Models for Managing Money

Abstract

The chapter reviews ways of computing equivalent present, future, or equal annual values that can be used to compare different time series of costs and benefits. It defines both simple and compound interest modeling, and the impacts of within-year compounding, inflation, and income taxes.

5.1 Introduction

This chapter serves two purposes. One is to introduce some methods used to convert costs and benefits at different time periods to equivalent values at other time periods, and the other is to show how to evaluate options for managing our own financial resources. All this involves modeling.

5.2 The Time Value of Money

Figure 5.1 illustrates the time value of money. For example, assume you have won a cash prize of $10,000. You can either receive it now, option A, or receive it in three years, option B. The offer is hypothetical, but play along. Which option would you choose, and why (Fig. 5.2)?

Fig. 5.1

The value of money can grow over time and the more time the more money. The initial investments shown are assumed to continue each year up to the age of 65 compounding at an annual rate of 5%

Fig. 5.2

Schematic of the two options for receiving $10,000. The amounts shown in blue are the equivalent values three years later for option A, or three years earlier for option B

If you’re like most people who prefer having more rather than less money, you would choose to receive the $10,000 now, option A. After all, three years is a long time to wait. Why would any rational person prefer being paid later when he or she could have the same amount of money now? For most of us, preferring to have money now than later is just plain instinctive. And why?

Having $10,000 now allows you to spend it now. If you do not need it, you can loan it to someone who does need it now, and for that loan, the receiver can promise to give it back to you later, plus some additional money, called interest. Indeed, that is what banks do with the money you ‘loan’ them to save for you.

Having money now rather than later is worth paying for by those who need that money now. Those who borrow money, say from a bank, usually have to pay it back later with interest. What they payback is more than what they borrow. This is true even for banks in countries where earning interest by individuals is considered unethical. Otherwise, how could those banks survive?

By receiving $10,000 today, you can increase the future value of your money by investing it and gaining interest over time. If you invested it in a savings bank for three years, you would have the $10,000 plus the interest earned that the bank pays you for the use of your money over that time. If you wait until the end of three years to get the $10,000 cash prize, all you will have is the $10,000. The interest the bank pays you is based on the amount you give to them to save for you and the time they have used it. The interest rate is usually expressed as a percent of that amount per unit time period, typically a year. The interest rate is commonly denoted by the fraction, or percent, i.

Example: Assume that the interest is paid at the end of each year based on the amount invested in the savings account at the beginning of the year. If the annual interest rate is 4.5%, then at the end of the first year your $10,000 becomes $10,000 (1 + 0.045) = $10,450. The interest earned that year is $450.

If the $10,450 in your investment account at the end of the first year remains for another year, at the end of that second year you would have that plus another year of interest: $10,450(1 + 0.045) = $10,920.25.

This value at the end of the second year is

$$\$ {1}0{,}000 \, \times \, \left( {{1} + 0.0{45}} \right) \, \times \, \left( {{1} + 0.0{45}} \right) \, = \, \$ {1}0{,}000 \, \left( {{1} + 0.0{45}} \right)^{{2}} .$$

Investing this amount for three years would give you

$$\$ {1}0{,}000\left( {{1} + 0.0{45}} \right)^{{3}} = \, \$ {11}{,}{411}.{66}.$$

The annual interest rate of 4.5% is a compound interest rate, as interest is reinvested and earns interest along with the initial investment, the principal. If the interest earned is removed from the savings account each year, the 4.5% interest rate is called a simple interest rate. The total amount one would accumulate after n years of investing at 4.5% simple interest rate per year would be $10.000 (1 + n(0.045), i.e., the $10,000 principal plus n years of $450 interest payments.

5.3 Computing Present Values of Future Cash Flows

If you received $10,000 today, the present value would of course be $10,000. If $10,000 were to be received in a year, the equivalent present value of the amount now at the beginning of the year would not be $10,000 but rather the amount if invested today would total $10.000 in a year. And that depends on the interest rate you can earn on that investment.

Letting P₀ be the present value (at the end of year 0) and F_n be the future value at the end of year n, the basic equation for finding either P₀, F_n, or the assumed constant compound interest rate i, is (Fig. 5.3)

$${\text{F}}_{{\text{n}}} = {\text{ P}}_{0} \left( {{1} + {\text{ i}}} \right)^{{\text{n}}} .$$

Fig. 5.3

Cash flow diagram showing present and equivalent future values

Finding a future amount at the end of period n, F_n, given a present amount, P₀, and period interest rate i, is called compounding into the future. The opposite is called discounting a future value to the present. The distinction between compounding and discounting is shown in Fig. 5.4.

Fig. 5.4

Distinguishing between compounding a present value into the future and discounting a future value to the present

One can use the above single payment compound amount equation to find that $8,762.97 invested today at an annual compound interest rate of 4.5% for three years will equal $10,000 at the end of that third year, assuming again that interest is paid and reinvested at the end of each year. $8,762.97 is the present value of $10,000 at the end of three years. $10,000 is the future value of $8,762.97 invested today. Both statements assume an annual compound interest rate of 4.5% with interest paid at the end of each year.

What if in option B the cash prize payment in three years is more than $10,000, the amount you would receive today in option A? Say you could receive either $10,000 today (option A) or $13,000 at the end of three years. Which would you choose? The decision is now less obvious. If you choose to receive $10,000 today and invest the entire amount, you may actually end up with an amount of cash at the end of three years that is less than $13,000. To decide which option is better you could compute either the future value of $10,000 three years from now and compare it to the $13,000, or compute the present value of $13,000, and compare it to the $10,000.

For example, if interest rates are currently 4%, using the above equation, the equivalent present value of $13.000 three years from now is $11,556.95. Thus, the choice is between $10,000 and $11,556.95. Most would choose to postpone prize payment for three years. If you really needed $10,000 today and could borrow it at an annual interest rate less than 9%, you would be able to pay off the debt in three years and still have some leftover.

5.4 Computing Equivalent Constant End-of-Period Amounts

Many benefit-cost calculations use annual costs and benefits. For example, if you want to borrow $200,000 to buy your first house, you typically go to a bank and get a loan. The bank tells you how much money you need to pay the bank, in equal payments, A, at the end of each year for a given number of years, to pay back the loan plus interest. To calculate this constant annual amount, A, paid at the end of each year, we find the sum of the present values of each of those annual payments of A and equate that sum to the original present value of debt of $200,000. If n is the number of years of payments

$${\text{P}}_{0} = { 2}00{,}000 \, = A/\left( {{1} + {\text{i}}} \right) \, + A/\left( {{1} + {\text{i}}} \right)^{{2}} + A/\left( {{1} + {\text{i}}} \right)^{{3}} + \, \cdots \, + A/\left( {{1} + {\text{i}}} \right)^{{\text{n}}} .$$

This is equivalent to

$${\text{P}}_{0} = A\left[ {\left( {{1} + {\text{i}}} \right)^{{\text{n}}} {-}{ 1}} \right]/\left[ {{\text{ i}}\left( {{1} + {\text{i}}} \right)^{{\text{n}}} } \right] \, {\text{or}}\,A = {\text{ P}}_{0} \left\{ {{\text{ i}}\left( {{1} + {\text{i}}} \right)^{{\text{n}}} /\left( {\left( {{1} + {\text{i}}} \right)^{{\text{n}}} - {1}} \right) \, } \right\}.$$

This is how the banks determine what you owe to pay back a loan with equal end-of-period payments over n time periods assuming an interest rate of i per period. The period most banks use is a month, not a year. If i represents an annual interest rate, the monthly rate is i/12 (Fig. 5.5).

Fig. 5.5

Cash flow diagram for a constant end-of-period cash flow equivalent to a present value of P_o

When one gives money to an organization’s endowment, they usually expect it will provide income to that organization forever. The end-of-year annual equal payment A from an endowment of P_o that can be paid forever can be calculated using the above equation when n goes to infinity. The result is the same as if simple interest were being used. The equal annual payment A = P_o(i).

5.5 Within-Year Compounding

If you are saving money in a bank savings account, the interest you earn each day is the minimum amount you have in your account that day times the daily interest rate. This daily rate is the annual ‘nominal’ rate (say 5%) divided by 365. This daily rate can be applied in any of the above equations, where instead of the time period being a year, it is a day.

Hence, F₁ at the end of a day = P₀ (1 + annual nominal interest rate/365).

This is daily compounding. Interest is earned and paid to the account each day.

F₃₆₅ at end of a year of daily compounding = P₀ at the beginning of the year times the factor (1 + annual nominal rate/365)³⁶⁵.

The future value after n years of daily compounding at a nominal annual rate of 5% is

$${\text{F}}_{{\text{n}}} = {\text{ P}}_{0} \left( {{1} + \, 0.0{5}/{365}} \right)^{{\text{365 n}}} .$$

If r is the nominal annual interest rate, but compounding occurs in each of m equal periods within a year, then the corresponding effective annual rate i that assumes compounding occurs only once in a year is

$$\left( {{1} + {\text{i}}} \right)^{{1}} = \, \left( {{1} + {\text{r}}/{\text{m}}} \right)^{{\text{m}}}\,{\text{or}}\,{\text{i}} = \, \left( {{1} + {\text{r}}/{\text{m}}} \right)^{{\text{m}}} - { 1}.$$

The annual effective rate i associated with within-year period compounding is clearly greater than the annual nominal rate r. For example, monthly compounding at a nominal annual interest rate of r is equivalent to annual compounding at an effective interest rate of (1 + r/12)¹² − 1.

Daily compounding, which many bank savings accounts offer, is almost equivalent to what is called continuous compounding ~ compounding every nanosecond! If the nominal annual rate of interest is r, the corresponding effective continuous compounding annual rate turns out to be e^r − 1, where e is the base of natural logarithms, e = 2.718281828. The factor (1 + i) becomes (1 + e^r − 1) or (e^r). Thus, for continuous compounding over n years, an investment of P₀ at the beginning of year 1 (or end of year 0) will yield

$${\text{F}}_{{\text{n}}} = {\text{ P}}_{0} ({\text{e}}^{{\text{r}}} )^{{\text{n}}}$$

at the end of n years.

5.6 Inflation

Prices of goods and services usually increase over time. This is called inflation. The actual rate of inflation varies depending on the item. The increase (or decrease) in home prices is not the same as, for example, the increase in university tuition. General consumer price index (CPI) inflation rates mentioned in the media are commonly based on the prices of a set of goods and services that are included in the CPI. The rate of inflation varies over time, of course, just like interest rates. The inflation rate is commonly designated by the letter f. Hence, assuming an annual inflation rate of f, something that costs $100 today will cost $100(1 + f)ⁿ at the end of n years from now. If there is no other reason to invest money, it is to keep up with inflation. Otherwise, even if you have the same amount of money now and n years from now, you will be poorer then in the sense you will not be able to buy as much then as you can now with that amount of money. Obviously, one tries to build wealth at a rate greater than the rate of inflation. Taking into account the effects of inflation, the ‘real’ uninflated rate of return, r, on any investment earning an interest rate of i is (Fig. 5.6)

$$\left( {{1} + {\text{r}}} \right) \, = \, \left( {{1} + {\text{i}}} \right)/\left( {{1} + {\text{f}}} \right)\,{\text{or}}\,{\text{r}} = \, \left( {{1} + {\text{i}}} \right)/\left( {{1} + {\text{f}}} \right) \, {-}{ 1}.$$

Fig. 5.6

Impact of 4% inflation on the purchasing power of today’s $1 over the next 25 years

The real rate of return, r, is often called the true or real time value of money. (Do not confuse this r with the r denoting the nominal annual interest rate applicable to within-year compounding).

To compute the inflation adjusted annual payments so that each payment has the same purchasing power, the real uninflated interest rate r can be used to compute the constant payment A, and then each A is inflated at the time of payment. Hence, instead of using

$$A = {\text{ P}}_{0} \left\{ {{\text{ i}}\left( {{1} + {\text{i}}} \right)^{{\text{n}}} / \, \left( {\left( {{1} + {\text{i}}} \right)^{{\text{n}}} - {1}} \right) \, } \right\},$$

use the real rate of return r in that equation in place of i to compute A and then inflate it at the time of payment.

$$A_{{\text{n}}} = {\text{ the actual payment at end of year n }} = A\left( {{1} + {\text{f}}} \right)^{{\text{n}}} .$$

5.7 Income Taxes

In addition to wanting the interest rate you are getting on your investments to be greater than the rate of inflation, you also want it to be greater than the inflation rate after you paid your income taxes on the interest earned. The net interest rate after taxes depends on the tax rate. Letting t be the tax rate, then the net interest rate after taxes is i(1 − t). This expression assumes you pay the taxes when the interest is earned. Even though this is rarely the case, it is a good enough assumption for most economic calculations we will be performing (Fig. 5.7).

Fig. 5.7

There are only two things that are certain in life: death and taxes. November 13th, 1789, Benjamin Franklin. http://www.clker.com/cliparts/4/9/f/1/1516760576154679115death-and-taxes-clipart.hi.png, http://www.clker.com/clipart-744320.html. Public domain

Thus, the future value, F_n, after taxes, on an investment of P₀ for n years at an annual before tax interest rate i will be

$${\text{F}}_{{\text{n}}} = {\text{ P}}_{0} \left( {{1} + {\text{i}}\left( {{1} - {\text{t}}} \right)} \right)^{{\text{n}}} .$$

If the investment is placed in a tax-deferred account, the income tax is paid only when the money is withdrawn, say at the end of n years. In this case, the after-tax amount will be

$${\text{F}}_{{\text{n}}} = {\text{ P}}_{0} \left( {{1} + {\text{i}}} \right)^{{\text{n}}} - \, \left[ {{\text{ P}}_{0} \left( {{1} + {\text{i}}} \right)^{{\text{n}}} - {\text{ P}}_{0} } \right] \, \left( {\text{t}} \right) \,{\text{or}} \,{\text{P}}_{0} \left[ { \, \left( {{1} + {\text{i}}} \right)^{{\text{n}}} \left( {{1} - {\text{t}}} \right) \, + {\text{ t }}} \right].$$

Obviously, if you can do this, tax-deferred investments offer more at the end of such investment periods than do accounts where taxes have to be paid each year. But this may depend on the tax rates that can differ over time as well.

In any event, unless the rate of interest one earns exceeds both the inflation and tax rates, the monetary gains recorded in bank statements over time will be losing purchasing power.

5.8 Comparing Alternatives

It is important to know how to calculate the value of money over time so that you can distinguish between the worth of alternative investments that offer different returns, or costs and benefits, at different times over different time periods. Remember that you cannot move money around over time without using the applicable interest rate unless, of course, it is 0. $100 today is not the same as $100 tomorrow. To compare different alternatives having different time streams of costs and benefits, we must move money around over time to compute equivalent present values, P₀, future values, F_n, or annual equal end-of-year values, A. When doing this comparison of alternatives, one must be considering what to do with the same amount of money invested (costs) over the same amount of time for all alternatives being compared.

For example, consider the following. There are two alternatives, A and B, that involve different initial investments. These initial investments along with the present values of the future net benefits are given in the table below. Both the net present values and the present benefit/cost ratios are also shown. You will see that based on an objective of maximizing net benefits alternative A is best. But based on the objective of maximizing the benefit/cost ratio alternative B is best (Table 5.1).

Table 5.1 Costs and benefits of two alternatives

Both the net benefit and the benefit/cost criteria should indicate the same best alternative. What is missing in this analysis?

In this example, the issue is how best to invest the $40 that is apparently available since alternative A is being considered. So the issue is what to do with the $40. The amount left over after investing 10 in alternative B is 30 and this plus 15 is the present value of the benefits. Thus, the benefit/cost ratio for alternative B is really 45/40 = 9/8. This is less than the benefit/cost ratio of 5/4 for alternative A, and hence based on both the net benefit and benefit/cost criteria, alternative A is best.

5.9 Investing for Retirement

Assume you can invest $5500/year of earned income into a tax-free account. In the US, it might be a Roth Individual Retirement Account (IRA). Also, assume that you can start investing at age 25 and you plan to retire 40 years later at age 65. Finally, assume that you can earn an average annual rate of interest of 8% over the 40-year period. Investing $5500 at the beginning of a year will result in 5500(1 + 0.08) = $5940 at the end of the year. Interest earned is $5500(0.08) = $440.00 and is tax free when it is withdrawn after you retire. At the beginning of the second year, you invest another $5500 in the account. At the end of two years of investing, you have

$$\left( {\$ {594}0 \, + \$ {55}00} \right)\left( {{1} + 0.0{8}} \right) \, = \, \$ { 12}{,}{355}.{2}0.$$

At the end of three years of investing $5500 at the beginning of each year

$$\left( {\$ {12355}.{2}0 \, + { 55}00} \right)\left( {{1} + 0.0{8}} \right) \, = \, \$ {19}{,}{283}.{62}.$$

Notice the model one can use to compute how much you will have, F_n, at the end of n years of investing P at the beginning of each year, at 8% per year, is

$$\begin{aligned} & {\text{F}}_{{1}} = {\text{ P}}\left( {{1} + 0.0{8}} \right), \hfill \\ &{\text{F}}_{{2}} = \, \left( {{\text{F}}_{{1}} + {\text{ P}}} \right)\left( {{1} + 0.0{8}} \right), \hfill \\ &{\text{F}}_{{3}} = \, \left( {{\text{F}}_{{2}} + {\text{ P}}} \right)\left( {{1} + 0.0{8}} \right) \ldots \hfill \\ \end{aligned}$$

and so on for each year y until y = n. This model can be written as

$${\text{F}}_{{\text{y}}} = \, \left( {{\text{F}}_{{{\text{y}} - {1}}} + {\text{ P}}} \right)\left( {{1} + 0.0{8}} \right){\text{ for y}} = { 1},{ 2},{ 3}, \, \ldots ,{\text{n}}\, {\text{and F}}_{0} = \, 0.$$

In this example, all the beginning-of-year investments, P, are $5500. There are more elegant ways of computing any F_n, but the above sequence of equations, solved sequentially for each time period y, works. At the end of 10 years of investing $5500 at 8% per year, you will have $86,050.18. After 30 years of investing, you will have $672,902.30.

Consider two options:

• (a) Invest $5500 at the beginning of each year starting at age 25 and stop after 10 years but keeping the total accumulated amount ($86,050.18) in the account earning 8%/year for the next 30 years. At the end of the next 30 years, at age 65, the amount in the account will be $86,050.18 (1 + 0.08)³⁰ = $865,893.40 for a total investment of 10($5500) = $55,000.

• (b) Start Investing $5500 at the beginning of each year beginning at age 35, for the next 30 years, using the same model as described above. The total amount at the end of the 30 years, at age 65, will be = $672,902.30, based on a total investment of 30($5500) = $165,000.

You invest more ($165,000 − $55,000) and get less ($865,893.40 − $672,902.30) using option ‘b’ than if you use option ‘a’. Of course, investing over the entire 40 years of your working life will give you a total of $865,893.40 + $672,902.30 = $1, 538,796.

That amount of money may seem like a lot, but will it be enough when you retire? At the end of 40 years, the price of what you might want to buy will be more than what it is now. For an annual inflation rate (fraction) of f, what you could buy for a dollar at age 25 after 40 years will cost (1 + f)⁴⁰ dollars. You can see that if the inflation rate f is say 3% per year, you will need $17,941.21 40 years from now to buy what $5500 could buy today. The message: Needing money for retirement is real. So is inflation. Hence, how to invest now to be ready to retire sometime in the future with the desired lifestyle is worth some thought and planning, and as the previous example shows, the sooner the better (Fig. 5.8)!

Fig. 5.8

Retirement. How much will you need to implement it?

Exercises

• 1. What is $1 invested today at 7% per year, compounded annually, worth at the end of 10 years?

• 2. How long will it take to double your investment if it is earning 10% per year

• 3. What is the value of $1 invested for a year if compounded at 1% per month?

• 4. What would be the answer to the previous question if an annual nominal interest rate of 12% were compounded continuously within the year?

• 5. Suppose after you graduate and begin receiving an income you start investing $6000 each year into a tax-free retirement account that earns 8% per year. You do this for only 10 years, and then just leave it in the account earning 8% interest for the next 30 years when you decide to retire. Alternatively, you only start investing $6000 per year into this tax-free account on the 11th year of employment and keep investing annually for the remaining 30 years. Which investment strategy will result in a higher retirement fund at the end of 40 years of employment?

• 6. How much money are you going to need when you retire to assure you can meet your standard of living for the remainder of your life? Specify all the assumptions you are making, taking into account taxes and inflation. How are you going to get that amount of money (i.e., your savings plan?).

• 7. One criterion for plan selection is maximum net annual benefits. The maximum benefit–cost ratio, or annual benefits divided by annual costs, is another criterion. Benefit–cost ratios should be no less than one if the annual benefits are to exceed the annual costs. Consider two projects, I and II:

  Project
  	I	II
  Annual benefits	20	2
  Annual costs	18	1.5
  Annual net benefits	2	0.5
  Benefit/cost ratio	1.11	1.3

  What additional information is needed before one can determine which project is the most economical project?

• 8. Bonds are often sold to raise money for infrastructure project investments. Each bond is a promise to pay a specified amount of interest, usually semiannually, and to pay the face value of the bond at some specified future date. The selling price of a bond may differ from its face value. Since the interest payments are specified in advance, the current market interest rates dictate the purchase price of the bond.

  Consider a bond having a face value of $10,000, paying $500 annually for 10 years. The bond or ‘coupon’ interest rate based on its face value is 500/10,000, or 5%. If the bond is purchased for $10,000, the actual interest rate paid to the owner will equal the bond or ‘coupon’ rate. But suppose that one can invest money in similar quality (equal risk) bonds or notes and receive 10% interest. As long as this is possible, the $10,000, 5% bond will not sell in a competitive market. In order to sell it, its purchase price has to be such that the actual interest rate paid to the owner will be 10%. In this case, what is the bond currently worth?

  The interest paid by some bonds, especially municipal bonds, may be exempt from state and federal income taxes. If an investor is in the 30% income tax bracket, for example, a 5% municipal tax-exempt bond is equivalent to about a 7% taxable bond. This tax exemption helps reduce local taxes needed to pay the interest on municipal bonds, as well as provides attractive investment opportunities to individuals in high tax brackets.

• 9. Assume a particular university’s tuition and fees are $C today.

  Assume the after-tax interest rate you can earn in the next 24 years is 5%.

  Assume the inflation rate of tuition and fees in the next 24 years will be 4%.

  Show how to determine how much would be enough to invest today to pay for four years of tuition and fees starting at the beginning of 20 years from now.

  Just set up the equations needed to find the answer. Drawing a picture may help.

• 10. You must pay back a bank debt, say of $1000, with interest, in 12 equal end-of-month payments. Each monthly payment contains both some of your debt and the monthly interest owed on the remaining debt. The bank tells you the annual interest rate is 5%. Describe how you could determine the annual interest rate you actually paid on the debt you owed.

• 11. You are considering taking flying lessons that if begun today will cost $10,000. Alternatively, you could wait a year to begin the lessons after paying the fee (that is likely to be higher) at that time.

  • (a) If you decide to wait a year and invest the $10,000 during the year, earning an annual interest rate i, describe how would you determine the extra money you would have at the end of the year after paying the inflated cost of lessons at that time?

  • (b) Assume you forgot to consider the fact that you will owe income taxes on the interest earned. Your income tax rate is t. How would your analysis change so as to include the tax payment?

• 12. You must pay back a bank debt, say of $1000, with interest, in 3 equal end-of-year payments. Each payment contains the interest on the debt at the beginning of the year and some of the principal.

  (As the debt decreases so do the interest payments in each successive A. The interest paid, I_y, at the end of a year y is based on the debt, P_y-1, at the beginning of that year.)

  The bank tells you the annual interest rate is 5%.

  Show how to compute the principal and interest contained in each of the three end-of-year payments ‘A’ using the following steps:

  • (a) Write the equation for solving for payments A:

  • (b) Show the equation for computing for the first interest payment, I₁:

  • (c) Given A and I₁, show the equation for computing for the remaining debt at beginning of 2nd year, P₁:

  • (d) Show the equation for computing for the interest paid in the 2nd payment:

  • (e) Given A, P_1, and I₁, solve for the remaining debt at beginning of 3rd year:

  You can deduct 30% of the annual interest payment from your income tax each year. Given all the interest payments I_y and A, show the equation you could use to compute the actual interest rate you are paying on your debt.