.32 A = 10 000 · = 4 264 USD 1 1 − 1/1.325 0.32 A = 12 000 · = 4 307 USD 2 1 − 1/1.328
Annual worth (6)
Recall our example: Machine 1: 10 000 USD and replacement after 5 years. Machine 2: 12 000 USD and replacement after 8 years. If the interest rate r = 4%, then 0.04 A = 10 000 · = 2 246 USD 1 1 − 1/1.045
Lecture 3 16 / 27 If on the other hand r = 32%, then 0.32 A = 10 000 · = 4 264 USD 1 1 − 1/1.325 0.32 A = 12 000 · = 4 307 USD 2 1 − 1/1.328
Annual worth (6)
Recall our example: Machine 1: 10 000 USD and replacement after 5 years. Machine 2: 12 000 USD and replacement after 8 years. If the interest rate r = 4%, then 0.04 A = 10 000 · = 2 246 USD 1 1 − 1/1.045 0.04 A = 12 000 · = 1 782 USD 2 1 − 1/1.048
Lecture 3 16 / 27 0.32 A = 12 000 · = 4 307 USD 2 1 − 1/1.328
Annual worth (6)
Recall our example: Machine 1: 10 000 USD and replacement after 5 years. Machine 2: 12 000 USD and replacement after 8 years. If the interest rate r = 4%, then 0.04 A = 10 000 · = 2 246 USD 1 1 − 1/1.045 0.04 A = 12 000 · = 1 782 USD 2 1 − 1/1.048 If on the other hand r = 32%, then 0.32 A = 10 000 · = 4 264 USD 1 1 − 1/1.325
Lecture 3 16 / 27 Annual worth (6)
Recall our example: Machine 1: 10 000 USD and replacement after 5 years. Machine 2: 12 000 USD and replacement after 8 years. If the interest rate r = 4%, then 0.04 A = 10 000 · = 2 246 USD 1 1 − 1/1.045 0.04 A = 12 000 · = 1 782 USD 2 1 − 1/1.048 If on the other hand r = 32%, then 0.32 A = 10 000 · = 4 264 USD 1 1 − 1/1.325 0.32 A = 12 000 · = 4 307 USD 2 1 − 1/1.328
Lecture 3 16 / 27 If the face value is the only cash flow the bond generates, then the bond is called a zero-coupon bond (ZCB).
A bond can generate cash during its life time. These are called coupons and can be thought of as interest rate payments. We will generally assume that each coupon payment is the same.
The interest rate used to discount the bond’s payment is called the bond’s yield to maturity, or yield for short.
Bonds
A bond is a fixed-income security that pays the holder an amount, called the face value or par value, at a specified time, called the maturity time.
Lecture 3 17 / 27 A bond can generate cash during its life time. These are called coupons and can be thought of as interest rate payments. We will generally assume that each coupon payment is the same.
The interest rate used to discount the bond’s payment is called the bond’s yield to maturity, or yield for short.
Bonds
A bond is a fixed-income security that pays the holder an amount, called the face value or par value, at a specified time, called the maturity time.
If the face value is the only cash flow the bond generates, then the bond is called a zero-coupon bond (ZCB).
Lecture 3 17 / 27 The interest rate used to discount the bond’s payment is called the bond’s yield to maturity, or yield for short.
Bonds
A bond is a fixed-income security that pays the holder an amount, called the face value or par value, at a specified time, called the maturity time.
If the face value is the only cash flow the bond generates, then the bond is called a zero-coupon bond (ZCB).
A bond can generate cash during its life time. These are called coupons and can be thought of as interest rate payments. We will generally assume that each coupon payment is the same.
Lecture 3 17 / 27 Bonds
A bond is a fixed-income security that pays the holder an amount, called the face value or par value, at a specified time, called the maturity time.
If the face value is the only cash flow the bond generates, then the bond is called a zero-coupon bond (ZCB).
A bond can generate cash during its life time. These are called coupons and can be thought of as interest rate payments. We will generally assume that each coupon payment is the same.
The interest rate used to discount the bond’s payment is called the bond’s yield to maturity, or yield for short.
Lecture 3 17 / 27 F denote the bond’s face value n denote the number of periods to the bond’s maturity m denote the number of coupon payments per year C denote the coupon payment (quoted per year) λ denote the yield
It assumed that F > 0 (unless n = ∞ in which case we set F = 0) and C ≥ 0.
Bond valuation (1)
We let P denote the price of the bond
Lecture 3 18 / 27 n denote the number of periods to the bond’s maturity m denote the number of coupon payments per year C denote the coupon payment (quoted per year) λ denote the yield
It assumed that F > 0 (unless n = ∞ in which case we set F = 0) and C ≥ 0.
Bond valuation (1)
We let P denote the price of the bond F denote the bond’s face value
Lecture 3 18 / 27 m denote the number of coupon payments per year C denote the coupon payment (quoted per year) λ denote the yield
It assumed that F > 0 (unless n = ∞ in which case we set F = 0) and C ≥ 0.
Bond valuation (1)
We let P denote the price of the bond F denote the bond’s face value n denote the number of periods to the bond’s maturity
Lecture 3 18 / 27 C denote the coupon payment (quoted per year) λ denote the yield
It assumed that F > 0 (unless n = ∞ in which case we set F = 0) and C ≥ 0.
Bond valuation (1)
We let P denote the price of the bond F denote the bond’s face value n denote the number of periods to the bond’s maturity m denote the number of coupon payments per year
Lecture 3 18 / 27 λ denote the yield
It assumed that F > 0 (unless n = ∞ in which case we set F = 0) and C ≥ 0.
Bond valuation (1)
We let P denote the price of the bond F denote the bond’s face value n denote the number of periods to the bond’s maturity m denote the number of coupon payments per year C denote the coupon payment (quoted per year)
Lecture 3 18 / 27 It assumed that F > 0 (unless n = ∞ in which case we set F = 0) and C ≥ 0.
Bond valuation (1)
We let P denote the price of the bond F denote the bond’s face value n denote the number of periods to the bond’s maturity m denote the number of coupon payments per year C denote the coupon payment (quoted per year) λ denote the yield
Lecture 3 18 / 27 Bond valuation (1)
We let P denote the price of the bond F denote the bond’s face value n denote the number of periods to the bond’s maturity m denote the number of coupon payments per year C denote the coupon payment (quoted per year) λ denote the yield
It assumed that F > 0 (unless n = ∞ in which case we set F = 0) and C ≥ 0.
Lecture 3 18 / 27 This can be simplified: Theorem (Bond pricing formula) The price of a bond is given by
F C 1 P = + 1 − (1 + λ/m)n λ (1 + λ/m)n C F − C/λ = + . λ (1 + λ/m)n
Bond valuation (2)
Using the present value formula we get
n F X C/m P = + . (1 + λ/m)n (1 + λ/m)k k=1
Lecture 3 19 / 27 Bond valuation (2)
Using the present value formula we get
n F X C/m P = + . (1 + λ/m)n (1 + λ/m)k k=1 This can be simplified: Theorem (Bond pricing formula) The price of a bond is given by
F C 1 P = + 1 − (1 + λ/m)n λ (1 + λ/m)n C F − C/λ = + . λ (1 + λ/m)n
Lecture 3 19 / 27 To do this we use the formula for a finite-life annuity on the sum of the present value of the coupons.
Here we have C λ A = and r = . m m We get
n X C/m C/m 1 = 1 − (1 + λ/m)k λ/m (1 + λ/m)n i=1 C 1 = 1 − λ (1 + λ/m)n
Bond valuation (3)
Proof. We need to show that the sum is equal to the expression in the Theorem.
Lecture 3 20 / 27 Here we have C λ A = and r = . m m We get
n X C/m C/m 1 = 1 − (1 + λ/m)k λ/m (1 + λ/m)n i=1 C 1 = 1 − λ (1 + λ/m)n
Bond valuation (3)
Proof. We need to show that the sum is equal to the expression in the Theorem.To do this we use the formula for a finite-life annuity on the sum of the present value of the coupons.
Lecture 3 20 / 27 We get
n X C/m C/m 1 = 1 − (1 + λ/m)k λ/m (1 + λ/m)n i=1 C 1 = 1 − λ (1 + λ/m)n
Bond valuation (3)
Proof. We need to show that the sum is equal to the expression in the Theorem.To do this we use the formula for a finite-life annuity on the sum of the present value of the coupons.
Here we have C λ A = and r = . m m
Lecture 3 20 / 27 C 1 = 1 − λ (1 + λ/m)n
Bond valuation (3)
Proof. We need to show that the sum is equal to the expression in the Theorem.To do this we use the formula for a finite-life annuity on the sum of the present value of the coupons.
Here we have C λ A = and r = . m m We get
n X C/m C/m 1 = 1 − (1 + λ/m)k λ/m (1 + λ/m)n i=1
Lecture 3 20 / 27 Bond valuation (3)
Proof. We need to show that the sum is equal to the expression in the Theorem.To do this we use the formula for a finite-life annuity on the sum of the present value of the coupons.
Here we have C λ A = and r = . m m We get
n X C/m C/m 1 = 1 − (1 + λ/m)k λ/m (1 + λ/m)n i=1 C 1 = 1 − λ (1 + λ/m)n
Lecture 3 20 / 27 The relation between the yield and the price is an inversely one: the higher the yield the lower the price, and vice versa.
Hence, with fixed F and C the curve
λ 7→ P(λ)
is decreasing. One can also show that this curve is convex.
Bond valuation (4)
Note that the yield is the IRR of the bond. It follows from the Main theorem of IRR that there exists a unique yield λ ∈ (0, ∞) as long as the price P > 0.
Lecture 3 21 / 27 Hence, with fixed F and C the curve
λ 7→ P(λ)
is decreasing. One can also show that this curve is convex.
Bond valuation (4)
Note that the yield is the IRR of the bond. It follows from the Main theorem of IRR that there exists a unique yield λ ∈ (0, ∞) as long as the price P > 0.
The relation between the yield and the price is an inversely one: the higher the yield the lower the price, and vice versa.
Lecture 3 21 / 27 Bond valuation (4)
Note that the yield is the IRR of the bond. It follows from the Main theorem of IRR that there exists a unique yield λ ∈ (0, ∞) as long as the price P > 0.
The relation between the yield and the price is an inversely one: the higher the yield the lower the price, and vice versa.
Hence, with fixed F and C the curve
λ 7→ P(λ)
is decreasing. One can also show that this curve is convex.
Lecture 3 21 / 27 This is an important class of bonds, since it is easy to solve for the yield if the price of the bond is observed in the market: " # F 1/n λ = m − 1 . P
Bond valuation (5)
When C = 0, then we have a zero-coupon bond. In this case F P = . (1 + λ/m)n
Lecture 3 22 / 27 Bond valuation (5)
When C = 0, then we have a zero-coupon bond. In this case F P = . (1 + λ/m)n
This is an important class of bonds, since it is easy to solve for the yield if the price of the bond is observed in the market: " # F 1/n λ = m − 1 . P
Lecture 3 22 / 27 This is the value of the of a perpetual bond or consol bond, which is a bond that like a perpetual annuitity gives the same payoff (in this case C) every period forever.
Bond valuation (6)
Note that as n → ∞ we get C P = . λ
Lecture 3 23 / 27 Bond valuation (6)
Note that as n → ∞ we get C P = . λ This is the value of the of a perpetual bond or consol bond, which is a bond that like a perpetual annuitity gives the same payoff (in this case C) every period forever.
Lecture 3 23 / 27 Replacing C with RF in the pricing formula yields
RF F − RF /λ P = + . λ (1 + λ/m)n
Bond valuation (7)
One often, this is not done in the book, though, introduce the coupon rate R defined by C R = . F
Lecture 3 24 / 27 Bond valuation (7)
One often, this is not done in the book, though, introduce the coupon rate R defined by C R = . F Replacing C with RF in the pricing formula yields
RF F − RF /λ P = + . λ (1 + λ/m)n
Lecture 3 24 / 27 This shows that if R = λ, then P = F .
In this case we pay F initially, get the coupons during the lifetime of the bond, and finally get the investment F back.
Hence the bond works like a bank account with interest rate R when R = λ.
Bond valuation (8)
The previous equation can be written
P R 1 − R/λ = + . F λ (1 + λ/m)n
Lecture 3 25 / 27 In this case we pay F initially, get the coupons during the lifetime of the bond, and finally get the investment F back.
Hence the bond works like a bank account with interest rate R when R = λ.
Bond valuation (8)
The previous equation can be written
P R 1 − R/λ = + . F λ (1 + λ/m)n
This shows that if R = λ, then P = F .
Lecture 3 25 / 27 Hence the bond works like a bank account with interest rate R when R = λ.
Bond valuation (8)
The previous equation can be written
P R 1 − R/λ = + . F λ (1 + λ/m)n
This shows that if R = λ, then P = F .
In this case we pay F initially, get the coupons during the lifetime of the bond, and finally get the investment F back.
Lecture 3 25 / 27 Bond valuation (8)
The previous equation can be written
P R 1 − R/λ = + . F λ (1 + λ/m)n
This shows that if R = λ, then P = F .
In this case we pay F initially, get the coupons during the lifetime of the bond, and finally get the investment F back.
Hence the bond works like a bank account with interest rate R when R = λ.
Lecture 3 25 / 27 This is a simplifying assumption, and we will often make it. In real-life, however, this is seldom the case.
Not only does this make the mathematics slightly more complicated, if we buy or sell a bond on the market we also need to take the accrued interest (AI) into account.
The accrued interest adjusts for the fact that if we buy a bond between to coupon payments, then the seller wants to be compansated for the fact that not interest has been paid out to him during this time.
Accrued interest (1)
So far we have considered the value of a bond at a time where a coupon have just been paid out.
Lecture 3 26 / 27 Not only does this make the mathematics slightly more complicated, if we buy or sell a bond on the market we also need to take the accrued interest (AI) into account.
The accrued interest adjusts for the fact that if we buy a bond between to coupon payments, then the seller wants to be compansated for the fact that not interest has been paid out to him during this time.
Accrued interest (1)
So far we have considered the value of a bond at a time where a coupon have just been paid out.
This is a simplifying assumption, and we will often make it. In real-life, however, this is seldom the case.
Lecture 3 26 / 27 The accrued interest adjusts for the fact that if we buy a bond between to coupon payments, then the seller wants to be compansated for the fact that not interest has been paid out to him during this time.
Accrued interest (1)
So far we have considered the value of a bond at a time where a coupon have just been paid out.
This is a simplifying assumption, and we will often make it. In real-life, however, this is seldom the case.
Not only does this make the mathematics slightly more complicated, if we buy or sell a bond on the market we also need to take the accrued interest (AI) into account.
Lecture 3 26 / 27 Accrued interest (1)
So far we have considered the value of a bond at a time where a coupon have just been paid out.
This is a simplifying assumption, and we will often make it. In real-life, however, this is seldom the case.
Not only does this make the mathematics slightly more complicated, if we buy or sell a bond on the market we also need to take the accrued interest (AI) into account.
The accrued interest adjusts for the fact that if we buy a bond between to coupon payments, then the seller wants to be compansated for the fact that not interest has been paid out to him during this time.
Lecture 3 26 / 27 The theoretical price if called the dirty price and the bond price adjusted for accrued interest is referred to as the clean price:
Dirty price = Clean price + AI.
It is the clean price that is quoted on the market.
See also ‘More on accrued interest’ on the course’s home page.
Accrued interest (2)
The accrued interest is defined by Number of days since last coupon AI = · Coupon amount, Number of days in current coupon period and this is the amount by which the bond price is adjusted.
Lecture 3 27 / 27 It is the clean price that is quoted on the market.
See also ‘More on accrued interest’ on the course’s home page.
Accrued interest (2)
The accrued interest is defined by Number of days since last coupon AI = · Coupon amount, Number of days in current coupon period and this is the amount by which the bond price is adjusted.
The theoretical price if called the dirty price and the bond price adjusted for accrued interest is referred to as the clean price:
Dirty price = Clean price + AI.
Lecture 3 27 / 27 See also ‘More on accrued interest’ on the course’s home page.
Accrued interest (2)
The accrued interest is defined by Number of days since last coupon AI = · Coupon amount, Number of days in current coupon period and this is the amount by which the bond price is adjusted.
The theoretical price if called the dirty price and the bond price adjusted for accrued interest is referred to as the clean price:
Dirty price = Clean price + AI.
It is the clean price that is quoted on the market.
Lecture 3 27 / 27 Accrued interest (2)
The accrued interest is defined by Number of days since last coupon AI = · Coupon amount, Number of days in current coupon period and this is the amount by which the bond price is adjusted.
The theoretical price if called the dirty price and the bond price adjusted for accrued interest is referred to as the clean price:
Dirty price = Clean price + AI.
It is the clean price that is quoted on the market.
See also ‘More on accrued interest’ on the course’s home page.
Lecture 3 27 / 27