171 W1 Theory of Interest PDF

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PSTAT 171 Theory of Interest

Ian Duncan, PhD FSA FIA FCIA FCA CSPA MAAA [email protected]

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Interest Theory deals with the time value of money. Under normal circumstances (which have been lacking since 2008) capital is scarce. Like every other commodity capital has a price, which in normal times is established in a market. The price that we pay for the use of capital is called “interest,” that is the price you have to pay the user of capital to lend it to you. Of course, because you will have to pay back the capital in the future you will want to put it to good use, and to earn a return on capital or yield (which should be higher than the price you pay for borrowing, or its not worth borrowing). Interest may also be thought of as the rate that balances consumption between periods. Said another way it is the reward that you get for postponing your gratification. You could enjoy something today - for you to be willing to postpone that enjoyment you would have to be rewarded. If you lend money to someone, you want to earn the price of money. You expect to be repaid the amount of the loan in the future, together with an amount that represents the price of the money, or the reward for postponing your enjoyment. So the interest charge on the loan is both the price of money AND the rate that you need to be paid to defer into the future your enjoyment of whatever it was that you were going to do with the money. From the perspective of the borrower, the money has a price (interest) and from the perspective of the lender the money has a price (reward, or interest). For example, you have all made the decision to attend university, a decision that has a cost to it (not just the residence and tuition that you pay, but the income that you forego). You expect to receive a greater reward in the future to make up for the deferral of your enjoyment today. We can actually analyze this decision when we have developed a few tools (it’s a bit complicated now but in a couple of classes it will seem easy). The following sketch should give you an idea:

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University

Now No University

The Future DEBT

Here is a simpler example: You could buy a Starbucks latte for $3.00 today; what would I have to pay you to defer your enjoyment of the Starbucks latte to tomorrow? You are unlikely to be willing to do it for $0.00. But maybe if I paid you another $1.00 you would consider it. If so, the implied interest rate is 33% daily ($4.00/$3.00 -rather a high rate, but its an example). So we have seen that interest is a price; both the price that you would have to pay to borrow money, and the price that you demand in order to defer your consumption into the future. Because we are looking at decisions made over time this price is also known as the time value of money. There are two types of interest: a. Simple Interest. If you deposit 100 in a bank at 6% interest, your capital will earn $6.00 at year-end, and you will have $106. Whether or not you choose to leave your money on deposit, another $6.00 will accrue by yearend 2, so now you will have $112.00 (assuming you haven’t withdrawn anything). b. Compound Interest is more common. By leaving your $6.00 on deposit at year-end 1, you will accrue “interest on the interest” by year-end 2. In 3

addition to the $6.00 that you earn in year 2 from your capital (also called principal), you will earn 0.06 * $6.00 or $0.36. So in the compound case you will have $1.1236, vs. $1.12 in the simple interest case. TIP #1: Something you will find useful in this course, and in actuarial work1 in general (which deals with values at different time periods) is the time-line. In the simple examples above, we could draw a time-line and visualize how interest is accruing: Simple $100

$106.00

$112.00

|_______________________________|________________________________| 0

1

2

Compound $100

$106.00

$112.36

|_______________________________|________________________________| 0

1

2

Of course, in the one-year case with simple interest there is no difference between the amount on deposit at the end of the year: the effective rate in both cases is 6%. The difference only arises in year 2, or (as we shall see later) if the interest is compounded during the year.

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I know that not all students taking this class are actuarial students; however, this material is required for actuarial examinations and the examples tend to be actuarial in nature. Of course actuarial work covers a broad spectrum, including Financial Mathematics so if you are an FMS major this material will be relevant. For any other majors finance and the time-value of money is everywhere so you will learn a lot of useful techniques in this course.

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In terms of formulae, simple interest accumulates to: 100 1  0.06  n  at the end of n years while the same amount at 6%

compounded annually accumulates to: 100 1.06  after n years. n

Similarly, we can reverse the equation and ask: how much would I have to invest now to have $100 at some point in the future?? In the one-year case, the answer is easy, and applies whether we are earning simple or compound interest. Let the amount be P (principal). We know that 100 amounts to 100 1.06  at the end of 1 year whether interest is simple or compound. So P amounts to P 1.06 and because we want to have 100 at the end of 1 year, P has to equal 100 /1.06 or 100(

1 ). 1.06

In this case, P = 94.34. You can check the math by multiplying 94.34x1.06 = 100. 1  Actuaries have a name for the term   by the way. We call it v . If you plan 1.06 

to be an actuary, you will get used to this particular symbol; if you are not an actuarial student, you will need to be familiar with v for the rest of this course (it 1  is also useful short-hand; it saves you writing out  .  1 i 

We will use v throughout this course to represent the present value today of an amount of 1 due one year hence. We will learn more actuarial symbolism as the course progresses.

What if we wanted to have 100 in 2 years’ time? Again, let the principal amount be P. If at simple interest, we know that we earn simple interest at P(0.06) annually and after 2 years I have interest of P (0.06)(2) or 0.12P. In total, I have 1.12P and this equals 100, so 100  P (1.12) => P  100 /1.12 or 89.29. 5

You can check this: simple interest in year 1 = (0.06)(89.29) or $5.36. After 2 years interest is $10.72, which (together with the principal of $89.29) gives us $100.01. Tip #2: always check your answer for reasonableness (is it in the right ball-park?) and accuracy using a different method (i.e. not simply repeating the same calculation). Always have a high-level idea of what answer you would expect so that you can check your actual answer against your expectation. (For example, if you add two integers, 2+3, you would expect an integer answer – so 5.5 would not be correct.) Tip #3: its difficult to recommend a simple rule regarding decimal places. When you are dealing with dollars, 2 is probably enough in most cases. I prefer fewer rather than more. What is more important is showing your work – as long as the work is correct the precise answer is less important. It is usually better to show more steps in your work than fewer. If you plan to do the Society of Actuaries’ FM exam, this is a multiple choice exam (as are the tests for this course now, thanks to the CoVID virus) so there will be some assistance available – the answers will give you a clue as to the accuracy necessary (and also provide a guide as to the range to expect). What if we earn compound interest? Well, now we know that after 2 years the original principal will have accumulated to P 1.06 2 and is equal to 100, so P  100 /1.062 or $89.00 . We can check this independently, as above by doing an

interest calculation, but more important, we know that the total interest earned when compounded is greater than simple interest, so the principal required to be invested should be smaller, which it is: $89.00 vs. $89.29. The two numbers, however, should be close because as we saw previously, the difference between simple and compound interest on $100 for 2 years at 6% is only $0.36. If we had been paying attention to the first year example, we could also have written this expression in actuarial terms as:

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P  100 /1.06 2 or 100 v2 . We would have derived the same answer because  1 v2   2  1.06

  1     1.1236  0.889996 = $89.00.   

Some actuarial notation Actuarial notation is different and important, as you will see in this course and in PSTAT 172, if you progress. It is useful because it allows actuaries to communicate precisely and at the same time in “shorthand.” Those of you intending to become actuaries or do the SOA exams should consult the SOA reference in the introductory material for the notation required for exam FM. By the way, you will sometimes see v written as v0.06 , just to be extra precise about the value we are talking about (especially when more than one interest rate is involved). There is also no shorthand term for1 i . We just write 1 i ! If you go on to become an actuary you will use this throughout your career – hardly a day goes by when I don’t do a calculation with 1 i or v !

Example 1 a. What is the present value of $20,000 payable in 15 years at 5% per annum compound? b. What will $5,000 amount to in 6 years at 5% annually compound? Write this problem in actuarial notation. c. How much will I have to invest to accumulate to $5,000 in 10 years time at 5% compound interest? Write this problem in actuarial notation. Answer: a. 9,620.34 b. 6,700.48. 5000(1+i) 6 0.05 c. 5000 v 10 0.05 = 3,069.57 7

Define: a( t ) the amount to which an initial investment of 1 grows by time t ≥ 0; and A (t ) the amount to which an initial investment of A (0) grows by time t ≥ 0.

Then for compound interest: a(t )  (1 i )t and A(t ) = A(0)(1  i )t

And for simple interest: a (t ) = (1  it ) and A(t ) = A(0)(1  it )

Effective Rate of Interest We can find the effective rate of interest over any time period. For the period t , t  1 the beginning amount is A (t ) and the amount at the end of the period is A (t  1) . Interest earned over the period is A(t 1) - A(t ) . The effective rate of

interest over this period is it 1 =

A (t  1)  A (t ) a (t  1)  a (t ) amount earned = = beginning amount A(t ) a (t )

Example 2 We assume that we earn 6% over the time interval [1,2]. Principal = 1. Note that this is year 2, so we begin the period with accumulated principal of 1.06. What is the effective rate of interest over the two years, assuming a. compound interest at 6% and b. simple interest at 6%?

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Solution a. Compound:

i2 =

a(2)  a(1) 0.0636 = = 0.06 1.06 a (1)

No surprise here: if the effective rate is 6% compounded annually, then measuring the effective annual rate earned over two years has to be 6%!

b. Simple:

i2 =

a (2)  a (1) 0.06 = = 0.0566 a (1) 1.06

Notice that the effective rate of interest in the case of compounded interest is the rate itself; in the simple interest case it is the equivalent compounded rate.

Example 3 a. A savings account offers simple interest of 10%. How long does it take to double your principal? b. Repeat the same calculation but assume compound interest. Solution a. Assuming simple interest, after n year the accumulated amount is a(n) = a (0)(1  ni ) and a (n ) / a (0) = 2.0

So (1  n (0.1)) = 2.0 and (0.1)n = 1.0 so n = 10. Again, no surprise here. In the general case, the number of years (n) n  b. Assuming compound interest a(n) = a (0)(1  i) n and a( n) / a(0) = 2.0 (1+ 0.1)n = 2.0 => n (ln 1.1) = ln 2 => n = ln 2.0/ln 1.1 And n = 7.2725. In the general case, (1+ i) n = 2.0 => n [ln (1  i )]= ln(2) 0.693  n=

ln (1 i ) 9

1 . i

The power of compound interest: note how much more quickly you double your money with compound interest!

Nominal and Effective Rates of Interest In cases where payments are made for periods less than a year, or when compounding is performed more frequently than annually (e.g. monthly or quarterly) rates are quoted as nominal annual rates and then converted to a periodic rate. For example, if you earn 10% annually, compounded quarterly, you will earn 2.5% for the first quarter, then 2.5% multiplied by the 1.025 in the second quarter, and 2.5% multiplied by (1.025) 2 in the third quarter, etc. The following time-line may help your understanding – important that you grasp when the interest is credited and then goes into the calculation for the next quarter. Compound

$1

$1.025 $1.025

($1.025) 2 $1.05062

($1.025) 3 $1.076891

($1.025) 4 $1.103813

|_______________|________________|________________|________________| 0

1/4

2/4

3/4

4/4

In this case, the nominal rate is 10% (0.10). The periodic rate (compounded quarterly) is 0.025 or 2.5%. The annual effective rate is (1+0.10/4) 4 = 1.103813. The effective annual rate is the equivalent one-year rate that would give you the same year-end accumulated amount as the nominal rate, compounded quarterly. Thus the yearly effective rate is 10.3813%. This is an example of a 10% nominal rate, compounded quarterly. A principal of 1 will accumulate to 1.103813 by the end of the year; so in terms of the definition, 10

a (t  1)  a(t ) 1.103813 1.00 = = 10.3813% a( t) 1.00

As we see: the effective rate is higher than the nominal rate when interest is compounded more frequently than annually. Effective rates are calculated as compound rates; so if simple interest is credited, the effective rate will be less than the nominal rate, as the next example shows. Example 4 Assume you earn a monthly rate of 1% compound. a. What is the annual nominal rate? b. What is the annual effective rate?

Solution a. Annual nominal rate 12%. b. Annual effective rate = (1.01)12 – 1 = 12.6825%.

You will encounter effective rates in bank and credit card advertisements. Here is a current example (from Citibank):

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In this case 14.74% (or 24.74%) are Annual Percentage Rates (APR) or effective annual rates. We now know that we can calculate the nominal annual rate that is equivalent to these APRs: 14.74% is the effective annual rate. For credit cards, the nominal rate is compounded monthly (accrued interest is calculated each month on your outstanding balance). So 1.1474 = (1+ i (12)/12) 12.

New notation: we denote a nominal interest rate compounded m times per year as i(m) . (1+ i(12) /12) = 1.14741/12 = 1.011524 i(12) /12 = 0.011524  i(12) = 13.8289%. So the annual nominal rate equivalent to the effective APR 14.74% is 13.829%. Moral of the story: not a good idea to accumulate credit card balances.

In the general case of m payments per year, the nominal rate is i(m) and the periodic rate is i (m)/m. The effective rate is: (1 

i(m ) m ) 1 m

 1 i = (1+

i( m) m ) => m[(1 i)1/m -1] = i (m ) m

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Example 5 a. Assume i(12) = 6%. What is the effective rate i? b. If the effective rate i is 5%, what is i(12) ? Answer:

a. 6.168%

b. 4.889%

Formulae:

a. effective rate i = (1.005)12 -1

b. = ((1.05) 1/12 -1 ) *12 => i(12) = 0.0489

Note: it is important to be comfortable with the concept of equivalent rates and rates compounded more frequently than once per year because most rates are. This is an important and tricky part of this course - make sure you understand the examples and the homework.

Discount Rates We saw earlier that we would have to invest an amount < $100 at interest of 6% to accrue $100 at the end of the period. Assuming a one-year period, we would need to invest $94.34, which would attract interest of $5.66 to add to $100. Terminology: The rate that is applied to $100 to determine the present value at time zero is called the discount rate. The discount factor (we have met before) is v; the discount rate, however, we define below: Important Definitions (memorize this) We define the discount, or present value factor as v. v = 1/1+i (as we saw above). Then the rate of discount or d = 1 - 1/1+i = 1-v = i/(1+i) = i v (these are all equivalent; just different ways of calculating d; make sure you know them all.)

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Discount rates can be confusing – accountants, for example, refer to the “discount rate” as the rate (that we now know is the interest rate) when they are talking about the discounting process, which leads to confusion between i (the annual interest rate used to calculate the discount) and d, the discount rate. In terms of the example above, the present value amount that we need to hold at 6% to accumulate to $1 is 100 (1/1.06) = 94.34. The interest rate in this calculation is 6%; the present value factor (v) = 0.9434 and the discount rate is d = 1-0.9434 or 0.0566. Multiplying by 100, the amount of the discount in this case is $5.66, which is the amount we can discount the $100 by to determine the principal required to accumulate to $100 by the end of the period. Note the difference between the interest and discount rates, and the fact that the present value and discount factors are the same. Some students are confused by the difference between interest and discount rates. The following real-world example may help to understand the difference: If you were running a store, you would mark up your products at some rate (to cover your costs and profit). Let’s call that rate “i” for now. Assume that the product costs $100; you mark it up at i=25%. So the marked up or retail price is now $125. To find the original cost of the item, however, we do not discount the retail price at 25% because that would imply the purchase price is (125*0.75 = 93.75) The original price divided by the selling price is 100/125, or 80%. 80% of course is = 100 - 20, hence the discount rate is 20%. So the discount rate equivalent to the mark-up (or interest rate) of 25% is 20%, which also happens to be 20% 

i 25% where i = 0.25 and 1  i = 1.25 which is our definition of d!  125% 1  i

As with the interest/discount case, the discount rate is lower than the corresponding interest rate (because it applies to a loaded price).

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Example 6 If the annual rate of interest is 10%, what is d? Solution Present value factor (v) = 1/1.10 = 0.9091 Then d = 1 – 0.9091 = 0.0909.

Nominal and effective rates of discount This works the same way as the nominal rate of interest, with one important difference. Define d(m) as the nominal rate of discount, compounded m-thly. Then 1 – d (= v) = (1 -

d (m ) m ) m

The key difference to remember is that to calculate the effective rate of interest, we add 1 to the nominal periodic rate (i(m)/m). To derive the effective discount rate we subtract the periodic discount rate from 1.

Example 7 Find the effective annual discount rate corresponding to a nominal rate of 8%, converted quarterly (“converted” is a synonym for compounded). Solution 

1  d = 1

d 4

(4)

4

 0.08 4 ) = 0.984 = 0.9224 => d = 0.0776  = (1 4 

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Example 8 Find the nominal rate convertible semi-annually corresponding to an effective annual rate of discount of 6% Solution d = 0.06 => 1 - d = 0.94. (2)

1 – d = (1 - d )2 = 0.94 2

(1 -

d(2) d (2) ) = 0.941/2 = 0.9695 => ( ) = 0.0305 ...