1 - Interest Rate Measurement

Question 1

What is the opening quote for this chapter?

Answer 1

The safest way to double your money is to fold it over and put it in your pocket.

Question 2

What is interest?

Answer 2

The time value of money. In the most common context, interest refers to the consideration or rent paid by a borrower of money to a lender for the use of the money over a period of time.

Question 3

What does the Federal Reserve Board set?

Answer 3

The “federal funds discount rate”, a target rate at which banks can borrow and invest funds with one another.

Question 4

What does Libor refer to?

Answer 4

The London Interbank Overnight Rate, which is an international rate charged by one bank to another for very short term loans denominated in US dollars.

Question 5

How are interest rates typically quoted?

Answer 5

As an annual percentage.

Question 6

What’s the formula for calculating the accumulated value or future value for compound interest?

Answer 6

Cn = C(1+i)^n

Question 7

If someone made an initial deposit on January 1st 2018, how would the interest accrue?

Answer 7

He would have interest added to his account on December 31 of 2018 and every December 31st after that for as long as the account remained open.

Question 8

For accumulated interest, the rate of interest may change from one year to the next. How would we incorporate that into the Cn = C(1+i)^n formula?

Answer 8

If the interest rate is i1 in the first year, i2 in the second year, and so on,
Cn = C(1+i1)(1+i2)…(1+in), where the growth factor for year t is (i+it) and the interest rate for year t is it. Note that “year t” starts at time t-1 and ends at time t.

Question 9

In practice, interest may be credited or charged more frequently than once per year. What’s the formula for calculating the accumulated value or future value for this compound interest?

Answer 9

Cn = C(1+j)^n after n compounding period.
It is typical to use i to denote an annual rate of interest, and, in this text, j will often be used to denote an interest rate for a period other than a year.

Question 10

What is the effective annual rate of interest?

Answer 10

The effective annual rate of interest earned by an investment during a one-year period is the percentage change in the value of the investment from the beginning to the end of the year, without regard to the investment behavior at intermediate point in the year.

Question 11

What are equivalent rates of interest?

Answer 11

Two rates of interest are said to be equivalent if they result in the same accumulated values at each point in time.

Question 12

When compound interest is in effect, and deposits and withdrawals are occurring in an account, how can the resulting balance at some future point in time be determined?

Answer 12

By accumulating all individual transactions to that future time point.

Question 13

When compound interest is in effect, and deposits and withdrawals are occurring in an account, the resulting balance at some future point in time can be determined by accumulating all individual transactions to that future time point. how would that look?

Answer 13

Cn = c(1+i)^n + d(1+i)^(n-t) - w(1+i)^(n-t)

where t is the period the deposit or withdrawal was made.

Question 14

What is the accumulation factor and accumulation amount function?

Answer 14

a(t) is the accumulated value at time t of an investment of 1 made at time 0 and defined as the accumulation factor from time 0 to time t. The notation A(t) will be used to denote the accumulated amount of an investment at time t, so that if the initial investment amount is A(0), the the accumulated value at time t is A(t) = a(0)⋅a(t).
A(t) is the accumulated amount function.

Question 15

Compound interest accumulation at rate i per period is defined with t as any real positive real number.
What is the formula for Compound Interest Accumulation?

Answer 15

At effective annual rate of interest i per period, the accumulation factor from time 0 to time t is
a(t) = (1+i)^t

Question 16

In practice, financial transaction can take place at any point in time, and it may be necessary to represent a period which is a fractional part of a year. How is a fraction of a year generally described?

Answer 16

A fraction of a year is generally described in terms of either an integral number of m months, or an exact number of d days. In the case that time is measured in months, it is common in practice to formulate the fraction of the year t in the form t = m/12, even though not all months are exactly 1/12 or a year. In the case that time is measured in days, t is often formulated as t = d/365

Question 17

When considering the equation X(1+i)^t = Y, given any three of the four variables X, Y, i t, it is possible to find the fourth. If the unknown variable is t, then how do we solve for it?

Answer 17

t = ln(Y/X) / ln(1+i)

Question 18

When considering the equation X(1+i)^t = Y, given any three of the four variables X, Y, i t, it is possible to find the fourth. If the unknown variable is i, then how do we solve for it?

Answer 18

i = (Y/X)^(1/t) -1

Question 19

When is simple interest often used?

Answer 19

When calculating interest accumulation over a fraction of a year or when executing short term financial transactions.

Question 20

What is simple interest accumulation function?

Answer 20

The accumulation function from time 0 to time t at annual simple interest rate i, where t is measured in years is
a(t) = 1+it

Question 21

How are fractions of years calculated for simple interest?

Answer 21

Like compound interest, t is either m/12 or d/365.

Question 22

What is a promissory note?

Answer 22

A short-term contract (generally less than one year) requiring the issuer of the note (the borrower) to pay the holder of the note (the lender) a principal amount plus interest on that principal at a specified annual interest rate for a specified length of time. At the end of the time period, the payment (principle and interest) is due.

Question 23

How are promissory notes calculated?

Answer 23

Promissory notes are calculated on the basis of simple interest. the interest rate earned by the lender is sometimes referred to as the “yield rate” earned on the investment.

Question 24

What’s an important note about yield rates?

Answer 24

As concepts are introduced throughout this text, we will see the expression “yield rate” used in a number of different investment contexts with differing meanings. In each case it will be important to relate the meaning of the yield rate to the context in which it is being used.

Question 25

What is a fixed-income investment?

Answer 25

A fixed-income investment is one for which the future payments are predetermined (unlike an investment in, say, a stock, which involves some risk, and for which the return cannot be predetermined).

Question 26

What is typical of fixed-income investments?

Answer 26

The inverse relationship between yield and price.

Question 27

How can we describe the inverse relationship between yield and price on a fixed-income investment?

Answer 27

The holder of a fixed income investment will see the market value of the investment decrease if the yield rate to maturity demanded by a buyer increases.

Question 28

How can the inverse relationship between yield and price on a fixed-income investment be explained?

Answer 28

By noting that a higher yield rate requires a smaller investment amount to achieve the same dollar level of interest payments.

Question 29

What is clear from the equations for simple interest vs. compound interest?

Answer 29

That accumulation under simple interest forms a linear function whereas compound interest accumulation forms an exponential function.

Question 30

Seeing that accumulation under simple interest forms a linear function whereas compound interest accumulation forms an exponential function, what should we expect? (try visualizing the graph (think inverse Solow model))

Answer 30

That simple interest accumulation is larger than compound interest accumulation for values of t between 0 and 1, but compound interest accumulation is greater than simple interest accumulation for values of t greater than 1.

Question 31

How is interest accumulation often based on a combination of both simple and compound interest?

Answer 31

Compound interest would be applied over the completed (integer) number of interest compounding periods, and simple interest would be applied from then to the fractional point in the current interest period.

Question 32

Express accumulated interest at an annual rate of 9% over a period of 4 years and 5 months using a combination of simple and compound interest.

Answer 32

(1.09)^4⋅[1 + 0.9(5/12))

Question 33

Regarding the accumulation of a single invested amount, how can we calculate the interest rate between two periods?

Answer 33

in+1 = (A(n+1)-A(n)) / (A(n)

Question 34

What does the relationship in+1 = (A(n+1)-A(n)) / (A(n) say?

Answer 34

That the effective annual rate of interest for a particular one-year period is the amount of interest for the year as a proportion of the value of the investment at the start of the year, or equivalently, the rate of investment growth per dollar invested.

Question 35

What’s another way of explaining the relationship in+1 = (A(n+1)-A(n)) / (A(n)?

Answer 35

Effective annual rate of interest for a specified one-year period = (amount of interest earned for the one-year period) / (value (or amount invested) at the start of the year)

Question 36

What is the one-period present value factor?

Answer 36

If the rate of interest for a period is i, the present value of an amount of 1 due one period from now is 1/(1+i).

Question 37

How is the factor 1/(1+i) denoted in actuarial notation and what is it called?

Answer 37

The factor 1/(1+i) is denoted v in actuarial notation and is called a present value factor or discount factor.

Question 38

How do we modify our present value factor in a situation involving more than one interest rate?

Answer 38

In a situation involving more than one interest rate, the symbol vi may be used to identify the interest rate i on which the present value factor is based.

Question 39

How is the present value factor particularly important in the context of compound interest?

Answer 39

Accumulation under compound interest has the form
A(t) = A(0)(1+i)^t. This expression can be rewritten as
A(0) = A(t)/(1+i)^t = A(t)(1+i)^(-1) = A(t)v^t
Thus, Kv^t is the present value at time 0 of an amount K due at time t when investment growth occurs according to compound interest.

Question 40

What does it mean that Kv^t is the present value at time 0 of an amount K due at time t when investment growth occurs according to compound interest?

Answer 40

This means that Kv^t is the amount that must be invested at time 0 to grow to K at time t, and the present value factor v acts as a “compound present value” factor in determining the present value.

Question 41

What can we learn about the relationship between accumulation and present value?

Answer 41

Accumulation and present value are inverse processes of one another.

Question 42

What is the present value of 1 due in one period as a function of i?

Answer 42

v = 1/(1+i)

Question 43

What is the present value of 1 due in t periods as a function of t?

Answer 43

v^t = 1 / (1+i)^t)

Question 44

What does the graph of v^t = 1 / (1+i)^t), the present value of 1 due in t periods as a function of t, illustrate?

Answer 44

That as the time horizon t increases, the present value of 1 due at time t decreases (if the interest rate is positive)

Question 45

What does the graph of the present value of 1 due in one period as a function of i, v = 1/(1+i), illustrate?

Answer 45

The classical “inverse yield-price relationship,” which states that at a higher rate of interest, a smaller amount invested is needed to reach a target accumulated value.

Question 46

What is a zero-coupon bond, sometimes called a stripped bond?

Answer 46

A bond with no coupons, only a payment on the maturity date)

Question 47

If simple interest is being used for investment accumulation, what is accumulated interest equal to?

Answer 47

A(t) = A(0)(1+it)

Question 48

If simple interest is being used for investment accumulation, what is the present value at time 0 of amount A(t) due at time t?

Answer 48

A(0) = A(t)/(1+it)

Question 49

What is important to note implicitly in the simple interest equations A(t) = A(0)(1+it) and A(0) = A(t)/(1+it)?

Answer 49

That simple interest accrual begins at the time specified as t=0.
The present value based on simple interest accumulation assumes that interest begins accruing at the time the present value is being found. There is no standard symbol representing present value under simple interest that corresponds to v under compound interest.

Question 50

What is a Treasury Bill or a T-Bill?

Answer 50

A debt obligation that requires the issuer to pay the owner a specified sum (the face amount or amount) on a specified date (the maturity date).

Question 51

How are Canadian T-Bills usually issued?

Answer 51

Canadian T-Bills are issued to mature in number of days that is a multiple of 7.

Question 52

When are Canadian T-Bills usually issued?

Answer 52

Canadian T-Bills are generally issued on a Thursday, and mature on a Thursday, mostly for period of (approximately) 3 months, 6 months, or 1 year.

Question 53

How are Canadian T-Bills valuated?

Answer 53

Valuation of Canadian T-Bills is algebraically identical to valuation of promissory notes described earlier.

Question 54

Given an accumulated amount function A(t), the investment grows from amount A(t1) at time t1 to amount A(t2) at time t2>t1. What does this imply?

Answer 54

An amount of A(t1)/A(t2) invested at time t1 will grow to amount 1 at time t2. In other words, A(t1)/A(t2) is a generalized present value factor from time t2 back to time t1.

Question 55

How are financial transactions usually represented algebraically?

Answer 55

When a financial transaction is represented algebraically it is usually formulated by means of one or more equations that represent the values of the various components of the transaction and their relationships.

Question 56

Along with the interest rate, what are the other components of the financial transaction represented algebraically?

Answer 56

Along with the interest rate, the other components of the transaction are the amounts disbursed and the amounts received.

Question 57

What are the amounts disbursed and the amounts received called?

Answer 57

These amounts are called dated cash flows.

Question 58

What would be a mathematical representation of the algebraic financial transaction?

Answer 58

A mathematical representation of the transaction will be an equation that balances the dated cash outflows and inflows, according to the particulars of the transaction.

Question 59

What must the equation balancing the dated cash outflows and inflows, according to the particulars of the transaction take into account?

Answer 59

The “time values” of these payments, the accumulated and present values of the payments made at the various time points.

Question 60

What is the equation balancing the dated cash outflows and inflows, according to the particulars of the transaction called?

Answer 60

Such a balancing equation is called an equation of value for the transaction, and its formulation is a central element in the process of analyzing a financial transaction.

Question 61

In order to formulate an equation of value for a transaction, it is first necessary to choose a reference time point or valuation date. At the reference time point the equation of value balances, or equates, which following two factors?

Answer 61

1. The accumulated value of all payments already disbursed plus the present value of a payments yet to be disbursed, and
2. The accumulated value of all payments already received plus the present value of all payments yet to be received.

Question 62

We see from Example 1.7 in the textbook that an equation of value for a transaction involving compound interest may be formulated at more than one reference time point with the same ultimate solution. Notice that Equation C: can be obtained from Equation A: by multiplying Equation A by (1+j)^7. What does this correspond to?

Answer 62

A change in the reference point upon which the equations are based, Equation A being based on t=0 and Equation C being based on t=7.

Question 63

In general, when a transaction involves only compound interest, how can an equation of value formulated at time t1 be translated into an equation of value formulated at time t2?

Answer 63

Simply by multiplying the first equation by (1+i)^(t2-t1).

Question 64

In Example 1.7, when t=7 was chosen as the reference point, the solution was slightly simpler than that required for the equation of value at t=0, in that no division was necessary. What’s the lesson?

Answer 64

For most transactions there will often be one reference time point that allows a more efficient solution of the equation of value than any other reference time point.

Question 65

What should you remember about quoted annual rates of interest?

Answer 65

Quoted annual rates of interest frequently do not refer to the effective annual rate.

Question 66

Quoted annual rates of interest frequently do not refer to the effective annual rate. Explain.

Answer 66

Say for credit cards, the interpretation of the phrase “payable monthly” is that the quoted annual interest rate is to be divided by 12 to get the one-month interest rate. The balance on statement 13 will have compounded for 12 months. The quoted rate of 24% is a nominal annual interest rate of interest equivalent to an effective annual rate of 26.82%

Question 67

What is Definition 1.7 - Nominal Annual Rate of Interest?

Answer 67

A nominal annual rate of interest compounded or convertible m times per year refers to an interest compounding period of 1/m years.
interest rate for 1/m period = (quoted nominal annual interest rate) / (m)

Question 68

When are nominal rates of interest frequently quoted in practice?

Answer 68

In situations in which interest is credited or compounded more often than once per year.

Question 69

In order to apply quoted nominal annual rate, what is it necessary to know?

Answer 69

The number of interest conversion periods in a year.

Question 70

What is the typical way to verify equivalence of rates?

Answer 70

To convert one rate to the compounding period of the other rate, using compound interest.

Question 71

Once the nominal annual rate and the compounding interest period are known, the corresponding compound interest rate for the interest conversion period can be found. What follows from this?

Answer 71

Then the accumulation function follows a compound interest pattern, with time usually measured in units of effective interest conversion periods.

Question 72

What is it necessary to do when comparing nominal annual interest rates with different coversion periods?

Answer 72

It is necessary to convert the rates to equivalent rates with a common effective interest period.

Question 73

There is standard actuarial notation for denoting nominal annual rates of interest, although this notation is not generally seen outside of actuarial practice. Describe the actuarial notation for the symbol: i.

Answer 73

In actuarial notation, the symbol i is generally reserved for an effective annual rate, and the symbol i^(m) is reserved for a nominal annual rate with interest compounded m times per year. Note that the superscript is for identification purposes and is not an exponent. The notation i^(m) is taken to mean that interest will have a compounding period of 1/m years and compound rate per period of (1/m)i^(m) = i^(m)/m.

Question 74

How would an annual interest rate of 24% 
1. Compounded monthly
2. Convertible monthly
3. Convertible 12 times
be denoted in actuarial  notation?

Answer 74

Here, m=12, so the nominal annual rate would be denoted as i^(12) = .24. The information indicated by the superscript “(12)” in this notation is that there are 12 interest conversion periods per year, and that the effective rate of 2% per month is 1/12 of the quoted rate of 24%

Question 75

What is the general relationship linking equivalent nominal annual interest rate i^(m) and effective annual interest rate i?

Answer 75

1 + i = [1+(i^(m))/m]^m

Question 76

Which 2 equations summarize other comparable relationships linking i and i^(m)?

Answer 76

i = [1+i^(m)/m]^m - 1
and
i^(m) = m[(1+i)^(1/m) -1 ]
Note that (1+i)^(1/m) is the 1/m -year growth factor, and (1+i)^(1/m) -1 is the equivalent effective 1/m -year compound interest rate.

Question 77

What should be clear from general reasoning about nominal annual rates of interest?

Answer 77

The more often compounding takes place during the year, the larger the year-end accumulated value will be, so the larger the equivalent effective annual rate will be as well.

Question 78

What is the equation for continuous compounding?

Answer 78

lim_m->infinity [(1+i)^(1/m) -1] = ln(1+i)

This is supposedly a consequence of l’Hospital’s Rule.

Question 79

What is the equation for continuous compounding?

Answer 79

lim_m->∞[(1+i)^(1/m) -1] = ln(1+i)

This is supposedly a consequence of l’Hospital’s Rule.

Question 80

What can be said about the relationship between the equivalent nominal annual rate and the frequency of compounding periods?

Answer 80

The more frequently compounding takes place (i.e., as m increases), the smaller is the equivalent nominal annual rate which bottoms out at continuously compounding interest rates.

Question 81

What should be remembered about the way that nominal interest rates are quoted?

Answer 81

A nominal rate, although quoted on an annual basis, might refer to only the immediately following fraction of a year. Thus when interest is quoted on a nominal annual basis, the actual rate may change during the course of the year, from one interest period to the next.

Question 82

In previous sections of this chapter, interest amounts have been regarded as paid or charged at the end of an interest compounding period, and the corresponding interest rate is the ratio of the amount of interest paid at the end of the period to the amount of principal at the start of the period. How are interest rates and amounts viewed int his way sometimes referred to as?

Answer 82

As interest payable in arrears (payable at the end of an interest period). This is the standard way in which interest rates are quoted, and it is the standard way by which interest amounts are calculated. In many situations it is the method required by law.

Question 83

Most often, interest is payable in arrears. Occasionally a transaction calls for interest payable in advance. How are these quoted?

Answer 83

In this case the quoted interest rate is applied to obtain an amount of interest which is payable at the start of the interest period.

Question 84

Let’s think about interest payable in advance. If smith borrows 1000 for one year at a quoted rate of 10% with interest payable in advance, the 10% is applied tot he loan amount of 1000, resulting in an amount of interest of 100 for the year. The interest is paid at the time the loan is made. Smith receives the loan amount of 1000 and must immediately pay the lender 100, the amount of interest on the loan. One year later he must repay the loan amount of 1000. What is the net effect?

Answer 84

The net effect is that Smith receives 900 and repays 1000 one year later. The effective annual rate of interest on this transaction is 100/900 = 11.11%. This 10% payable in advance is called the rate of discount for the transaction.

Question 85

What is the rate of discount?

Answer 85

The rate used to calculate the amount by which the year end value is reduced to determine the present value.

Question 86

How is the effective annual rate of discount another way of describing investment growth in a financial transaction?

Answer 86

In the example just considered we see that an effective annual interest rate of 11.11% is equivalent to an effective annual discount rate of 10%, since both describe the same transaction.

Question 87

What is Definition 1.8 - Effective Annual Rate of Discount?

Answer 87

In terms of an accumulated amount function A(t), the general definition of the effective annual rate of discount from time t=0 to time t=1 is
d = [A(1) - A(0)] / A(1)

Question 88

How is the definition that

d = [A(1) - A(0)] / A(1) in contrast with the definition for the effective annual rate of interest?

Answer 88

The effective annual rate of interest has the same numerator but has a denominator of A(0).

Question 89

What is another way of interpreting how the definition that

d = [A(1) - A(0)] / A(1) is in contrast with the definition for the effective annual rate of interest?

Answer 89

Effective annual interest measures growth on the basis of the initially invested amount, whereas effective annual discount measures growth on the basis of the year-end accumulated amount.

Question 90

Should we use the effective annual rate of interest or the effective annual rate o discount in our analysis of financial transactions?

Answer 90

Either measure can be used in the analysis of a financial transaction.

Question 91

What does the rewriting of the equation d = [A(1) - A(0)] / A(1) into
A(0) = A(1)*(1-d) show us?

Answer 91

That 1-d acts as a present value factor. The value at the start of the year is the principal amount of A(1) minus the interest payable in advance, which is d*A(1).

Question 92

On the other hand, on the basis of effective annual interest we have
A(0) = A(1)*v. What does this imply?

Answer 92

We see that for d and i to be equivalent rates, present values under both representations must be the same, so we must have
1/(1+i) = v = 1-d
or equivalently,
d = i/(1+i), or i = d/(1-d)

Question 93

What are equivalent rates of interest and discount i and d?

Answer 93

Equivalent rates of interest and discount i and d are

d = i/(1+i) and i = d/(1-d)

Question 94

How is is the relationship between equivalent interest and discount rates for period of other than a year similar?

Answer 94

Suppose that j is the effective rate of interest for a period of other than one year. Then dj = j/(1+j) where dj is the equivalent effective rate of discount for that period.

Question 95

How can the present value of 1 due in n years can be represented using discount rates?

Answer 95

v^n = (1-d)^n, so that present values can be represented in the form of compound discount.

Question 96

The present value of 1 due in n years can be represented using discount rates represented in the form v^n = (1-d)^n, so that present values can be represented in the form of compound discount. What does this mean in relation to describing the behavior of an investment?

Answer 96

This underlines the fact that the concepts of discount rate and compound discount form an alternative to the concept of interest rate and compound interest in describing the behavior of an investment.

Question 97

From a practical point of view, what will A(0) in d = [A(1) - A(0)] / A(1) not be less than?

Answer 97

0

Question 98

If A(1)>A(0), an effective rate of discount can be no larger than 1 (100%). What should we note about an effective discount rate of d=1?

Answer 98

That an effective discount rate of d = 1 (100%) implies a present value factor of 1-d = 1-1 = 0 at the start of the period (an investment of 0 growing to a value of 1 at the end of a year would be a very profitable arrangement).

Question 99

What do we see in the equivalence between i and d?

Answer 99

That lim as i–>∞ of d = 1, so that very large effective interest rates correspond to equivalent effective discount rates near 100%.

Question 100

What is Definition 1.10 - Nominal Annual Rate of Discount?

Answer 100

A nominal annual rate of discount compounded m times per year refers to a discount compounding period of 1/m years,
[discount rate for 1/m period] =
= (quoted nominal discount rate) / m

Question 101

What are the actuarial symbols for discount rates?

Answer 101

In actuarial notation, the symbol d is generally used to denote an effective annual discount rate, and the symbol d^(m) is reserved for denoting a nominal annual discount rate with discount compounded (or convertible) m times per year.

Question 102

What is the notation d^(m) is taken to mean?

Answer 102

The notation d^(m) is taken to mean that discount will have a compounding period of 1/m years and compound rate per period of
(1/m)*d^(m) = d^(m)/m

Question 103

How does the relationship between equivalent nominal and effective annual discount rates parallel the relationship between nominal and effective annual interest rates?

Answer 103

The 1/m -year present value factor would be

1 - [d^(m)/m].

Question 104

The relationship between equivalent nominal and effective annual discount rates parallels the relationship between nominal and effective annual interest rates. Describe the notation d^(4) = .08 to illustrate what is meant by this.

Answer 104

For instance, the notation d^(4) = .08 refers to a 3-month discount rate of .08/4 = .02, and a 3-month present value factor of 1-.02 = .98. This would be compounded 4 times during the year to an effective annual present value factor of (.98)^4 = .9224. This annual present value factor could then be described as being equivalent to an effective annual discount rate of 7.76%.

Question 105

With d^(m) in effect, there would generally be m compounding periods during the year, so what would the equivalent effective annual present value factor be?

Answer 105

(1 - d^(m)/m)^m

Question 106

With d^(m) in effect, there would generally be m compounding periods during the year, so the equivalent effective annual present value factor would be (1 - d^(m)/m)^m. If d is the equivalent effective annual rate of discount, then which relationship do we have?

Answer 106

1-d = (1 - d^(m)/m)^m

Question 107

How can we express d given the relationship

1-d = (1 - d^(m)/m)^m?

Answer 107

d = 1 - (1 - d^(m)/m)^m

Question 108

How can we express d^(m) given the relationship

1-d = (1 - d^(m)/m)^m?

Answer 108

d^(m) = m*[1 - (1-d)^(1/m)]?

Question 109

What is d^(m) as m goes to infinity?

Answer 109

lim as m–> infinity of m*[1 - (1-d)^(1/m)] =

= -ln(1-d)

Question 110

What happens to D^(m) as m increases to infinity?

Answer 110

As m increases, d^(m) increases with upper limit d^(infinity). Thus if m>n then d^(m)>d^(n) for equivalent rates. This is the opposite of what happens for equivalent nominal interest rates.

Question 111

As m increases, d^(m) increases with upper limit d^(infinity). Thus if m>n then d^(m)>d^(n) for equivalent rates. This is the opposite of what happens for equivalent nominal interest rates. How can we justify this?

Answer 111

This can be explained by noting that interest compounds on amounts increasing in size whereas discount compounds on amounts decreasing in size.

Question 112

What is one important relationship between equivalent rates i and d?

Answer 112

In general, for equivalent rates i and d it is always the case that d^(infinity) = i^(infinity), equal to the force of interest.

Question 113

Financial transactions occur at discrete time points. Many theoretical financial models are based on events that occur in a continuous time framework. The famous Black-Scholes option pricing model (which will briefly be reviewed in Chapter 9) was developed ont he basis of stock prices changing continuously as time goes on. In this section we describe a way to measure investment growth in a continuous time framework. How will we approach measuring continuous growth of an investment?

Answer 113

Continuous processes are usually modeled mathematically as limits of discrete time processes, where the discrete time intervals get smaller and smaller.

Question 114

What is d^(m) as m goes to infinity?

Answer 114

lim as m–> ∞ of m*[1 - (1-d)^(1/m)] =

= -ln(1-d)

Question 115

What happens to D^(m) as m increases to infinity?

Answer 115

As m increases, d^(m) increases with upper limit d^(∞). Thus if m>n then d^(m)>d^(n) for equivalent rates. This is the opposite of what happens for equivalent nominal interest rates.

Question 116

As m increases, d^(m) increases with upper limit d^(∞). Thus if m>n then d^(m)>d^(n) for equivalent rates. This is the opposite of what happens for equivalent nominal interest rates. How can we justify this?

Answer 116

This can be explained by noting that interest compounds on amounts increasing in size whereas discount compounds on amounts decreasing in size.

Question 117

What is one important relationship between equivalent rates i and d?

Answer 117

In general, for equivalent rates i and d it is always the case that d^(∞) = i^(∞), equal to the force of interest.

Question 118

What is i^(infinity) given that it is equal to A’(t)/A(t)?

Answer 118

i^(infinity) is a nominal annual interest rate compounded infinitely often or compounded continuously. i^(infinity) is also interpreted as the instantaneous rate of growth of the investment per dollar invested at time point t and is called the force of interest at time t.

Question 119

What should we note about A’(t) in the equation

A’(t)/A(t)?

Answer 119

Note that A’(t) represents the instantaneous growth of the invested amount at time point t (just as A(t+1) - A(t) is the amount of growth in the investment from t to t+1), whereas A’(t)/A(t) is the relative instantaneous rate of growth per unit amount invested at time t (just as [A(t+1) - A(t)] / A(t) is the relative rate of growth from t to t+1 per unit invested at time t).

Question 120

The force of interest may change as t changes. How is the actuarial force of interest at time t usually denoted?

Answer 120

The actuarial notation that is used for the force of interest at time t is usually

Question 121

Given m*[A(t+1/m) - A(t)] / A(t), what happen if we let m go to infinity?

Answer 121

i^(∞) =
= lim as m–>∞ of i^(m) =
= lim as m–>∞ of m*[A(t+1/m) - A(t)] / A(t)

Question 122

Given m[A(t+1/m) - A(t)] / A(t),
i^(∞) =
= lim as m–>∞ of i^(m) =
= lim as m–>∞ of m[A(t+1/m) - A(t)] / A(t)
How can this be reformulated to help our calculations?

Answer 122

This limit can be reformulated by making the following variable substitution. Define the variable h to be h=1/m, so that h–>0 as m–>(∞). The limit can then be written in the form
i^(∞) =
= (1/A(t))lim as h–>0 of [A(t+h) - A(t)]/h =
= (1/A(t))(dA(t)/dt) =
= A’(t)/A(t)

Question 123

What is i^(∞) given that it is equal to A’(t)/A(t)?

Answer 123

i^(∞) is a nominal annual interest rate compounded infinitely often or compounded continuously. i^(∞) is also interpreted as the instantaneous rate of growth of the investment per dollar invested at time point t and is called the force of interest at time t.

Question 124

The force of interest may change as t changes. How is the actuarial force of interest at time t usually denoted?

Answer 124

The actuarial notation that is used for the force of interest at time t is usually δt instead of i^(∞).

Question 125

What is required for the force of interest to be defined?

Answer 125

In order for the force of interest to be defined, the accumulated amount function A(t) must be differentiable (and thus continuous, because any differentiable function is continuous).

Question 126

Continuous investment growth models have been central to the analysis and development of financial models with important practical applications. Which ones most notably?

Answer 126

Most notably for models of investment derivative security valuation such as stock options.

Question 127

What is Definition 1.11 - Force of Interest?

Answer 127

For an investment that grows according to accumulated amount function A(t), the force of interest at time t, is defined to be
δt = A’(t)/A(t)

Question 128

What is the expression for δt is accumulation is based on simple interest at annual rate i?

Answer 128

A(t) = A(0)[1+it], so that
A’(t) = A(0)i. Then
δt = A’(t)/A(t) =
= i/(1+it)

Question 129

What is the expression for δt is accumulation is based on compound interest at annual rate i?

Answer 129

A(t) = A(0)*(1+i)^t, so that 
A'(t) = A(0)*(1+i)^t * ln(1+i). Then 
δt = A'(t)/A(t) Therefore
δt = ln(1+i)

Question 130

In the case of compounding interest growth, what happens to the force of interest?

Answer 130

The force of interest is constant as long as the effective annual interest rate is constant.

Question 131

In the case of simple interest, what happens to the force of interest?

Answer 131

In the case of simple interest, δt decreases as t increases.

Question 132

How can the force of interest be used to describe investment growth with
δt = A’(t)/A(t) = d/dt ln[A(t)]?

Answer 132

Integrating this equation from time t=0 to time t=m we get 
∫δtdt [from 0-n] =
= ∫d/dt * ln[A(t)]dt =
= ln[A(t)] - ln[A(0)] =
= ln[A(n)/A(0)]

Question 133

How can we rewrite the expression

∫δtdt [from 0-n] = ln[A(n)/A(0)]?

Answer 133

exp[∫δtdt (from 0-n)] =
= A(n)/A(0)
or 
A(n) = A(0)*exp[∫δtdt]
or
A(0) = A(n)*exp[-∫δtdt ]

Question 134

What is the general form of the accumulation factor from time t=n1 to time t=n2 (where n1

Answer 134

e^[∫δtdt (from n1-n2)]

Question 135

What is the general form of the present value factor from time t=n1 to time t=n2 (where n1

Answer 135

e^[-∫δtdt (from n1-n2)]

Question 136

Given the equations
e^[∫δtdt (from n1-n2)],
what does the accumulation factor for that period simplify to in the case in which δt is constant with value δ from time n1 to time n2?

Answer 136

e^[(n2-n1)δ]

Question 137

Given the equations
e^[-∫δtdt (from n1-n2)],
what does the present value factor for that period simplify to in the case in which δt is constant with value δ from time n1 to time n2?

Answer 137

e^-(n2-n1)δ

Question 138

For simple interest accumulation at annual interest rate i, with accumulation function
A(t) = A(0)*(1+it), the force of interest is δt = i/(1+it). For an investment of 1 made at time n1, what is the growth factor for the investment to time n2?

Answer 138

exp(∫i/(1+it) dt from n1 to n2 =
= exp[ln(1+n2i) - ln/91+n1i)] =
= (1+n2i)/(1+n1i)

Question 139

In practice however, when simply interest is being applied, it is assumed that simple interest accrual for an investment begins at the time that the investment is made, so that n investment made at time n1 will grow by a factor of 1+(n2-n1)*i to time n2, which is not the same as the growth factor found using the force of interest δt = i/(1+it). Why is that?

Answer 139

This force of interest has a starting time of 0, and later deposits must accumulate based on the force of interest at the later time points, whereas in practice, each time a deposit or investment is made, the clock is reset at time 0 for that deposit, and simple interest begins anew for that deposit.

Question 140

Another identity involving the force of interest is based on the relationship
d/dt * A(t) = A(t)δt.
A(t)δt is the instantaneous amount of interest earned by the investment at time t. What does integrating both from time 0 to time n result in?

Answer 140

∫A(t)*δt dt [from 0-n] =
= ∫(d/dt * A(t) dt =
= A(n) - A(0)
This is the amount of interest earned from time 0 to time n.

Question 141

Given δt = .08 + .005t, the accumulated value over five years of an investment of 1000 made at time 0 is 1588.04 and an equivalent investment made at time 2 is 1669.46. Why are these two 5 year investments different?

Answer 141

The accumulations are different as a result of the non-constant force of interest.

Question 142

It was previously shown that if the effective annual interest rate i is constant then δt = ln(1+i). Let’s now suppose the force of interest δt is constant with value δ from time 0 to time n. Then A(n) =
= A(0)e^(∫δt dt [from 0 to n]) =
= A(0)e^(nδ) =
= A(0)(e^δ)^n.
This form of accumulation is algebraically identical to compound interest accumulation of the form A(n) = A(0)(1+i)^n, where e^δ = 1+i, or equivalently, where δ = ln(1+i). A constant effective annual interest rate i is equivalent to constant force of interest δ according to which relationship?

Answer 142

1+i = e^δ,
or equivalently,
δ = ln(1+i)

Question 143

The explicit use of the force of interest does not often arise in a practical setting. For transactions of very short duration (a few days or only one day), a nominal annual interest rate convertible daily, i^(365), might be used. This rate is approximately equal to the equivalent force of interest. What’s an example of when this would be used?

Answer 143

Major financial institutions routinely borrow and lend money among themselves overnight, ir order to cover their transactions during the day. The interest rate used to settle these one day loans is called the overnight rate. The interest rate quotes will be a nominal annual rate of interest compounded every day (m=365).

Question 144

Why are investors concerned with the level of inflation?

Answer 144

It is clear that a high rate of inflation has the effect of rapidly reducing the value (purchasing power) of currency as time goes on.

Question 145

Why is it not surprising that periods of high inflation are usually accompanied by high interest rates?

Answer 145

Because the rate of interest must be high enough to provide a “real” return on investment.

Question 146

What is the real rate of interest?

Answer 146

The real rate of interest refers to the inflation-adjusted return on an investment.

Question 147

What is Definition 1.12 - Real Rates of Interest?

Answer 147

With annual interest rate i and annual inflation rate r, the real rate of interest for the year is
i_real =
= (value of amount of real return (yr-end dollars) / (value of invested amount (yr-end dollars) =
= (i-r)/(1+r)

Question 148

What is the simple and commonly used measure of the real rate of interest?

Answer 148

i-r, where i is the annual rate of interest and r is the annual rate of inflation.

Question 149

As a precise measure of the real growth of an investment, or real growth in purchasing power, i-r is not theoretically correct. Why not? Why is it (i-r)/(i+r) and not just i-r?

Answer 149

The interest earned on an investment (which was paid in beginning-of-year dollars) is paid out in end-of-year dollars, so that to regard the interest paid as a percentage of the amount invested, we must measure the real return of that amount and the amount invested in equivalent dollars to take into accurately measure the growth in purchasing power.

Question 150

Explain further why it is (i-r)/(i+r) and not just i-r that we should use.

Answer 150

In general, with annual interest rate i and annual inflation rate r, an investment of 1 at the start of a year will grow to 1+i at year end. Of this 1+i, an amount of 1+r is needed to maintain dollar value against inflation, i.e., to maintain the purchasing power of the original investment of 1. The remainder of (1+i)-(1+r) = i-r is the “real” amount of growth in the investment, and this real return is paid at year end. The investment of 1 at the start of the year has an inflation-adjusted value of 1+r at year end in end-of-year dollars.

Question 151

What should we notice about the relationship

i_real = (i-r)/(1+r)?

Answer 151

That the lower the inflation rate r, the closer 1+r is to 1, and do the closer i-r is to (i-r)/(1+r). On the other hand, if inflation is high then the denominator 1+r becomes an important factor in (i-r)/(1+r).

Question 152

What is one more point to note about quoted inflation rates?

Answer 152

That inflation rates are generally quoted as the rate that has been experienced in the year just completed, whereas interest rates are usually quoted as those to be earned in the coming year.

Question 153

That inflation rates are generally quoted as the rate that has been experienced in the year just completed, whereas interest rates are usually quoted as those to be earned in the coming year. What does this mean for any meaningful analysis?

Answer 153

That in order to make a meaningful comparison of interest and inflation, both rates should refer to the same one-year period. Thus, it may be more appropriate to use a projected rate of inflation for the coming year when inflation is considered in conjunction with the interest rate for the coming year.