课程资料：Chapter 2：Risk Free Asset

• For a given compounding method with interest rate r the
effective rate re is one that gives the same growth factor over a one year period under annual compounding.
• 2.2 Money Market
• 2.2.1 Zero-Coupon Bonds • 2.2.2 Coupon Bonds
2.1 Time Value of Money
• The way in which money changes its value in time is a complex issue of fundamental importance in finance.
1 r
k r 2 ...
k
k
rm
2!
m!
1 1
(1 1 ) ... (1 k 1)
1 r
k r 2 ...
k
k rk
2!
k!
(1 r )k k
• The first inequality holds because each term of the sum on the lefthand side is no greater than the corresponding term on the righthand side. The second inequality is true because the sum on the right-hand side contains m-k additional non-zero terms as compared to the sum on the left-hand side. This completes the proof.
2 chapter
Risk-Free Assets
main content
• 2.1 Time Value of Money
• 2.1.1 Simple Interest • 2.1.2 Periodic Compounding • 2.1.3 Streams of Payments • 2.1.4 Continuous Compounding
•
V (t) = (1+tr)P
• If the principal P is invested at time s, rather than at time 0,then the value at time t ≥ s will be
•
V (t) = (1+(t − s)r)P.
Simple Interest
2.2.1 Zero-Coupon Bonds
• The simplest case of a bond is a zerocoupon bond, which involves just a single payment. The issuing institution (for example, a government, a bank or a company) promises to exchange the bond for a certain amount of money F, called the
denoted by P, plus all the interest earned since the money was deposited in the account.
• The value of the investment at time t, denoted by V (t), is given by
are to be made once a year for n years, the first
one due a year hence. Assuming that annual
compounding applies, we shall find the present
value of such a stream of payments.
period. In this formula t must be a whole multiple of the
period 1 .The numbe（ r 1 r ）tm is the growth factor .
m
m
Exercise
• To show that V (t) increases as the compounding frequency m increases,the others remaining unchanged.we need to verify that if m < k, then
2.1.4 Continuous Compounding
2.1.5 How to Compare Compounding Methods
• Frequent compounding will produce a higher future value than less frequent compounding if the interest rates and the initial principal are the same.
er 1 re
2.2 Money Market
• The money market consists of risk-free (more precisely, default-free) securities. An example is a bond, which is a financial security promising the holder a sequence of guaranteed future payments. Risk-free means here that these payments will be delivered with certainty.
– What is the future value of an amount invested or borrowed today?
– What is the present value of an amount to be paid or received at a certain time in the future?
（1 r )tm (1 r )tk
m
k
• which can be verified directly using the binomial formula:
（1
r )m
1 1
(1 1 ) ... (1 m 1)
1 r
m r 2 ...
m
m rm
m
2!
m!
1 1
(1 1 ) ... (1 m 1)
2.1.1 Simple Interest
• Suppose that an amount is paid into a bank account, where
it is to earn interest. The future value of this investment consists of the initial deposit, called the principal and
face value, on a given day T, called the maturity date.
Return:
Some Concepts
• Future Value, Growth Factor
• Discounted Value(present), Discount Factor
• In practice simple interest is used only for short-term investments and for certain types of loans and deposits.
PA(r, n) 1 (1 r)n r
Perpetuity
• When n→∞
2.1.4 Continuous Compounding
• Formula for the future value at time t of a principal P
attracting interest at a rate r > 0 compounded m times a
2.1.3 Streams of Payments
• An annuity is a sequence of finitely many
payments of a fixed amount due at equal time
intervals. Suppose that payments of an amount C
பைடு நூலகம்
time between two consecutive payments measured in years
will be 1 ,the first interest payment being due at time 1 .
Each intmerest payment will increase the principal by a mfactor
year can be written as:
V(t) [(1
r
m
) r ]tr P
m
• In the limit as m→∞, we oe btain
V(t) etr P
• This is known as continuous compounding.The corresponding
growth factor is etr.
• There are many kinds of bonds like treasury bills and notes, treasury, mortgage and debenture bonds, commercial papers, and others with various particular arrangements concerning the issuing institution, duration, number of payments, embedded rights and guarantees.
of
1
+
r m
.Given
that
the
interest
rate
r
remains
unchanged,
after t years the future value of an initial principal P will
become
•
V (t) =（1 r ）tm P
m
• because there will be tm interest payments during this
• In particular, in the case of periodic comproeunding with
frequency m and rate r the effective rate satisfies
（1
r ）m m
1
re
• In the case of continuous compounding with rate r
• The interest already earned can be reinvested to attract even more interest.
2.1.2 Periodic Compounding
• In general, if m interest payments are made per annum, the
PV
C 1 r
C (1 r)2
...
C (1 r)n
• It is sometimes convenient to introduce the following seemingly cumbersome piece of notation:
PA(r, n)
1 1 r
1 (1 r)2
...
1 (1 r)n
• This number is called the present value factor for an annuity. It allows us to express the present value of an annuity in a concise form:
•
PA(r, n) × C