FIN Chapter 5 The Time Value of Money

Interest compounded quarterly

FVn = PVo (1 + i nom / 4)^4n

the relationship between compound value and present value can be shown by rewriting to solve for PVo

FVn = PVo(1+i)^n. or PVo = FVn [ 1/(1+i)^n]. where 1/(1+i)n is the reciprocal of the compound value factor.

Compound Interest Equation

FVn=PVo(1+i)^n Where PVo is the initial deposit, i, is the annual interest rate, n is the number of years and FVn is the future value that will accumulate from the annual compounding of PVo.

FVn

Future value of an investment: denotes the principal plus interest accumulated at the end of n years. FVn = PVo + I $1000 loan for 9 months at a rate of 8 percent per annum. How much will he have to repay at the end of the 9 month period? (combining the previous equations above) FVn = PVo + (PVo x i x n) FV3/4 = $1000 + (1000 x 0.08 x 3/4) = $1060.00 How about an investment? $1,000 venture at 10% simple interest each year for two (2) years. How much money will she have at the end of the second year? Assuming two 10 percent simple interest payments, the future value would be: FV2 = PVo + (Pvo x i x 2) = $1,000 + ($1,000 x 0.10 x 2) = $1,200.00

Present Value of $1,000 received 20 years in the future discounted at 10%

PVo = FV20(PVIF0.10,20) = $1,000(0.149) = $149

Capital Recovery Problem

Present value of an annuity interest factors can also be used to find the annuity amount necessary to recover a capital investment, given a required rate of return on that investment.

Loan Amortization and Capital Recovery Problems

Present value of an annuity interest factors can be used to solve loan amortization problem, where the objective is to determine the payments necessary to pay off, or amortize, a loan, such as a mortgage.

An offer to pay you $255.20 in five years if you deposit X dollars today at an annual 5 percent interest rate.

Pvo = FV5 (1/FVIG0.05,5) = $255.20 (1/1.276) = $200

Present Value of Deferred Annuities

Suppose that you wish to provide for the college education of your child. They will start college 5 years from now and yo wish to have $15,000 available for her at the beginning of each year in college. How much must be invested today at a 12% annual rate of return in order to provide the 4 year, $15,000 annuity? See attached photo. The initial Present Value of Deferred Annuity = $28,950.00

Compounding Periods and Effective Interest Rates

The frequency with which interest rates are compounded (annually, semiannually, quarterly, and so on) affects both the present and future values of cash flows as well as the effective interest rates being earned or charged.

Term of the loan

The length of time over which the loan can be repaid

Discounting

The process of finding the present value of a cash flow or a series of cash flows; discounting is the reverse of compounding.

Solving for Interest Rate

When the PVo and FVn are given. $1,000 promising a $1,629 return after 10 years FVIFi,10 = FV10/PVo = $1,629/$1,000 = 1.629 When reading across a 10 year row in a table, 1.629 is found in the 5 percent column, thereby the investment yields a 5 percent compound rate of return.

Future Value of an Annuity Due

a deposit of $1,000 at the beginning of each year for the next 3 years, earning 6% interest, compounded annually, what is the worth of the the account at the end of 3 years? PMT1 is compounded for 3 years, PMT2 for two years, and PMT3 for one year. by multiplying the FVIGA for 3 years and 6% (3.184) by 1 plus the interest rate (1+0.06). This yields a FVIFA for an annuity due of 3.375 and the future value of the annuity due (FVANDn) calculated as: FVANDn = PMT[FVIFAi,n(1+i)] FVAND3 = $1,000(3.375) = $3,375

future value interest factor (FVIF)

a factor multiplied by today's savings to determine how the savings will accumulate over time

Perpetuity

a financial instrument that promises to pay an equal cash flow per period forever: that is , an infinite series of payments. "an infinite annuity"

annuity

a recurring payment, at set intervals, for a given amount of time. a payment or receipt of equal cash flows per period for a specified amount of time

An annuity is a series of equal payments that occur over a number of time periods.

a. ordinary annuity payments are made at the end of each period. b. Annuity due payments are made at the beginning of each period.

annuity due

an annuity whose payments occur at the beginning of each period. earns one more interest cycle. ex: of annuities due are life insurance premiums and apartment rentals.

n

denotes the number of periods

annual effective interest rate

designated as i eff

annual nominal interest rate

designated i nom

Present Value Calculations

determine the value today (present value) of some amount to be received in the future.

The future value of an annuity calculation

determines the future value of an annuity stream of payments

i

interest rate

As a general rule, lowercase letters are used to denote

percentage rates and lengths of time

The present value of an annuity calculation determines the

present value of an annuity stream of payments

Nominal Interest Rate

rates you see in the market, i.e. bond rate, bank rate, etc.

Discounting con'td

reducing the value of future cash flows to their present value.

PVIFA tables to simplify calculations

see book tables

T

tax rate

Principal

the amount of money borrowed or invested.

Present Value and Discounting

the current value for a future amount based on a certain interest rate and a certain time period

The higher the discount rate

the lower the present value

rate of interest

the percentage on the principal that the borrower pays the lender per time period as compensation for forgoing other investment or consumption opportunities.

Interest

the return earned by or the amount paid to someone who has forgone current consumption or alternative investment opportunities and "rented"money in a creditor relationship.

Loan Amortization example

$10,000 loan. Period is for 4 years at an interest rate of 9%. Installments must be made in 4 equal, annual, end of year payments that include both principal and interest on the outstanding balance. n = 4, PVANo = $10,000, i = 0.09 $10,000 = PMT(PVIFA0.09,4) 10,000 = PMT(3.240) PMT = 3,086.42

Time Value of Money Problems may be solved using

1. mathematical formulas 2. special interest factor tables provided at the end of the text book. 3. financial calculators 4. spreadsheet software

Present Value of an Annuity Due

5 year annuity of $1,000 each year, discounted at 6%. What is the present value of this annuity if each payment is received at the beginning of each year? The first payment received at the beginning of year 1 (end of year 0) is already in its present value form and therefore requires no discounting. PMT2 is discounted for one period, PMT3 is discounted for two periods, PMT4, is discounted for three periods, PMT5 is discounted for four periods. the correct annuity due interest factor for this problem may be found in the table, by mulitiplying the present value of an ordinary annuity interest factor for 5 years and 6% by 1 plus the interest rate (1+0.06). this yields a PVIFA for an annuity due of 4.465. The present value of this annuity due (PVANDo) is calculated as: PVANDo = PMT[PVIFAi,n(1+i)] PVANDo = $1,000 (4.465) = $4,465

Compound Interest Formula

A = P(1 + r/n)^(n x t), r is the rate, n is the number of times compounded, t is time

sinking fund problem

An annuity amount that must be invested each period (year) to produce a future value. End goal $5million 9.5% return per annum over 5 year period. What needs to be the initial investment/deposit? n = 5, FVAN5 = $5,000,000 and i = 0.095 yields: $5,000,000 = PMT(FVIFA0.095,5) $5,000,000 = PMT ((1+0.095)^5-1/0.095) PMT = $827,182

ordinary annuity

An annuity that pays at the end of each period.

Present Value of an Uneven Payment Stream

Consider an investment that is expected to produce a series of unequal payments (cash flows) over the next n periods. The present value of this uneven payment stream is equal to the sum of the present values of the individual payments. Algebraically, the present value can be represented as: PVo = PMT1/(1+i) + PMT2/(1+i)^2 + PMT3/(1+i)^3 and so on, or using summation notation, as: PVo = n SUM t=1 x PMTt / (1+i)^t = n SUM t=1 x PMTt(PVIFi,t) where i is the interest rate (required rate of return) and PVIFi,t is the appropriate interset factor from table II. Payments can be either positive (inflows) or negative (outflows). see attached photo.

A general formula for computing future values can be developed by combining the equations above

FV2 = PVo(1+i)(1+i). or FV2 = PVo(1+i)^2

Future Value of an Ordinary Annuity by multiplying the annuity payment

FVANn = PMT(FVIFAi,n) = $1,000 (3.184) = $3,184

FVIF formula

FVIF i,n = (1 + i)^n. May also be written as: FVn = PVo(FVIFi,n). where i = the nominal interest rate per period and n = the number of periods.

Future Value of an Ordinary Annuity Formula

FVIFA i,n = (1 + i)^n - 1 / i

Compound interest for any number of periods during a year may be computed by means of the following equation

FVn = PVo (1 + i nom / m)^mn m is the number of times during the year the interest is compounded and n is the number of years.

Calculating interest compounded semiannually

FVn = PVo(1 + i nom/2)^2n

Future value at then end of period n for any payment compounded at interest rate i

FVn = PVo(1+i)^n

Table 5.7 contains a number of present values, PVo, for $1000 received one year in the future discounted at a nominal interest rate of 10 percent with several different compounding frequencies.

For example, the present value (PVo) of $1000 compounded quarterly (m = 4) at a nominal interest rate (i nom) of 10 percent per year is: PVo = $1000 / (1 + 0.10/4) ^4x1 $905.95

FV

Future value

Simple Interest Formula

I = prt (Interest = Principal X Rate X Time)

Effective Interest

In contrast to the nominal interest, is the actual rate of interest earned by the lender and is generally the most economically relevant definition of interest rates.

Nominal and real interest rates

Nominal not adj for inflation. Real interest rate is adj and is a measure of the purchasing power of interest earned or paid= Nominal rate-Inflation rate Nominal interest rates and inflation move together.

Ordinary Annuity vs. Annuity Due

Ordinary = in arrears (start later) Annuity due = start now (-1 = ordinary) Ordinary Annuity makes interest payments at the end of the interval Annuity Due makes interest payments at the start of the interval, therefore the last annuity payment earns interest.

The Present Value of an Annuity can be determined by multiplying the annuity payment, PMT, by the appropriate interest factor, PVIFAi,n

PVANo = PMT(PVIFAi,n) With the appropriate table reference to determine the interest factor for i = 6%, and n = 5, the present value of an annuity can be calculated as: PVANo = PMT(PVIFA0.06,5) = $1,000(4.212) =$4212

PVIFAs can also be computed as:

PVIFAi,n = 1 - 1/(1+i)^n / i

Because determining the reciprocals of the compound value interest factors, 1/(1+i)^n, can be tedious, present value interest factors (PVIFs) are used to simplify.

PVIFi,n = 1 / (1+i)^n and can be written in the following form: PVo = FVn(PVIF1,n)

For example, assume an energy company series E preferred stock promises payments of $4.50 per year forever and that an investor requires a 10% rate of return on this type of investment. How much would the investor be willing to pay for this security? In this example, the value of a $4.50 perpetuity at a 10% required rate of return is:

PVPERo = $4.50/0.10 = $45

Example: a financial instrument that promises to pay an infinite stream of equal, annual payments of PMTt = PMT where t = 1,2,3....

PVPERo = Infinity Sum over t=1 x PMT/(1+i)^t. See attached photo.

An exmination of the PVIFA interest factors for 10 percent (in the table IV) indicates thta the value in the 10 percent column increases as the number of years increases, but at a decreasing rate. For example, the PVIFA factor for 10 percent and 10 years is 6.145, whereas the factor for 10 percent and 20 years is only 8.514 (much less than twice the 10 year factor). The limiting value in any column of Table IV is 1 divided by the interest rate of that column, i. In the case of a 10 percent perpetuity, the appropriate interest factor is 1/0.10, or 10. Thus the equation can be rewritten as:

PVPERo=PMT/i

The present value of a sum to be received at the end of year n, discounted at the rate of i nom percent and compounded m times per year, is:

PVo- FVn / (1+i nom / m)^mn

PMT

Payments

Additional Present Value types

Perpetuities, uneven cash flows and deferred annuities.

PV

Present Value

Solving for Interest and Growth Rates

Present value interest factors can also be used to solve for interest rates. If you borrow $5,000 today with a promise to pay back $6,250, 4 years from now. PVo = FV4(PVIFi,4) $5000 = $6250(PVIFi,4) (PVIFi,4) = $5,000 / $6250 = 0.800 When one read across the 4 year row of the Table, 0.800 is found between the 5 percent (0.823) and 6 percent (0.792) columns. Interpolating between these two values yields: i = 5% + 0.823 - 0.800 / 0.823 - 0.792 x (1%) = 5.74% thus the effective interest rate on the loan is 5.74% per year, compounded annually.

Solving for the Interest Rate

Present value of an annuity interest factors can also be used to solve for the rate of return expected from an investment. PVANo = PMT(PVIFAi,5) $100,000 = $26,378(PVIFAi,5) PVIFAi,5 = 3.791 By reviewing the table we see that 3.791 occurs in the 10% column

Solving for the Number of Compounding Periods

The future value interest factor tables can also be used to determine the number of annual compounding periods (n). How long it would take for $1,000 invested at 8 % to double, search the 8 percent column in the table to locate a future value interest factor of 2.000. The closes value to this figure is 1.999. Reading to the left of this figure, it can be seen that the original $1,000 would be worth nearly $2,000 in 9 years. FVn = PVo(FVIF0.08,n) FVIF0.08,n = FVn/PVo = $2,000 / $1,000 = 2.00

Net Present Value (NPV)

an investment is equal to the present value of the expected future cash flows generated by the investment minus the initial outlay of cash. NPV = Present value of future cash flows minus initial outlay

The future cash flows are

discounted to the present at a required rate of return that reflects the perceived risk of the investment.

The dollar in the future is worth less than a dollar you receive today

due to uncertainty

Another common present value application is the calculation of the compound rate of growth of an earnings or a dividend stream.

earnings of $4.01 per share with an expected growth to $5.63 in 5 years. over the 5 year period, what is the expected compounded annual rate of growth in earnings? $4.01 = $5.63(PVIFi,5) PVIFi,5 = 0.712 From the table we find this approximate present value interest factor in the 5 year row under the 7 percent interest, or growth rate, column. The forecast compound annual rate of growth is said to be 7%.

Interest rate may be thought of as a

growth rate as the higher the compound interest rate, the faster the growth rate of the value of the initial principal.

In some circumstances, interest on an account is compounded semiannually instead of annually

half the nominal annual interest rate, i nom / 2 is earned at the end of six months. The investor earns additional interest on the interest earned before the end of the year or (i nom / 2)PVo.

Given the annual nominal rate of interest (i nom), the effective annual rate of interest (i eff) can be calculated as follows:

i eff = (1 + i nom / m)^m - 1 where m is the number of compounding intervals per year.

In general, the rate of interest per period (where there is more than one compounding period per year), i m, which will result in an effective annual rate of interest, i eff, if compounding occurs m times per year, can be computed as:

i m = (1 + i eff) ^1/m -1

compound interest

interest earned on both the principal amount and any interest already earned but not withdrawn during earlier periods

simple interest

interest paid on the principal alone

Present Value

mirror image of future value - the value today of a cash flow or a series of cash flows to be received in the future.

As a general rule, uppercase letters are used to denote

money or dollar amounts.

Five Basic Keys which re utilized for solving time value of money problems on a financial calculator

n, i, PV, PMT, FV

Effect of Compounding Periods on Present and Future Values

no definition

Real Interest Rate

nominal interest rate - inflation rate

the more frequently an annual nominal rate of interest is compounded,

the greater is the effective rate of interest being earned or charged. thus, if you were given the choice of receiving (1) interest on an investment, where the interest is compounded annually at a 10% rate, or (2) interest on an investment, where the interest is compounded semiannually at a 5% rate every 6 months, you would choose the second option.

The more frequent the compounding, the greater the future value of the deposit

the greater the effective interest rate.

Compound Interest

the interest paid not only on the principal but also on any interest earned but not withdrawn during earlier periods. ex: $1,000 deposit in savings account paying 6 percent interest compounded annually, the future (compound) value at the end of one year (FV1) is: FV1 = PVo (1 + i) = $1,000 (1 + 0.06) = $1,060.00 If left in the account for another year the accumulated interest in the account for another year its second year calculation would be: FV2 = FV1 (1 + i) = $1,060 (1 + 0.06) = $1,123.60 If for a 3rd year FV3 = FV2 (1 +i) = $1,123.60 (1+0.06) = $1,191.02

Simple Interest

the interest paid or earned on the principal only. The amount of simple interest is equal to the product of the principal times the rate per time period times the number of time periods. I = PVo x i x n I = the simple interest PVo = the principal amount at time 0, or the present value i = the interest rate per time period n = the number of time periods. What is the simple interest on $100 at 10 percent per annum for six months? PVo = $100, i = 10 percent or (0.10) n = 6/12 or (0.5) I = $100 x 0.10 x 0.5 = $5.00 How about a $30,000 at a 10 percent annual interest rate, what would be the first months interest payment? I = $30,000 x 0.10 x 1/12 = $250

Future Value of an Ordinary Annuity

the last cash payment will not earn any interest. If an amount PMT is deposited in an account at the end of each year for n years, and if the deposits earn an interest rate, i, compounded annually, what will be the value of the account at the end of n years? 3 year ordinary annuity of $1,000 per year and deposits the money in savings account at the end of each year. The account earns an interest rate of 6% compounded annually. What is the account worth at the end of the 3 year period? PMT = $1,000, i = 6%, n = 3 FV3rd = PMT3 (1 + 0.06)^0 = $1,000(1) =$1,000 Because its an ordinary annuity the last deposit, PMT3, will not earn interest. The second deposit, PMT2, made at the end of year 2, will be in the account for 1 year before the 3 year period. thus, FV2nd = PMT2 (1 + 0.0g)^1 = $1,000 (1.06) =$1,060 The first deposit, PMT1, made at the end of year 1, will be in the account earning interest for two full years before the end of the 3 year period. its future value is: FV1st = PMT1 (1 + 0.06)^2 = $1,000 (1.124) = $1,124 the sum of the three figures is the FV of the annuity: FVAN3 = FV3rd + FV2nd + FV1st = $1,000 + $1,060 + $1,124 = $3184 The Future Value of an Annuity Interest Factor (FVIFA) is the sum of the future value interest factors presented in the table. FVIFA0.06,3 = FVIF0.06,2 + FVIF0.06,1 + FVIF0.06,0 = 1.124 + 1.060 + 1.000 = 3.184

The net present value rule is

the primary decision making rule used throughout the practice of financial management. a. The net present value of an investment is equal to the present value of the future cash flows minus the initial delay. b. The net present value of an investment made by a firm represents the contribution of that investment to the value of the firm and, accordingly, to the wealth of shareholders.

Present Value of an Ordinary Annuity PVANo

the sum of the present value of a series of equal periodic payments. to find the PVAN ($1,000) received at the end of each year for 5 years discounted at a 6% rate, the sum of the individual present values would be: PVANo = $1,000(PVIF0.06,1) + $1,000(PVIF0.06,2) + $1000 (PVIF0.06,3) + $1,000(PVIF0.06,4) + $1,000(PVIF0.06,5) = $1,000 (0.943 + 0.890 + 0.840 + 0.792 + 0.747) = $4212

Future Value

the value at some point in the future of a deposit made today, or of a series of deposits to be made in the future.

the net present value of an investment made by a firm represents the contribution of that investment to

the value of the firm and accordingly to the wealth of shareholders