Chapter 10: Interest Rates and Time Value of Money

[Pages:25]Chapter 10: Interest Rates and Time Value of Money

Presented To: Dr. John Banko Dr. Craig Tapley

Written By: Meghan Fierko Shannon Lauer Daniela Scarpetta

University of Florida FIN4403 ? Honors Finance

November 16, 2009

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"In signing a 10-year, $252 million free-agent contract with the Texas Rangers, [Alex] Rodriguez...became the highest-paid player in the history of baseball." i Even though this may have been the highest quoted professional sports contract, is it really worth $252 million today? Sports reporters and news analysts commonly misinterpret the value of a contract by totaling the annual salary payments. However, one must consider the financial effects such as inflation, fluctuating interest rates, and loss of reinvestment opportunities to calculate the real present value of his contract.

According to USA Todayii, Alex Rodriguez's (A-Rod's) contract details are the following (excluding the $10 million signing bonus), as shown in Figure 10.1 below:

Figure 10.1

The fundamental principle of Time Value of Money states that a dollar today is worth more than a dollar tomorrow because of compounding interest effects. Because his salary is paid annually as opposed to a lump sum, A-Rod loses the opportunity to invest his entire signing contract in 2001.

How can we best approximate the present value of A-Rod's quoted $252 million contract, excluding the $10 million signing bonus?

We first discount the amounts paid every year back to 2001 at a pre-determined interest rate. In this example, we will use a 6% discount rate to calculate the present value of the future payouts. Next, to calculate the present value of the deferred payments, we first need to compound the deferred payments to their payout date using the stated 3% interest rate; according to the contract, the deferred payments will accrue at 3% interest rate starting in 2001, until the date they are paid out. Then, we discount their future value to 2001 using our 6% discount rate.

Now that all payout amounts from 2001-2020 are in PV terms as of 2001, we add these values to find the total present value of the contract. The contract is worth $178,393,182.90 in 2001, a stark contrast to the reported $242 million. This means that sports analysts and reporters overvalued A-Rod's contract by $63,606,817.10!

This example demonstrates the importance of accurately applying time value of money principles, which we will explore further in this chapter.

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Why do people invest money? Since interest rates enable peoples' money to grow, investors know that a dollar today is worth more than a dollar tomorrow. Interest rates are the cost of money that a lender pays a borrower for their investment. As stated in the previous chapter, interest rates are determined by risk, inflation, changing consumption preferences, and production opportunities. The most basic form of interest rate that exists is the real-risk free rate of interest (i*). This rate is commonly referred to as the short-term rate on a U.S Treasury Security if inflation were non-existent. The real risk-free rate sets a minimum for the interest rate that companies use as the cost of their money. Because companies face factors such as inflation, defaults, and maturity and liquidity risk, the real-risk free rate of interest is only one component of the quoted, or nominal, interest rate that is used to value an investment, as shown in the formula below.

Nominal interest rate =

i* + Inflation Premium + Default Risk Premium + Liquidity premium + Maturity Risk Premium

This nominal interest rate can also be referred to as the annual percentage rate (APR), which we will use in later sections to find the future and present value of an investment.

SECTION 10.1: FUTURE VALUE OF A SINGLE TIME DEPOSIT

Suppose that your uncle deposited $100 in your bank account the day you were born and that the bank paid you an annual simple nominal interest rate of 6% each year. You are now ten years old and would like to withdraw the money from your account. If you made no further deposits, how much money would you have at the end of 10 years? In other words, what is the future value, FV, at year 10 of the original $100 deposit?

Since your account grows by 6% each year, at the end of year 1 you will have $100 plus the additional yearly 6% interest.

(1+ interest rate)*Principal = FV1

(1+.06)*100 = $106 = FV1

As you can see in Figure 10.2, you would earn $6 of interest the first year. Your bank would give another $6 in interest the following year, which means that in two years you have earned an extra $12 just by leaving your money in the bank.

The general formula to find the FV for year n using a simple interest rate, i, is:

Principal + (i*Principal)*n = FVn

100 + (.06*100)*2 = $112 = FV2

Now, assume that this bank offers to pay you an annual compounding nominal interest rate of 6%. How would this effect your investment?

Compounding interest builds upon your accumulated interest, in other words, you are earning interest on interest. Therefore, in this example, in year 2 your account continues to grow by 6% on the already

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compounded $106. This exponential growth means you have now received $6.36 worth of interest in your second year, already $0.36 more than your previous simple interest accumulation! In order to see the difference between simple vs. compounding interest rates at each year, reference Figure 10.2 below. Figure 10.2

We can see from the figure above that the FV for investments that have compounding interest is calculated as:

Future Value = (Present Value)(1 + Interest Rate)Number of years which simplifies to: FVn = (PV)(1+i)n

In Excel, the function that allows us to determine the future value of an investment using compounding interest rates is as follows:

=FV(rate,nper,pmt,pv,type*) *For this example we will not use type. To find the FV at year 7 ($150.36), the Excel function would read:

=FV(.06,7,0,-100) = $150.36 = FV7 Rate = Interest rate, in decimal form, .06 Nper = Number of years, 7 Pmt = Yearly cash inflows, for this example there are none, 0 PV= Initial deposit, -100* *When solving for PV or FV, both the FV and PMT amounts must be negative, or the PV amount must be negative. In this case, we chose to make our PV negative because it is a cash outflow.

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CONTINUOUS COMPOUNDING Another type of compounding process is continuous compounding. Like compounding, continuous interest also builds on accumulated interest. However, instead of compounding at the end of the year, interest is compounding infinitely. This formula is represented by:

PVcontinuous = Pert If we were to use the example above, P is our $100 principal, e represents the infinite compounding, and r and t correspond to our interest rate and time, respectively. The letter e stands 2.718281825. Excel represents this number by the EXP function as seen below in Figure 10.3. Figure 10.3

Most banks and institutions do not rely on continuous compounding and instead compound interest monthly, quarterly, semi-annually or annually, which we will explore later in this chapter.

SECTION 10.2: PRESENT VALUE OF A SINGLE TIME DEPOSIT Suppose your uncle deposited money into your bank account ten years ago and it is now worth $179.08. The bank gave him a 6% annual interest rate but he cannot remember how much he initially deposited. Just like we were able to figure out the future value of our investment, we can discount our future value back 10 years and determine how much your uncle invested at year 0. We will now be solving for the present value, PV, which is represented by the formula:

FV PV (1 i)n Using the values from the above example, we see that the initial deposit equals: $179.08 PV0 (1.06)10 $99.997 $100.00 Present value is important because it can help us decide between two investments or it can help us determine if a bond or stock is over or undervalued.

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We can demonstrate the PV concept with a new example: If a bank offers you $2,000 today or $3,000 four years from now at 5% interest rate, which offer should you accept? Your choice depends on which offer gives you the highest PV as of today. Therefore, we need to discount the $3,000 back four years to compare it to the other offer of $2,000 received today. Using the PV formula, we find that the PV today of the $3,000 offer equals $2,468.11, [$3,000/(1.05)4]. Consequently, it is better to take the $3,000 investment four years from now because it is worth $468.11 more than getting the $2,000 today ($2,468.11 - $2,000). If the year in which you receive the $3,000 is paid out in a different year, we can see how the PV differs. Through this process, we can determine the time in which the $3,000 offer is no longer the better deal, as seen in Figure 10.4. Figure 10.4

In Excel, the function that allows us to determine the present value of an investment using the discounting interest rate can be written as:

=PV(rate,nper,pmt,fv,type*) *For this example we will not be using type. To find the present value of $3,000 paid out in a specified year, such as in year 3 for the current example, the Excel function would read:

=PV(0.05,3,0,3000) = -$2,591.51 = PV0 Rate= Interest rate in decimal form, 0.05 Nper= Number of years, 3 Pmt= Yearly cash inflows, for this example there are none, 0 FV= The value after last time period that you expect to receive interest for, 3000 Our present value will be negative because Excel requires either FV or PV to be negative (an outflow).

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SECTION 10.3: SOLVING FOR N AND I We are able to solve for variables such as n and i as long as we have the other variables in our present and future value equations. Note that you will be able to use this same principle to find missing variables of the equations for annuities later in the chapter. For example, if you deposit $2,468.11 today, at what interest rate would your money need to be invested so your original deposit is worth $4,000 at the end of five years? The Excel function that allows us to solve for this interest rate is:

=RATE(nper,pmt,pv,fv,type,guess*) *For this example, we will not use type or guess. Solving the Excel function, we find that your deposit must earn an interest rate of 10.138% for it to be worth $4,000 in five years, as shown in Figure 10.5

=RATE(5,0,-2468.11,4000) = 10.138% Figure 10.5

Now assume that our interest rate declines to 4%. How many years would it take the $2,468.11 investment to grow to $4,000? The Excel function that allows us to solve for number of years is:

=NPER(rate,pmt,pv,fv,type*) *For this example we will not use type.

=NPER(.04,0,-2468.11,4000) = 12.311 years

SECTION 10.4: ANNUITIES (PV and FV) In the previous sections, we examined present and future value calculations of single time deposits. We will now explore the valuation of an investment with multiple cash payments.

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An annuity provides a stream of fixed payments across equally spaced periods, allowing investors to obtain a steady flow of income. There are two types of annuities, ordinary and due, which distinguish between the timing of payments. An ordinary, or regular, annuity results when payments are made at the end of the period, while an annuity due makes payments at the beginning of the period. If you invest $100 for the next four years and your first deposit occurs one year from today, your investment would be an ordinary annuity. On the other hand, if you make your first $100 deposit today, your investment would be considered an annuity due.

Ordinary Annuity:

Annuity Due:

t= 0 1 2

3

4

t= 0 1 2 3

4

CF = $100 $100 $100 $100

CF = $100 $100 $100 $100

We see various examples of annuities in everyday life. Mortgage and student loan payments are typically made at the end of the month and are therefore ordinary annuities. Alternatively, rental payments and insurance premiums are usually annuities due, with payments occurring at the beginning of each month.

PRESENT VALUE OF AN ANNUITY:

Finding the PV of an annuity consists of discounting each payment back to the point in time you are solving for, and then summing these present values. The general formula can be represented as follows, where PVAn stands for the present value of an annuity at period n:

n

PVAn

t1

PMT (1 i)t

The PV of an ordinary annuity is valued one period before the first payment, while the PV of an annuity due is valued at the time of the first payment. The following image depicts the points in time in which the present values are calculated for both ordinary annuities and annuities due.

Period 1 Year 1

t=

0

1

2

3

4

CF=

PMT

PMT

PMT

PMT

PV Ordinary (yr0) PV Due (yr1)

As you can see, if payments start at the end of the first year (t=1), the PV of an ordinary annuity will be calculated at year 0. The PV of an annuity due will be found at year 1, when the first payment occurs. Note the difference between period 1 and year 1. Period 1 represents the time elapsed during the first

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