MATH3090/7039: Financial mathematics Lecture 2
Financial instruments and valuation
Single cashflow (money market securities)
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Multiple cashflows (debt capital market securities )
Yield to maturity (YTM)
Effective annual yield
Recommended Resources for This Week
ASX Bonds and Understanding ASX Interest Rate Securities. Bond Market and in particular Bond Valuation.
Brealey, Myers and Allen, Principles of Corporate Finance, McGraw-Hill/Irwin.
(Part 1 Value lays the foundation of finance and financial mathematics.)
Financial instruments and valuation
Single cashflow (money market securities)
Multiple cashflows (debt capital market securities )
Yield to maturity (YTM)
Effective annual yield
Financial markets and instruments
Recall from L1.18-L1.23 that financial markets can be classified in terms of instruments traded.
• Equity/share markets • Debt markets
◦ Money markets (short-term, and very liquid) ◦ Debt capital markets (long-term, not liquid)
• Foreign exchange (FX) markets • Derivative security markets
• Insurance markets
Some also include commodity markets in this classification.
Recall from L1.26 that
• A financial security is a contract between market participants which represents future cashflow obligations.
• We view a financial security only in terms of its potential future cashflow structure: size, timing and risk of the cashflows.
There are two main valuation or pricing frameworks: • Discounted cashflow (DCF) valuation.
• Valuation by arbitrage and replication.
Discount cashflow (DCF) valuation
The today value of a financial security is
• the present value of its expected future cashflows
• discounted at an appropriate required rate of return.
Leads to the risk-neutral approach to asset pricing.
We will learn it today, and revisit in the second module of the course.
Valuation by no-arbitrage and replication
Application of the law of one price. In the absence of arbitrage, the following must hold
• Assets with equal future cashflows/payoff must have the same price.
• The value of an asset is the value of any portfolio which replicates the asset’s future cashflow structure.
Leads to the partial differential equation (PDE) approach to asset pricing.
We will cover this in the second module of the course.
Today we begin on DCF valuation framework for a simple case. We assume that
• all the cashflows are certain (i.e. no risk), and hence, no expectation involved.
• the required interest rate is given (r) We first look at valuing
• Single cashflow in money market securities
• Multiple discrete cashflows in debt capital market securities
Financial instruments and valuation
Single cashflow (money market securities)
Multiple cashflows (debt capital market securities )
Yield to maturity (YTM)
Effective annual yield
Single cashflow (money market securities)
These securities are characterised by a single cashflow called the face value or future value, payable at maturity. No other cashflows.
We are interested in finding the present (today’s) value of this cashflow.
We investigate two main pricing conventions • Simple interest convention.
• Discount rate convention.
Simple interest convention: notation
• T is the number of years until maturity. • F is the face value or future value.
• P is the present value.
• r is the annual simple interest rate.
• I is the accumulated interest earned on P from today until T.
Simple interest convention: formulas
Simple interest formulas:
I = PrT, The term β(t) = 1
P = F β(t) i.e.
F = P(1 + rT) and P = F . 1+rT
is the discount process.
today value = future value × discount process.
Example: T = 1 (years), P = 100, 000, and r = 5%/year. What is the future value F?
Accumulation process
Express F (t) = P (1 + rt) as a function of time t to maturity. Define the accumulation process
A(t) = 1 + rt so β(t) = 1 . A(t)
We also see that
F(t) = PA(t) and P = F(t)β(t).
If P = 1, then the accumulation process is sometimes call the bank account process.
Simple interest convention: example
A Treasury bill (T-Bill) is a short-term debt obligation backed by the U.S. government with a maturity of less than one year.
Investors can buy T-bills directly in the primary market (government) or the secondary market.
Pay P today, and receive the face value F > P at maturity.
Numerical example: What is the price of a 90 day T-bill with the face value of $100, 000 and which is yielding 5% simple interest?
Solution: Note that T = 90 365
is the number of years until maturity.
P = F = 100,000 =$98,782.14.
1+rT 1+0.05 90 365
In this example, we view r as the required rate of return on an investment which pays a single cashflow F in T years time. We then use r for discounting.
This may not be the case in general, when other factors, such as liquidity, risk, are present.
Two views of the same problem.
• The present value P of a single cashflow F received in T years
time is simply P = F under the simple interest convention. 1+rT
(We discount F back to t = 0 under the required rate of return r.)
• Alternatively, the value in T years time of a single cashflow P received today given the required rate of return r is
F =P(1+rT).
• It always helps to draw a timeline.
• We use the convention that there are 365 days in a year.
• Always express T in terms of the number of years to maturity. • The annual simple interest rate r is in decimal form.
• We can solve for any given parameter from the others.
Discount rate convention
• d be the (annual) discount rate. • D be the total discount.
The discount rate formulas are: D=FdT, P =F(1−dT)
and F = The term β(t) = 1 − dt is the discount process. We write
P = F β(t) ie today value = future value × discount process.
• Always draw a timeline.
• The price P is expressed as a discount to the face value.
◦ Imagine buying something at a discount.
• There are 365 days in a year and always express T in years to
• The discount rate d is always in decimal form.
• We can solve for any given parameter from the others.
What is the price of a 90-day T-bill if instead it is priced as a 5% discount to the face value $100, 000?
P =F(1−dT)=100,0001−0.05 90 =$98,767.12. 365
Simple interest ↔ discount rate
Assume only one cashflow at maturity. Derive the following
conversion formulas
r=d andd=r.
1−dT 1+rT Solution: we find r and d such that
P= F =F(1−dT). 1+rT
What is the intuition here?
Financial instruments and valuation
Single cashflow (money market securities)
Multiple cashflows (debt capital market securities )
Yield to maturity (YTM)
Effective annual yield
Multiple cashflows (debt capital market securities )
For simplicity, we assume that all cashflows are one-year apart over time, and T is a multiple of one year.
Debt capital market securities are characterised by
• A final payment of the principal F at maturity.
• A series of cashflows of sizes C1,C2,…,CT at years 1,2,…,T, respectively. These cashflows can be referred to loan/interest/coupons payments.
Annuity: all cashflow C1 , C2 , . . . , CT are of equally-sized cashflows C and are equally spaced (one-year apart) over time.
Valuation: sum over all present values of cashflows. We first need to recall the compound interest formulas.
Compound interest
• T is the number of years until maturity.
• F is the face value or future value.
• P is the present value.
• y is the annual compound interest rate or yield. • C is the annuity cashflow in annual terms.
• n is the number of compounding periods in a year. An illustration with 1, 2, . . . , n compounding periods.
Compound interest: formulas
Compound interest formulas:
• The term β(t) = 1+ ny nt
P = Fβ(t) ie today value = future value×discount process. • It may be simpler to write
F =P(1+y)T
and remember to adjust y and T for the compounding frequency.
F=P 1+n and
P=1+nynT.
is called the discount process, so
Compound interest: accumulation process
Express F (t) = P 1 + ny nt as a function of time t to maturity. Define the accumulation process
It follows that
ynt A(t)= 1+n so
F(t) = PA(t) and
1 β(t)=A(t).
P = F(t)β(t).
Continuous compounding
Under continuous compounding, the number of compounding periods n → ∞. The formulas become:
F =PeyT and P =Fe−yT. (It can be shown by l’Hopital’s rule that
lim 1 + y nT = eyT . n→∞ n
The continuous accumulation and discount processes respectively are
A(t) = eyt and β(t) = e−yt = 1 . A(t)
Continuous compounding: nonconstant rate
Let y(t) be the nonconstant instantaneous interest rate (yield) at time t.
The accumulation process is
t y(s)ds
β(t) = exp −
These formulas automatically cover the case of constant y.
A(t) = exp The discount process is
y(s)ds = A(t).
Continuous compounding: stochastic rate
• Above we are assuming that the nonconstant rate y(t) is a deterministic function of time.
This modeling assumption is only reasonable for a short period of time.
• In more advanced financial models, the rate y is allowed to be a stochastic process or random over time.
This explains the terms accumulation and discount processes.
• We will cover some very simple stochastic interest rate models later in the course.
DCF valuation: multiple discrete cashflows
View y as your required annual rate of return on an investment which pays cashflows Ct in t years time, t = 1,…,T, over a period of T years.
The present value of the cashflow stream C1 , . . . , CT is • Discrete compounding (constant): P = T Ct
• Continuous compounding (constant): P = Cte−yt t=1
• Continuous compounding (nonconstant):
DCF valuation: continuous cashflows
Suppose we receive a continuous cashflow stream and let C(t) be the instantaneous annual cashflow rate at time t.
• Constant continuously compounded yields P = • Non-constant continuously compounded yields
P= C(t)exp − y(s)ds dt 00
C(t)e−ytdt
DCF valuation: general case
We may have discrete Ct and continuous C(t) cashflow.
The general discounted cashflow (DCF) equation combines them.
• Constant continuously compounded yields
T T Cte−yt +
C(t)e−ytdt
• Nonconstant continuously compounded yields
T t T t
Ct exp − y(s)ds + C(t)exp − y(s)ds dt 000
Valuing zero coupon bonds
• Characterised by only the face value F payable at maturity T . • They are long-dated securities (not money market instruments). • They could be purchased in the primary or secondary markets. • The convention is to price them by compound interest:
β(T)= (1+y)T or β(T)=e or β(T)=exp − depending on the interest rate convention being used.
1 −yT T y(s)ds
Valuing zero coupon bonds: example
Suppose a company issues a zero with face value F = $100, 000 and which matures in T = 5 years. Calculate the price based on (i) a 5% discrete annual yield, (ii) a 5% continuous annual yield, and (iii) a nonconstant yield of y(t) = 0.045 + 0.002t2.
(i) P = 100,000 = 100,000 = $78, 352.62.
(1+y)T 1.055
(ii) P = 100, 000e−yT = 100, 000e−0.05×5 = $77, 880.08.
P = F exp − y(s)ds = 100, 000 exp − (0.045 + 0.002t2)ds
T5 00
0.002 3 =100,000exp −0.045×5− 3 5
= $73, 467.04.
Annuities: discrete compounding
Annuity: sequence of equally spaced and equally sized cashflows at the end of each year.
We are interested in the present value (P ) and the future value (F ) of an annuity.
Under discrete compounding, the annuity formulas are:
C1+ nynT −1 C1−1+ ny−nT F=n ny and P=n ny
where n is the number of compounding periods in a year.
It is sometimes simpler to write these formulas as (1+y)T −1
F=C y and 1−(1+y)−T
and remember that C, y, and T must be adjusted to reflect the compounding frequency.
Annuities: derivation
Derived using the notion of a geometric sequence: use
• future value, use 1+a+a2 +…+aT−1 = (1−aT)
and let a = 1 + y
• present value, use a+a2 +…+aT−1 +aT = a(1−aT)
andleta= 1 (1+y)
Annuities: continuous compounding
Assuming annual cashflow payments C.
Under continuous compounding the annuity formulas become:
eyT −1 1−e−yT F = C ey − 1 and P = C ey − 1 .
Derived the same way by letting a = ey and then a = e−y.
Annuities: example
WhatistheF andP ofanannuityof$100overT =20yearsat5%, under (i) discrete compounding and (ii) continuous compounding?
(i) F = C(1+y)T −1 = 100(1+0.05)20−1 = $3,306.60. y 0.05
P = C1−(1+y)−T = 1001−(1+0.05)−20 = $1,246.22. y 0.05
(ii) F = CeyT −1 = 100e(0.05)20−1 = $3,351.37. ey −1 e0.05 −1
P = C1−e−yT = 1001−e−(0.05)20 = $1,232.90. ey −1 e0.05 −1
A bond is characterised by a promise to pay the face value F at maturity and an annuity cashflow of coupon payments C.
With the convention that the compounding convention is typically semiannual, i.e. n = 2, the value B is
C1−1+ y−2T F B=2+.
2 y2 1+y2T 2
Bonds: example
A 30 year $100, 000 government bond has a coupon rate of 6% payable semiannually and yields 5%. Calculate the price. Solution:
C1−1+ y−2T F B=2+
2 y2 1+y2T 2
6,0001−1+ 0.05−2×30 100,000 =2+
2 0.05 1 + 0.05 2×30 22
= $115, 454.33.
Perpetuities and preference shares
A preference share is not really a debt instrument. It is a kind of share (equity) which pays an annuity cashflow forever.
If 0 < a < 1
lim a(1−aT)= a T→∞ 1−a 1−a
sorecallingthat0