time value of money

.

......

Time Value of MoneyQTM1300: Quantitative Methods for Business

Dr. Ji Li

Babson College

November - December, Fall 2010

. . . . . .

. . . . . .

Simple Interest and Compound InterestSinking Funds, Annuities, and Bonds

More on FinanceNotations and Formulas

Simple InterestCompound Interest

.. Time Value of Money...1 Simple Interest and Compound Interest


...2 Sinking Funds, Annuities, and BondsSinking FundsAnnuitiesAmortization ScheduleMore Examples

...3 More on FinanceSinking Funds and Annuities: New FormulasPerpetuitiesNet Present Value

...4 Notations and FormulasNotationsSimple Interest and Compound InterestSinking Funds, Annuities, and PerpetuitiesNet Present Value

Dr. Ji Li Time Value of Money

. . . . . .




.. Simple Interest

An investment of PV dollars growing with simple interest rate ofr after t years is worth FV dollars:

FV = PV (1 + r t).


. . . . . .




.. Simple Interest


FV = PV (1 + r t).

.Example..

......

The Megabucks Corporation is issuing 10-year bonds payingan annual rate of 6.5%. If you buy $10,000 worth of bonds, howmuch interest will you earn every six months? How muchinterest will you earn over the life of the bonds?


. . . . . .




.. Simple Interest


FV = PV (1 + r t).

.Example..

......A stock fund costs $900 in July 2001 and sells for $892 in July2002. What is the annual percentage loss of this stock?


. . . . . .




.. Simple Interest


FV = PV (1 + r t).

.Example..

......

You hear the following on your local radio station’s business news:

The economy last year grew by 1%. This was the second yearin a row in which the economy showed a 1% growth.

Because the rate of growth was the same two years in a row, this representsa simple interest growth, right?


. . . . . .




.. Example

Suppose that $20, 000.00 is invested in a bank account. Assume that there isno other deposits or withdrawals. How much is in the account after 10 yearsif

(a) the bank pays 6% simple interest rate once a year?

(b) the bank pays 6% interest rate compounded annually?

(c) the bank pays 6% interest rate compounded monthly?


. . . . . .




.. Example





.Answer to (a)..

......

The account earns20, 000 × 0.06 = $1, 200

interest every year. In 10 years, the account becomes

FV = PV (1 + rt) = 20, 000(1 + 0.06 × 10) = $32, 000.


. . . . . .




.. Example





.Answer to (b)..

......

In 10 years, the account becomes

FV = 20, 000 (1 + 0.06)10 = $35, 816.95.


. . . . . .




.. Example





.Answer to (c)..

......

In 10 years, the account becomes

FV = 20, 000(

1 +0.0612

)12×10

= $36, 387.93.


. . . . . .




.. Compound Interest

An investment of PV dollars earning interest at an annual rateof r compounded (reinvested) m times per year for a period of tyears is worth FV dollars:

FV = PV(

1 +rm

)mt.


. . . . . .






FV = PV(

1 +rm

)mt.

.Example..

......

Determine the amount of money you must invest at 5% peryear, compounded monthly, so that you will be a millionaire in30 years.


. . . . . .






FV = PV(

1 +rm

)mt.

.Effective Rate (APY)..

......

You deposit $100,000 in an account earning interest of 5%compounded annually. What is the APY (Annual PercentageYield) of your account?


. . . . . .






FV = PV(

1 +rm

)mt.

.Constant Dollars..

......

You deposit $100,000 in an account earning interest of 5%compounded annually. What is the APY (Annual PercentageYield) of your account?

Suppose also that inflation is running 3% when you make thedeposit. How much money will you have two years from now?


. . . . . .






FV = PV(

1 +rm

)mt.

.Bonds Part I..

......How much do you have to pay for a 20-year zero coupon bondwith maturity value of $100,000 and a yield of 5.65% annually?


. . . . . .






FV = PV(

1 +rm

)mt.

.Bonds Part II..

......

How much do you have to pay for a 20-year zero coupon bond with maturityvalue of $100,000 and a yield of 5.65% annually?Once purchased, bonds can be sold in the secondary market. How muchmoney would you have received if you sold your bond 5 years before maturityto an investor looking for a return of 5% annually?


. . . . . .






FV = PV(

1 +rm

)mt.

.Bonds Part III..

......

How much do you have to pay for a 20-year zero coupon bond with maturityvalue of $100,000 and a yield of 5.65% annually?Once purchased, bonds can be sold in the secondary market. How muchmoney would you have received if you sold your bond 5 years before maturityto an investor looking for a return of 5% annually?What is your annual yield on your 15-year investment?


. . . . . .



Sinking FundsAnnuitiesAmortization ScheduleMore Examples







. . . . . .




.. Example

Suppose you make a deposit of $1000 at the end of everymonth into an account earning 5% interest per year,compounded monthly. What will be the value of the investmentat the end of 30 years?


. . . . . .




.. Sinking Funds

A sinking fund is worth FV dollars if you make a payment of PMT atthe end of each compounding period into an account earning interestat an annual rate of r compounded (reinvested) m times per year fort years:

FV = PMT(1 + r/m)mt − 1

r/m.


. . . . . .




.. Sinking Funds


FV = PMT(1 + r/m)mt − 1

r/m.

.Example..

......

At the end of each month you deposit $100 into an account earning3% annual rate compounded monthly. How much is the accountworth in the end of one year?


. . . . . .




.. Sinking Funds


FV = PMT(1 + r/m)mt − 1

r/m.

.Retirement Account..

......

Your retirement account has $10,000 in it and ears 5% interest peryear compounded monthly. Every month for the next 20 years you willdeposit $500 into the account. How much money will be there at theend of those 20 years?


. . . . . .




.. Sinking Funds


FV = PMT(1 + r/m)mt − 1

r/m.

.Example..

......

If $2,000 is deposited in an account at the end of each year for thenext 12 years, how much will be in the account at the time of the finaldeposit if interest is 5% compounded annually?


. . . . . .




.. Example

Suppose you deposit an amount PV now in an account earning5% interest per year, compounded monthly. Starting 1 monthfrom now, the bank will send you monthly payments of $5,000.What must PV be so that the account will be drawn down to $0in exactly 10 years?


. . . . . .




.. Annuities

An annuity is an account earning compound interest from whichperiodic withdrawals are made. If the starting balance is PV dollars,you receive a payment of PMT at the end of each compoundingperiod, and the account is down to $0 after for t years, then

PV = PMT1 − (1 + r/m)−mt

r/m.


. . . . . .




.. Annuities


PV = PMT1 − (1 + r/m)−mt

r/m.

.Example..

......

At the end of each month you want to withdraw $100 from an accountearning 3% annual rate compounded monthly. How much is it worthright now if you want the account to last for 5 years?


. . . . . .




.. Annuities


PV = PMT1 − (1 + r/m)−mt

r/m.

.Car Loan Part I..

......

Mira bought a car worth $30,000 with an initial payment of $6,000.How much does she have to pay in the end of each month for the5-year car loan with an interest rate of 4% compounded monthly?


. . . . . .




.. Annuities


PV = PMT1 − (1 + r/m)−mt

r/m.

.Car Loan Part II..

......

Mira bought a car worth $30,000 with an initial payment of $6,000 ona 5-year car loan with an interest rate of 4% compounded monthly.After making monthly payments over 3 years, she decided to end theloan earlier. How much does she have to pay in her last payment?


. . . . . .




.. Amortization Schedule

Mira bought a car worth $30,000 with an initial payment of$6,000 on a 5-year car loan with an interest rate of 4%compounded monthly.

How much interest does she have to pay in the end of thefirst month?How much outstanding principal is left after Mira makes thefirst payment?How much interest does she have to pay in the end of thesecond month?How much interest does Mira have to pay in total?


. . . . . .




.. Example

Find the monthly payment on the mortgage if you are buying a$300,000 apartment with a down payment of $60,000 for 30years at 9% interest rate compounded monthly.

Find the total amount you will pay in interest.

Produce an amortization schedule for the first 12 payments.


. . . . . .




.. Bonds

Suppose that a corporation offers a 20-year bond paying coupon interestrate 4.5% with semiannual coupons. If someone pays $5,000 for bonds witha maturity value of $5,000, he will receive a coupon every 6 months for 20years for the interest. At the end of the 20 years, he will get the $5,000 back.

How much is each coupon worth?

Think of the bonds as an investment that will pay the owner a certainamount every 6 months for 20 years, at the end of which it will pay$5,000. How much will a bond trader be willing to pay for the bond ifhe’s looking for a yield (also called “rate of return”) of 7%?

Another trader is looking for 6% yield on her investment. How much willshe pay for the same bond?


. . . . . .




.. Multi-Step Example

You wish to provide yourself with an income of $5,000 every 6months, starting 15 and a half years from now and ending 35years from now.

You deposit $25,000 in the account now, and a guaranteedinheritance of $10,000 which you will receive 10 years fromnow.

You know that these sums will not provide the income you want,so you plan to make periodic deposits to the account at the endof every 6 months for 15 years to make up the difference.

How much should the periodic deposits be if all interest iscomputed at 6% compounded semiannually?


. . . . . .




.. The Effect of Inflation

(Turner Example 4, Finance.xlsx) An entrepreneur borrows $10,000 underthe following terms:

Loan interest rate: 12%

Term: 6 years

Payment schedule: Monthly

Determine the cost of the loan in today’s dollars if inflation average 5% overthe term of the loan.

There are two steps involved to solve this problem:

Step 1: Find the monthly payment PMT by ignoring the inflation rate.

Step 2: Find the present value using the PMT found in an annuitysituation.


. . . . . .



Sinking Funds and Annuities: New FormulasPerpetuitiesNet Present Value







. . . . . .




.. Example on Annuities

Sally has just received a loan to finance the purchase of apretty blue convertible. The amount of the loan is $25,000.

Sally is required to transfer to the lending institution a fixedamount at the end of each month starting at the end of the firstmonth after she receives her loan.

The interest rate on the loan is 9% compounded daily, and theterm of the loan is three years.

What will be Sally’s monthly payment?


. . . . . .




.. Sinking Funds and Annuities — New Formulas

Suppose that the number of compounding periods per year, m, is differentfrom the number of payments per year, ppy . Then the interest rate perpayment period becomes

j =(

1 +rm

)m/ppy

− 1

and the sinking fund and annuity formulas become

Sinking Fund FV = PMT((1 + j)ppy·t − 1

j

),

Annuity PV = PMT(

1 − (1 + j)−ppy·t

j

),

In calculator, input the following:

N = ppy · t

I % = j %


. . . . . .




.. Examples on New Formulas

Sinking Fund FV = PMT((1 + j)ppy ·t − 1

j

),

Annuity PV = PMT(

1 − (1 + j)−ppy ·t

j

),


. . . . . .






j

),

Annuity PV = PMT(

1 − (1 + j)−ppy ·t

j

),

.Example..

......

Mira bought a car worth $30,000 with an initial payment of$6,000. How much does she have to pay in the end of eachmonth for the 5-year car loan with an interest rate of 4%compounded daily?


. . . . . .






j

),

Annuity PV = PMT(

1 − (1 + j)−ppy ·t

j

),

.Example..

......

Find the monthly payment on the mortgage if you are buying a$300,000 apartment with a down payment of $60,000 for 30years at 9% interest rate compounded daily.


. . . . . .




.. Perpetuities: Example 1

(Turner Example 5)Determine the initial deposit that must be placed in an account that bears 5%interest compounded monthly in order to withdraw $1,000 every month forever.

Now in the annuity formula

PV = PMT1 − (1 + r/m)−mt

r/m,

by setting t −→ ∞, the perpetuity formula follows

PV = PMT1

r/m.


. . . . . .





Determine the initial deposit that must be placed in an account that bears5.49% interest compounded monthly in order to withdraw $10,000 every 6months forever, starting a month from now.

j =(

1 +rm

)m/ppy

− 1 PV = PMT(

1 − (1 + j)−ppy·t

j

)

Setting t −→ ∞, the perpetuity formula is

PV = PMT1j=

PMT(1 + r

m

)m/ppy

− 1


. . . . . .





PV = PMT1j=

PMT(1 + r

m

)m/ppy

− 1

.Example..

......

Determine the initial deposit that must be placed in an accountthat bears 2.75% interest compounded daily in order towithdraw $8,000 quarterly forever, starting 3 months from now.


. . . . . .




.. Issuing a Bond

(Turner Example 6, Finance.xlsx) A company floats a $10,000,000 bondissue with a 20 year term. The interest rate on the bond is 3%. How much iseach bond interest payment (to the bond holders) made semiannually?

The company has set up a sinking fund for the accumulation and dispersionof all funds related to the bond issue and wants to make equal quarterlypayments to the fund. Note that the fund will be used both to pay the bondinterest due each six months and the bond face value of $10,000,000 twentyyears hence. Determine the minimum quarterly payment to the fund thatwould meet the needs of the company if the interest on the fund is 8%compounded quarterly.


. . . . . .




.. Net Present Value

Assume r is the APR and m is the number of compounding periods per year.The net present value NPV of a sequence of equally-spaced end-of-periodpayments (negative) and income (positive) with same compounding andpayment/income periods, with an initial cash flow v0 and the payment/incomesequence v1, v2, . . . , vn is

NPV = v0 +v1

1 + r/m+

v2

(1 + r/m)2 + · · ·+ vn

(1 + r/m)n .


. . . . . .






NPV = v0 +v1

1 + r/m+

v2

(1 + r/m)2 + · · ·+ vn

(1 + r/m)n .

.Example 1..

......

Suppose you are considering an investment in which you pay $10,000 oneyear from today and receive an annual income of $3,000, $4,200, and $6,800at the end of the three years that follow, respectively. Assuming an annualinterest rate of 10%, what is the net present value of this investment?


. . . . . .






NPV = v0 +v1

1 + r/m+

v2

(1 + r/m)2 + · · ·+ vn

(1 + r/m)n .

.Example 1: Continue..

......

Suppose you are considering an investment in which you pay $10,000 oneyear from today and receive an annual income of $3,000, $4,200, and $6,800at the end of the three years that follow, respectively.Determine the Internal Rate of Return, or the interest rate per period whichwould provide a net present value of $0, after all four years.


. . . . . .




.. NPV: Example 2

You run a takeout pizza business and believe you could improve your returnby buying a van to add delivery service.

You can buy a van today for $15,000 and belive you will use it for five yearsand then sell it for $5,000. After expenses, you estimate that your businesswill make $4,000 annually by adding the delivery service, starting a year fromnow.

You could make 7.5% by investing in a U.S. Treasure bill over five years, butyou decide that the added risks of the pizza delivery business mean youshould earn at least twice that rate. So you set the annual interest rate at15%.

Should you buy the van?


. . . . . .



NotationsSimple Interest and Compound InterestSinking Funds, Annuities, and PerpetuitiesNet Present Value







. . . . . .




.. Notations

PV — Present Value

FV— Future Value

r — Nominal Rate (also called APR)

t — Number of years

m — Number of compounding periods per year

n = mt — Total number of compounding periods

iper = r/m — Interest rate per compounding period

PMT — The amount of payment

ppy — Number of payments per year in an ordinary annuity

j =(

1+rm

)m/ppy

−1 — Interest rate per payment period for an annuity

NPV — Net Present Value


. . . . . .




.. Simple Interest


FV = PV (1 + r t).


. . . . . .






FV = PV(

1 +rm

)mt.


. . . . . .




.. Sinking Funds

A sinking fund is worth FV dollars if you make a payment ofPMT at the end of each compounding period into an accountearning interest at an annual rate of r compounded(reinvested) m times per year for t years:

FV = PMT(1 + r/m)mt − 1

r/m.


. . . . . .




.. Annuities

An annuity is an account earning compound interest fromwhich periodic withdrawals are made. If the starting balance isPV dollars, you receive a payment of PMT at the end of eachcompounding period, and the account is down to $0 after for tyears, then

PV = PMT1 − (1 + r/m)−mt

r/m.


. . . . . .




.. New Formulars for Sinking Funds and Annuities

Suppose that the number of compounding periods per year, m,is different from the number of payments per year, ppy . Thenthe interest rate per payment period becomes

j =(

1 +rm

)m/ppy

− 1

and the sinking fund and annuity formulas become


j

),

Annuity PV = PMT(

1 − (1 + j)−ppy ·t

j

),


. . . . . .




.. Perpetuities

A perpetuity is basically an annuity that lasts for ever.

PV = PMT1

r/m.

If the compounding periods is not the same as paymentperiods, then we use the following formula

PV = PMT1j=

PMT(1 + r

m

)m/ppy

− 1


. . . . . .





Assume r is the APR and m is the number of compoundingperiods per year. The net present value NPV of a sequence ofequally-spaced end-of-period payments (negative) and income(positive) with same compounding and payment/incomeperiods, with an initial cash flow v0 and the payment/incomesequence v1, v2, . . . , vn is

NPV = v0 +v1

1 + r/m+

v2

(1 + r/m)2 + · · ·+ vn

(1 + r/m)n .
