Lecture 1: Interest Rate
Measurement
Goals:
• Review interest, and introduce notation
• Study how interest factors into calculation
• Savings, Treasury Bills and Promissory notes
• Introduce rate of discount, nominal rate of interest and discount
• Learn force of interest and applications
Suggested Textbook Readings: Chapter 1
Practice Problems: All exercises in §1.1-§1.7 without an asterisk
Lecture 1: Interest Rate Measurement
2
Compound and Simple Interest
Interest is the time value of money.
Example 1: Suppose $1000 is deposited into a saving account which earns 9% interest
per year with interest credited annually. How much money will be in the account after 3
years?
Interest Function An initial deposit of C earning interest at an annual rate i, credited
annually, for n years will have value
A(n) = C(1 + i)n .
The rate of interest may vary from one year to the next.
Example 2: An income fund earned following rates of interest from 2002 to 2006.
2002
6.91%
2003
5.39%
2004
5.17%
2005
5.02%
2006
2.73%
If $5000 is deposited in the fund at the beginning of 2002. How much will be in the
account at the end of 2006? What is the average rate of interest earned over the 5
years?
MATH 384 Fall 2012
Lecture 1: Interest Rate Measurement
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Compound Interest
Example 3: Suppose $1000 is deposited into a saving account which earns 9% interest
per year compounded quarterly. How much money will be in the account after 3 years?
If an annual rate i is compounded m times a year, then after t years, an initial inverstment
of C = A(0) would have value
mt
i
A(t) = A(0) 1 +
m
Effective Annual Rate of Interest is the percentage in the value of the investment
from the beginning of the year to the end of the year.
A(1) − A(0)
A(t + 1) − A(t)
or
A(0)
A(t)
Simple Interest
To calculate interest outside of a normal payment/credit schedule or over a fraction of
a interest period, simple interest is often used.
Promissory Note is a short term contract requiring the issuer to pay the holder a
principal plus interest at a specified annual interest rate for a specified period of time
(usually less than a year). The interest rate earned by the holder is called yield rate.
Example 4: Smith borrows $480 from Brown and give Brown a promissory note stating
that he will pay Brown $500 in 40 days. What is the yield rate for Brown on the note?
Assume 365 days a year.
MATH 384 Fall 2012
Lecture 1: Interest Rate Measurement
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Present Value
There are times that we may know the future value A(n) of an investment after n years
and wish to find the present value A(0).
Present value factor for compound interest v =
1
.
1+i
Then A(0) = A(n)v n .
Example 5: Ted wants to save $10,000 to buy a car in 2 years. He deposited $4000
five years ago in an account earning an effective annual rate of 5%. How much does he
need to deposit now to have $10,000 in the account in 2 years?
Present value for simple interest A(0) =
A(n)
.
1 + it
Treasury Bills
A Treasury Bill is a debt obligation that requires the issuer to pay the owner a spcified
sum (the face amount) on a specified date (the maturity date).
Example 6: A 35-day T-Bill is sold with a yield rate of 3.5% and face value of $200.
What is the price of the bill?
MATH 384 Fall 2012
Lecture 1: Interest Rate Measurement
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Effective Annual Rate of Discount
In previous sections, interest amounts are paid at the end of the interest period. This
usually is referred to as “interest payable in arrears”. Sometimes we need to consider
interest payable in advance. In this case, the annual discount rate, usually denoted by
d, is applied to obtain an amount of interest which is payable at the start of the period,
that is, the interest on a loan is calculated and deducted at the time the loan is made.
For example, payday loans.
Example 7: Smith ask Brown for another $300 to borrow for a year. Brown agrees,
but only if Smith pays interest rate of 15% in advance. How much does Smith receive at
the beginning of the loan? What interest rate has Smith paid on the loan?
Effective annual discount rate from time t = 0 to time t = 1 is
d=
A(1) − A(0)
A(1)
It can be written as A(0) = (1 − d)A(1). During the n-th year the effective rate of
discount is
A(n + 1) − A(n)
dn =
A(n + 1)
In general from time t = 0 to time t = n the amount function is
A(0) = A(n)(1 − d)n
Example 7: continued
Smith borrows $300 from Brown and pay back in one year. Compare the two scenarios.
• Annual interest rate of 15%.
• Annual discount rate of 15%.
MATH 384 Fall 2012
Lecture 1: Interest Rate Measurement
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Connections between Annual Interest Rate i and Annual Discount Rate d
For every effective annual interest rate i, there is a corresponding effective annual discount rate d.
i
d
d=
i=
1+i
1−d
Simple Discount
Simple discount is usually applied for periods less than one year, and is calculated
linearly.
A(0) = (1 − dt)A(t)
With an annual discount rate of d, based on simple interest, the present value of 1
payable t years (t is a fraction of a year) from now is 1 − dt
Example 8: (U.S. Treasury Bill) If a 182-day U.S. Treasury Bill with face value 100
is sold for $99.292. What is the annual discount rate d? Assume 360 days in a year.
What is the annual effective interest rate?
MATH 384 Fall 2012
Lecture 1: Interest Rate Measurement
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Nominal Rate of Interest and Discount
If an interest rate j is credited m times a year, we define the nominal rate of interest
i(m) to be j · m. Similarly the nominal rate of discount, denoted by d(m) , is c · m if c is
the discount rate in m1 year.
Given the nominal rate i(m) , the effective interest rate is
m
i(m)
1+
−1
m
Similarly given the nominal discount rate d(m) , the effective discount rate is
m
d(m)
1− 1−
m
Example 9: Four banks each offers a fund with the following rate.
Bank A: effective interest rate 12%
Bank B: i(4) = 11.5%
Bank C: effective discount rate 11.2%
Bank D: d(4) = 11%
Which offers the best rate?
MATH 384 Fall 2012
Lecture 1: Interest Rate Measurement
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Continuous Compounding
At nominal rate i(m) , the accumulated value of an initial investment of a dollar after a
year is
m
i(m)
a(m) = 1 +
m
With fixed i(m) , the more often compounding taking place during the year, the larger
the accumulated value is.
Example 10: Suppose that the interest is compounded continuously. Find the accumulated value a(m) when m → ∞.
Force of Interest
If the interest is compounded continuously, the accumulated amount function A(t) is a
continuous function of t. The nominal rate is called the force of interest and denoted
by δt (sometimes i(∞) ). The notation i(∞) makes sense, since we can think of the force
of interest as the limit as the number of times we credit the compound interest goes to
infinity. that is, δ = i(∞) = lim i(m) .
m→∞
Force of Interest
For an investment that grows according to accumulated amount function A(t), the force
of interest at time t, is defined to be
δt =
A0 (t)
A(t)
MATH 384 Fall 2012
Lecture 1: Interest Rate Measurement
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Example 11: (Example 1.13, page 40) Derive an expression for δt if accumulation
is based on
1. simple interest at annual rate i, and
2. compound interest at annual effective rate i.
Force of Interest and Accumulation function
We can recover the accumulation function from the force of interest δt .
We may also calculate the interest earned from time t = 0 to time t = n from the force
of interest δt .
The accumulation function from time 0 to time n is
A(n) = A(0)e
Rn
and the present value is
A(0) = A(n)e−
0
δt dt
Rn
0
δt dt
MATH 384 Fall 2012
Lecture 1: Interest Rate Measurement
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Example 12: Find the accumulation function if the force of interest is
1. δt = k where k is a constant.
√
2. δt = k/ t.
Inflation, Real Rate of Interest
Inflation is the growth in prices from year to year. It is generally measured via the rate
of change in the price of a ”basket of goods”. The rate of inflation is denoted by r.
Real rate of interest With annual interest rate i and annual inflation rate r, the real
rate if interest for the year is
ireal =
value of amount of real return (yr-end dollars)
i−r
=
value of invested amount (yr-end dollars)
1+r
Note that when r is small, i − r is a good estimate for the real rate of interest.
MATH 384 Fall 2012