the time value of money · strayer university. fin 534 – the time value of money 2 abstract the...

Running head: THE TIME VALUE OF MONEY 1

The Time Value of Money

Ma. Cesarlita G. Josol

MBA - Acquisition

Strayer University

FIN 534 – THE TIME VALUE OF MONEY 2

Abstract

The paper presents computations about applications on time value of money, bonds, bonds

valuation and bonds’ interest rates. It is essential for financial managers to have a good

understanding of the concept of time value of money (TMV) and its impact on stock prices.

Companies do not only sustain their operation through assets, but also through debts. Bond is a

form of debt where corporations derive its capital for its operations. Hence, financial managers

also give specific attention to bonds, valuation of bonds and how its interest rates will affect their

financial decision-making. This paper focuses on time value analysis applications like

calculation of present value of uneven cash flow using four different procedures: the step-by-step

approach, the formula approach, using the financial calculator and by use of spreadsheets. The

calculations will show how to do the discounting method to find the present value. Other

applications of the time value concept is also presented here such as the concept of future value,

the effective annual rate, compounding and interest rates. The paper will also present the

calculation of the bond price. It will also provide how the change in market interest rates would

result in a discounted bond and a premium bond. I will also present the calculation of yield to

maturity, the total return, current yield and the capital gains yield a bond. There will be nine (9)

questions presented in this paper, of which I provided solutions on a step-by-step approach.

Keywords: Time value of money, present value, future value, bonds, bond valuation


The Time Value of Money

If you are offered a $1,000 now and a $1,100 in two years, what will you choose? In a

situation where you have extra money to spare for investment what is your decision on accepting

the offer to get money now or in two years. Think you really don’t need the money now, and you

can afford to save for a rainy day. I am going to take the $1,000 and invest in a fixed rate that

grows my money in a certain period, rather than choosing to get the $1,100 offered to me in two

years. This is the concept of time value of money (TVM). For me the time value of money is a

concept where you are going to decide how you will give value to your cash and decide to invest

and get the value of the money over time. In businesses, it is essential for managers – financial

managers, to have a thorough understanding of the time value of money (TVM). A good analysis

and understanding of how money will be valued at a certain future is important in sustaining the

operations of a business.

Another vital information is to get a good grasp of the concept of debts or borrowing.

This paper presents calculations of bond valuations, bond interest rates, and changes of bond

valuation over time. There is a relationship that exist between the market interest rate and the

annual coupon rate of bonds that would result in either discounted bonds or premium bonds

The following nine (9) questions provides scenarios where I calculate solutions by using

the applications of the concept time value of money and bonds.

What is the present value of the following uneven cash flow stream - $50, $100, $75, and

$50 at the end of Years 0 through 3? The appropriate interest rate is 10%, compounded

annually?

This question involves an annuity with uneven cash flow instead of constant payment.

There are two important cases of uneven cash flows: (1) where the cash flow stream consists of a


series of annuity payments plus an additional final payment, and (2) the uneven or irregular cash

flow stream (Brigham, & Ehrhardt, 2014). This problem is an example of the uneven cash flow

stream. In calculations involving cash flows , payment (PMT) is used in situations where the

cash flow are constant and thus an annuity is involved; if cash flows are different in different

time periods, the term CFt is used which means cash flow in period t.

There are three ways to calculate for the net present value (NPV): (1) the step-by-step, (2)

the use of the financial calculator, and (3) Excel spreadsheet

Variables: I = 10%; CF0 = -$50; CF1 = $100; CF2 = $75; CF3 = $50; N = 3

Cash Flow Stream

0 1 2 3

($50) $100 $75 $50

1. The step-by-step method

The Step-by-Step Method

Periods(N) 0 1 2 3

Cash Flow (CF) ─$ 50.00 $100.00 $ 75.00 $ 50.00

PVs of the CFs ─$ 50.00 $ 90.91 $ 61.98 $ 37.57

PVt of the Irregular CF Stream $ 140.46

Calculation of the above step-by-step method:

PVt = CF0 / (1 + I)0 + CF1 / (1 + I)1 + CF2 / (1 + I)2

+ CF3 / (1 + I)3

PV0 = CF0 / (1 + I)0 = (─ $50) / (1 + 0.10)0 = (─ $50) / (1) = ─$50.00

PV1 = CF1 / (1 + I)1 = ($100) / (1 + 0.10)1 = ($100) / (1.1) = $90.91

PV2 = CF2 / (1 + I)2 = ($75) / (1 + 0.10)2 = ($75) / (1.21) = $61.98

PV3 = CF3 / (1 + I)3 = ($50) / (1 + 0.10)3 = ($50) / (1.331) = $37.57

PVt = $50.00 + $90.91 + $61.98 + $37.57 = $140.46


Sum of the individual Present Values (PVs) = $140.46

PVt = $140.46

2. Using the financial calculator – Texas Instrument BA II Plus

2. Financial Calculator (using Texas Instruments BA II Plus )

Step 1: Press 2nd and CE│C

Step 2: Press CF

Step 3: key in 50 (+/-) enter and press down arrow once , CO1

appears on screen

Step 4: Key in 100 , press enter and press down arrow twice, C02

appears on screen

Step 5: Key in 75 and press enter and press down arrow twice, C03

appears on screen

Step 6: Key in 50 press enter

Step 7: Press NPV key and the I appears, key in 10 , press enter

Step 8: Press down arrow once, NPV= 0.00 appears on screen

Step 9: Press CPT

Screen gives the net present value =NPV= $ 140.46

3. Excel Spreadsheet

1 A B C D E F

2 Inputs:

3 Interest Rate= I = 10%

4 Table of Cash Flows

5 Periods 0 1 2 3

6 Cash Flow -50 100 75 50

Calculation:

Using the NPV function = NPV(I, CFS)

Fixed Inputs NPV = NPV(0.10,100,75,50) = 190.46

Cell references NPV = NPV(C3,D6:F6) = 190.46

NPV = 190.46 – 50 = $140.46


The Net Present Value is the present value of the expected future cash flows less the cost

of the investment (Microsoft Excel, n.d.). The NPV function in Excel only calculates the present

value from period 1 to period 3 in this problem. Then we need to subtract the -$50 which is an

outflow (Microsoft Excel, n.d.). The result of the NPV is equal to $140.46.

We sometimes need to find out how long it will take a sum of money (or something else,

such as earnings, population, or prices) to grow to some specified amount. For example, if a

company’s sales are growing at a rate of 20%, how long will it take sales to double? In order

to know how long will it take for a company’s sales to double with the given interest rate of

20%, we need to find the number of years, N. For example the company’s sales is $1,000,000;

the interest rate given here is 20%. How long will it take for the $1,000,000 to double?

In order to find the number of years, N, we can use three procedures: (1) Using the

financial calculator, (2) Excel spreadsheet, (3) by working with natural logs

Variables: FV = $2,000,000; PV = ─ $1,000,000; I = 20%; N=?

Method 1: using the financial calculator Texas Instrument BA II Plus

Using the N,I/Y,PV, PMT, FV

1. Clear all values, press 2nd CE│C , press 2nd FV

2. Key in 2,000,000 then press FV

3. Key in 1,000,000 the +│─ button , then press PV

4. Key in 20 , press I│Y

5. Press 2nd P│Y, key in 1 and press enter

6. Press 2nd then QUIT

7. Press CPT and press N key

8. N = 3.80


It will take 3.8 years for the sales of $1,000,000 to double with the given interest rate of 20%

Method 2: using the Excel spreadsheet

A B C

1

2 Present Value ($1,000,000)

3 Future Value $2,000,000

4 Interest Rate 20%

NPER 3.8018

In Excel, the NPER function is used to determine the period, N. NPER returns the

number of periods for an investment based on a periodic constant payments and a constant

interest rate. NPER in this example is NPER = NPER(I,PMT,PV,FV). Providing the data

NPER = NPER(0.20,0,-1,000,000,2,000,000) = 3.8018 or 3.8 years.

Method 3: Using the log solution by finding the natural logs using the financial calculator

and solve N (Brigham, & Ehrhardt, 2014).

Interest Rate, I = 20%

Sales = $1,000,000

$2,000,000 = $1,000,000(1+I)N

$2,000,000 = $1,000,000(1 + 0.20)N

$2,000,000 ÷ $1,000,000 = (1 + 0.20)N

2 = (1 + 0.20)N

ln 2 = N[ln(1.20)]

N = ln(2) / ln(1.20)

N = 0.6931471806 / 0.1823215568

N = 3.8018 = 3.80


Will the future value be larger or smaller if we compound an initial amount more

often than annually ─ for example, every 6 months, or semiannually ─ the stated interest

rate constant? Why? The future value of an investment would be large if the initial amount will

compounded more than annually. There will be higher future values when an initial investment is

compounded more frequently. Interest will be earned on interest more often the more frequent

compounding occurs (Brigham, & Ehrhardt, 2014). The effective annual rate (EAR) also known

as the effective percentage rate (EFF%) will increase due to frequent compounding; hence the

future value and the EFF% will increase as the frequency of the compounding increase

(Brigham, & Ehrhardt, 2014). The biggest increase occurs when compounding goes from annual

to semi-annual (Brigham, & Ehrhardt, 2014).

Comparison of annual compounding and semi-annual compounding:

Variables: PV = $100; Interest rate = 8%; N = 1; M = 2

Compounding: FVN = PV(1 + IPER)Number of Periods = PV(1 + INOM / M)MN

Compounding annually:

FV1 = $100(1 + 0.08 / 1) = $108

Compounding semi-annually:

FV2 = $100(1 + 0.08 / 2)2 = $108.16

What is the effective annual rate (EAR or EFF%) for a nominal rate of 12%,

compounded semi-annually? Compounded quarterly? Compounded monthly? Compounded

daily? The effective (equivalent) annual rate (EAR or EFF%) is the annual (interest once a year)

rate that produces the same final result as compounding at the periodic rate for M times per year

(Brigham, & Ehrhardt, 2014). Given a nominal rate of 12%. The EAR, also known as EFF% is

found from the following equation:


EAR = EFF% = (1 + IPER )M – 1.0

EAR = EFF% = (1 + INOM / M)M ─ 1.0

Where: INOM is the nominal rate; IPER is the periodic rate; M is the number of periods per

year; N is the number of years and the INOM = 12%

1. Compounded semi-annually:

M = 2

EAR = EFF% = (1 + IPER)M ─ 1.0

EAR = EFF% = (1 + INOM / M)M ─ 1.0

EAR = EFF% = (1 + 0.12 / 2)2 – 1.0

EAR = EFF% = (1 + 0.06)2 ─ 1.0

EAR = EFF% = (1.06)2 ─ 1.0

EAR = EFF% = 1.1236 ─ 1.0

EAR = EFF% = 12.36%

2. Compounded quarterly:

M = 4

EAR = EFF% = (1 + 0.12/4)4 – 1.0

EAR = EFF% = (1 + 0.03)4 – 1.0

EAR = EFF% = (1.03)4 – 1.0

EAR = EFF% = 12.5509%

3. Compounded monthly:

M = 12

EAR = EFF% = (1 + 0.12/12)12 ─ 1.0

EAR = EFF% = (1 + 0.01)12 ─ 1.0


EAR = EFF% = (1.01)12 ─ 1.0

EAR = EFF% = 1.12682503 ─ 1.0

EAR = EFF% = 12.6825%

4. Compounded daily

M = 365

EAR = EFF% = (1+0.12/365)365 ─ 1.0

EAR = EFF% = (1+0.000327671233)365 ─ 1.0

EAR = EFF% = 1.127475 ─ 1.0

EAR = EFF% = 12.7475%

Suppose the on January 1 you deposit $100 in an account that pays a nominal (or

quoted) interest rate of 11.33463%, with interest added (compounded) daily. How much will you

have in your account on October 1, or 9 months later? To solve this problem we focus on

fractional time periods. Here I assume a 365 days in a year so M = 365

Given: Nominal interest rate = 11.33463%; Period = 9 months or 9/12

1. Computing for the Periodic Rate

IPER = INOM / M = 0.1133463/365 = 0.0003105378082 per day

Computing for the number of days = (9/12)(365) = 273.75 = 274 days

Amount in account on October 1 = $100 (1 + IPER)Number of Periods

= $100(1.0003105378082)274

= $108.8797799

Amount in account on October 1 = $108.88

What would be the value of the bond described below if, just after it had been issued, the

expected inflation rate rose by 3 percentage points, causing investors to require a 13% return?


Would we now have a discount or a premium bond? A firm issues a 10-year par value bond with

a 10% annual coupon and a required rate of return is 10%.

N = 10; I│YR = 13%; PMT = $1,000(10%)= $100

First I will calculate the bond price on what is originally given which is $1,000.

Input

10.00 10.00 100.00 1000.00

N I│YR PV PMT FV

($1,000.00)

Output

Then calculating using raised interest rate of 13% using the financial calculator returns a

PV = ─$837.21, therefore the bond price is $837.21, which is lower than the original bond price.

Input

10.00 13.00 100.00 1000.00

N I│YR PV PMT FV

($837.21)

Output

This is a discount bond. The coupon rate remains after the issuance of the bond however

the interest rates in the market move up and down (Brigham, & Ehrhardt, 2014). An increase in

the market interest rates (rd) will cause the price of the bond to fall (Brigham, & Ehrhardt, 2014).

This bond is a discount bond as this fixed-rate bond’s price fell below its par value due to the

increased market interest rate over the annual coupon rate.

What would happen to the bond’s value if inflation fell and rd declined to 7%? Would we

now have a premium or a discount bond? In the same situation, that a firm issues a 10-year,

$1,000 par value bond with a 10% annual coupon and a required rate of return is 10%. So the

original bond price is $1,000.

First calculating for the bond price at the given 10% interest rate.


Input

10.00 10.00 100.00 1000.00

N I│YR PV PMT FV

($1,000.00)

Output

Then computing for the bond price with a rd = 7%

Input

10.00 7 100.00 1000.00

N I│YR PV PMT FV

($1,210.71)

Output

The bond price is $1,210.71 which is above the original bond price. Bond prices rise

when the market interest falls or when rd falls. When a going interest rate falls below the coupon

rate, a fixed-rate bond’s price will rise above its par value and it is called a premium bond

(Brigham, & Ehrhardt, 2014). This bond that has a 7% market interest rate returns a bond price

of $1,210.71 is a premium bond.

What is the yield to maturity on a 10-year, 9% annual coupon, $1,000 par value bond

that sells for $887.00? That sells for $1,134.20? What does a bond selling at a discount or a

premium tell you about the relationship between rd and the bond’s coupon rate? In this problem

the given for the described bond is N=10 years, Coupon rate = 9%, Par Value = $1,000, Bond

Price = $887.00. The yield to maturity (YTM) is the same as the market rate of interest, rd .

To solve for the yield to maturity (YTM) for the bond price of $887.00, using the

financial calculator, I entered all the required values in the financial calculator to solve the yield

to maturity (YTM)


Input

10 ($887.00) 90.00 1000.00

N I│YR PV PMT FV

10.91

Output

The yield to maturity (YTM) is 10.91%

To solve for the yield to maturity (YTM) for the bond price of $1,134.20, using the

financial calculator:

Input

10 ($1,134.20) 90.00 1000.00

N I│YR PV PMT FV

7.08

Output

The yield to maturity (YTM) is 7.08%

A bond selling at a discount means that the rate of interest (rd) is above the bond’s coupon

rate that would result in the bond’s fixed-rate price falling below the par value. While a bond

selling at a premium means that the rate of interest (rd) falls below the bond’s coupon rate, as a

result the fixed-rate’s bond price will rise above its par value. When rd goes up above the coupon

rate the bond price goes down below the par value, and when rd goes down, below the bond’s

coupon rate, the bond price goes up above the par value. Whenever the rd equals the coupon rate,

a fixed-rate bond will sell at its par value.

What are the total return, the current yield, and the capital gains yield for the discount

bond in the previous question #8 at $887.00? At $1,134.20 (Assume the bond is held to maturity

and the company does not default on the bond). The bond described in this problem is a 10-year,

9% annual coupon, $1,000 par value bond. As I stated before when rd rose above the coupon rate,


the bond would sell below the par value hence the bond will sell at a discount, this bond is called

the discount bond. The discount bond price of $887.00 is a result of the rise of the rd to 10.91%.

Using this bond price , I will compute the total return, the current yield, and the capital gains for

this discount bond from Year 2(with N=9 years) and Year 1 (with N=10 years).

The question did not specify as to what year or what period the total return, current yield,

and the capital gains be calculated, I will provide two periods. First I will calculate for the

required data for year 2 and then I will calculate at the maturity date which is the year 10, where

N = 0. This is assuming the bond is held to maturity and company does not default on the bond.

This calculation will hold through for the discount bond and the premium bond.

1. Calculations for the Discount bond of $887.00

a) Computing for the total return, current yield, and the capital gains for the discount

bond of $887 for year 2 where N = 9

Variables: N = 10 years; Par = 1,000; Annual coupon rate = 9%; YTM for discount bond

or rd = 10.91%

Computing for the bond price with N = 9 years to maturity

Discount bond price for year 2 = $893.87

Input

9 10.91% 90.00 1000.00

N I│YR PV PMT FV

($893.87)

Output

Computing for the capital gain for the year for Year 2 and Year 1:

Capital Gain = $893.87 - $887.0 = $6.87

Computing for the following:

Current yield = PMT / Discount Bond Price at Year 1


= $90 / $887.00

= 0.1015

Current yield = 10.15%

Capital gains yield = Capital Gain between Year 2 and Year 1 / Discount Bond Price at

Year 1

Capital gains yield = $6.87 / $887.00 = 0.0077 = 0.77%

Total rate of return, or yield = Current yield + Capital gains yield

Total rate of return, or yield = 0.1015 + 0.0077 = 0.1092

Total rate of return, or yield = 10.15% + 0.77% = 10.92%

Or

Total rate of return, or yield = PMT + Capital gain / Discount bond price

Total rate of return, or yield = ($90 + $6.87) / $877.00 = $96.87 / $887.00 = 10.92%

b) Computing for the total return, current yield, and the capital gains for the discount

bond of $887.00 at maturity which is year 10 where N = 0

Discounted bond price at maturity = $1,000

Input

0 10.91% 90.00 1000.00

N I│YR PV PMT FV

($1,000.00)

Output


Capital Gain at maturity = $1,000.00 - $887.00 = $113.00


Current yield = PMT / Discount Bond Price at Year 1

= $90 / $887.00


= 0.1015


Capital gains yield = Capital Gain between Year 10 and Year 1 / Discount Bond

Price at Year 1

= ($1,000.00 ─ $887.00) / $887.00

Capital gains yield = $113.00 / $887.00

= 0.1274

Capital gains yield = 12.74%


Total rate of return, or yield = 0.1015 + 0.1274 = 0.2289

Total rate of return, or yield = 10.15% + 12.74%

Total rate of return, or yield = 22.89%

Or


Total rate or return, or yield = ($90.00 + $113.00) / $887.00

= $203.00 / $887.00

Total rate of return, or yield = 22.89%

2. Calculations for the Premium bond of $1,134.20

a) Calculating the total rate of return, the current yield, and the capital gains yield with the

premium bond price of $ 1,134.20:

Variables: N = 10; Annual coupon rate = 9%; YTM or rd = 7.08%; Par value = $1,000

Computing for the premium bond price in Year 2 where N = 9

Premium bond price in Year 2 = $1,124.67


Input

9 7.08% 90.00 1000.00

N I│YR PV PMT FV

($1,124.67)

Output


Capital Gain = $ $1,124.67 ─ $1,134.20 = ─ $9.53


Current yield = PMT / Premium Bond Price at Year 1

= $90 / $1,134.20

= 0.0794


Capital gains yield = Capital Gain between Year 2 and Year 1 / Premium Bond Price at

Year 1

Capital gains yield = ─$9.53 / $1,134.20

= ─ 0.01

Capital gains yield = ─ 1.00%


Total rate of return, or yield = 0.0794 + (─ 0.01) = 0.07

Total rate of return, or yield = 7%

Or


Total rate or return, or yield = [$90 + (─ $9.53)] / $1,134.20

= $80.47 / $1,134.20

= 0.07


Total rate of return, or yield = 7%

b) Computing for the total return, current yield, and the capital gains for the premium

bond of $1,134.20 at its maturity in year 10 where N = 0

Premium bond price in Year 10 or at its maturity = $1,000.00

Input

0 7.08% 90.00 1000.00

N I│YR PV PMT FV

($1,000.00)

Output

Computing for the capital gain at maturity which is Year 10 and Year 1:

Capital Gain = $ $1,000 ─ $1,134.20 = ─ $134.20


Current yield = PMT / Premium Bond Price at Year 1

= $90 / $1,134.20

= 0.0794


Capital gains yield = Capital Gain between Year 10 and Year 1 / Premium Bond Price at

Year 1

= $1,000 ─ $1,134.20 / $1,134.20

Capital gains yield = ─$134.20 / $1,134.20

= ─0.1183

= ─ 11.83%


Total rate of return, or yield = 0.0794 + (─ 0.1183)

= ─ 0.0389


Total rate of return, or yield = ─ 3.89%

Or


Total rate or return, or yield = [$90 + (─ $134.20)] / $1,134.20

= ─ $44.20/ $1,134.20

= ─0.0389

Total rate of return, or yield = ─ 3.89%


References

Brigham, E., & Ehrhardt, M. (2014). Financial Management (14th ed.). Mason, OH: Cengage

Learning

Microsoft Excel as a Financial Calculator Part III (n.d.). Time Value of Money & Financial

Calculator Tutorials.com. Retrieved from http://www.tvmcalcs.com/

calculators/excel_tvm_functions/excel_tvm_functions_page3

the time value of money · strayer university. fin 534 – the time value of money 2 abstract the...

Documents