Chapter 5

Question 1

Using the time line:

Answer 1

you can find one variable if you have 4 others in the group (PV,r,n,FV).

FV = PV(1 + r)t
FV = future value
PV = present value
r = period interest rate, expressed as a decimal
t = number of periods

But, notice that we so far we dealt only with single cash flows…

Question 2

Introduction

Answer 2

Many business situations involve multiple cash flows

Annuity problems deal with regular, evenly-spaced cash flows

Car loans and home mortgage loans
Saving for retirement
Companies paying interest on debt
Companies paying dividends

Question 3

Ordinary Annuity: Future Value

You deposit $100 at the end of each year in a bank account offering 5% interest rate. How much will you have at the end of 3 years? Question: FV of ordinary annuity of PMT=$100 for n=3, r=5%

Answer 3

FVA3=PMT+PMT(1+r)^1+PMT(1+r)^2

FVA=100+100(1+).05)^1+1+0.05

Question 4

Ordinary Annuity: Present Value

What lump sum payment today would be equivalent to having a 3-year ordinary annuity of $100? You earn 5% in your investments. Question: PV of ordinary annuity of PMT=$100 for n=3, r=5%

Answer 4

PVA3=PMT/(1+r)+PMT/(1+r)^2+PMT/(1+r)^3

PVA3=100/(1.05)+100/(1.05)^2+100/(1.05)^3

Question 5

Ordinary Annuity: Formulas

Answer 5

Present Value:
PVA= Sigma(PMT/1+r)^t
PVA=PMT[1/r-1/r*(1+r)^n]

Future Value:
FVA=PMT[Sigma(1+r)^n-t]
FVA=PMT*[(1+r)^n-1/r]

Question 6

Suppose you want to retire at 60 years old with $1 million at the bank. You are told you can earn 10% per year. You are 20 years old now, and plan to invest the same amount at the end of each year. What is this amount?

Answer 6

Let’s solve it:
The inputs are PV=0, FV=1,000,000, r=10%, n=40 and what I need is PMT

FVA=PMT[(1+r)^n-1/r]
1,000,000=PMT[(1.01)^40-1/0.1]
PMT=2,259.41

*Can use financial calculator

Question 7

Annuities and the Financial Calculator

Answer 7

In the previous chapter, the level payment button PMT was always set to zero
Now, for annuities, we can use the PMT key to input the annuity payment
Example: suppose that $100 deposits are made at the end of each year for five years. If interest rates are 8 percent per year, the future value of the annuity is:

Input: N=5; I/YR=8; PMT=-100
Output: FV=586.66

Question 8

Ordinary Annuity: Solving FV

You deposit $100 at the end of each year in a bank account offering 5% interest rate. How much will you have at the end?
Question: FV of ordinary annuity of PMT=$100 for n=3 years, r=5%

Answer 8

Inputs: N=3; I/YR=5; PV=0; PMT=-100
Output: FV=315.25

Question 9

Ordinary Annuity: Solving PV

What lump sum payment today would be equivalent to having a 3-year $100 ordinary annuity? You can earn 5% in your investments.
Question: PV of ordinary annuity of PMT=$100 for n=3 years, r=5%

Answer 9

Inputs: N=3; I/YR=5; PMT=100; FV=0
Output: PV=-272.32

Question 10

Special Annuity: Perpetuity

Answer 10

Perpetuity: a stream of equal payments expected to continue forever

Example: British bonds sold in 1815 promising to pay $100 per year in perpetuity

0–r—1(100)—-2(100)—-n(100)—–

Question 11

Special Annuity: Perpetuity Cont.

Answer 11

Luckily, computing the PV of a perpetuity is simple:
PV(Perpetuity)=PMT/r

That is, if your opportunity cost rate is 10% then the British bonds were worth:
0—r=10%—1(100)—2(100)—-n(100)—-

PV(Perpetuity)=100/0.10=1,000

Question 12

Perpetuity: Challenge

Answer 12

Can you derive the formula to compute the present value of a perpetuity?

Hint: derive first a general formula to compute the value of an ordinary annuity, and then see what happens as n (the number of periods) increase

PVA=PMT[1/3-1/r(1+r)^n]

Question 13

What if the Flows are Not Fixed?

What lump sum payment today would be equivalent to having the stream of payments below? You earn 5% in your investments

Answer 13

0--r=5%--1(100)--2(150)--3(280)
100*1/1.05=95.24
150*1/1.05^2=136.05
280*1/1.05^3=241.87
PVA3=473.16

*Methodologically, nothing changes; using the calculator, though, you won’t use the PMT feature.

Question 14

Applications: Example 2

You decide to finish your $1,000 debt with the credit card by paying the minimum amount of $20 each month. The interest rate on the credit card is 1.5% per month. How long will you need to pay off the debt?

Answer 14

Let’s solve it:
The inputs are PV=1,000, FV=0, PMT=-20, r=1.5 and what I need is n.
You can solve it algebraically, starting from:

PVA=PMT[1/r-1/r(1+r)^n]
1,000=20[1/0.015-1/0.015(1+0.015)^n]

Or you go directly to the calculator and find n=93.11. Thus, you need more than 93 months to liquidate the debt.

Question 15

Applications: Example 3

Tuition costs for your newborn will be $90,000 in 18 years. You plan to deposit $1,000 at the end of each year in your bank account. How much interest rate do you need in order to cover her tuition?

Answer 15

Let’s solve it:
The inputs are PV=0, FV=90,000, PMT=-1,000, n=18 and what I need is r.
Look at the algebra:
FVA=PMT*[(1+r)^n-1/r]
90,000=1,000*[(1+r)^18-1/r]

but you cannot solve using the formula!
It is trial and error, or use a financial calculator, or Excel. The result is r=16.65%

Question 16

Ordinary Annuities vs. Annuities Dues

Answer 16

So far we have worked problems where the payment occurs at the end of each period. This is called an ordinary annuity.

Sometimes, however, the annuity payments occur at the beginning of each period. These are called annuities due.

In calculating the future value of an annuity due, we recognize that the payments all occur one period sooner than for an ordinary annuity, and therefore earn an extra period of interest. We can adjust the FV as follows:

FVA>NN<*(1+i)

Question 17

Compounding

Answer 17

So far we have assumed that interest is compounded once a year. This is called annual compounding

But a bond usually pays interest semiannually; car loans require monthly payments, etc

Thus, sometimes we need to account for interest that compounds more often than the stated period

Question 18

Annual Compounding

You deposit $100 in a bank paying 5% interest rate. How much will you have at the end of two years?

Answer 18

0(-100)–r=5%–1—2(FV2=?)

FVn=PV(1+r)^n
FV2=100(1.05)^2=110.25

Question 19

Semiannual Compounding

You deposit $100 in a bank paying 5% interest rate, semiannually. How much will you have at the end of two years?

Answer 19

0(-100)—r=2.5%–1–2–3–4(FV4=?)
*6 month periods

Actual interest rate per period (periodic rate)
    rPER=stated rate/number of payments per year=5%/2=2.5%
Actual number of periods
    # years* periods per year=2*2=4

Question 20

Semiannual Compounding: Solution

You deposit $100 in a bank paying 5% interest rate, semiannually. How much will you have at the end of two years?

Answer 20

0(-100)—r=2.5%–1–2–3–4(FV4=?)
*6 month periods

FVn=PV(1+rPER)^n=PV(1+rNOM/2)^n
FV4=100(1.025)^4=110.38

Question 21

You deposit $100 in a bank paying 5% interest rate, quarterly. How much will you have at the end of two years?

Answer 21

0(-100)-1-2-3-4-5-6-7-8(FV8=?)
*3 month periods
r=1.25

FVn=PV(1+rPER)
FV8=100(1.0125)^8=110.45

Question 22

APR and Compounding

You deposit $100 in a bank paying 5% APR. How much will you have at the end of two years?

Answer 22

Annual Compounding:
FVn=PV(1+rPER)^n
FV2=100(1.05)^2
FV2=110.25

Semiannual:
FVn=PV(1+rNOM/2)^n
FV4=100(1.05/2)^4
FV4=110.38

Quarterly:
FVn=PV(1+rNOM/4)^n
FV8=100(1.05/4)^8
FV8=110.45

Question 23

Equivalent Annual Rate

What is the rate that, w^hen compounded annually, would be equivalent to 5% compounded quarterly?

Answer 23

FVn=PV(1+r)^n
110.45=100(1+r)^2
rEAR=5.094%

rEAR is the equivalent annual rate, or the effective rate

Question 24

Equivalent Annual Rate: Solving Algebraically

Answer 24

The equivalent annual rate rEAR is the rate that compounded annually will lead to the same FV as compounding the nominal rNOM through m periods per year

FV through the actual compounding
FVmn=PV(1+rNOM/m)^mn

FV through the annual compounding
FVn=PV(1+rEAR)^n

Set them eaqual and solve the equation to get:
rEAR=(1+rNOM/m)^m - 1

Question 25

Types of Interest Rates

Answer 25

Nominal, stated, quoted rate (rNOM): a rate that ignores compounding effects. Usually given as APR (Annual Percentage Rate)

Periodic rate (rPER) – amount of interest charged each period, e.g. monthly or quarterly
rPER=rNOM/m, where m is number of compounding periods per the “stated period”

Effective or equivalent rate: the actual rate of interest actually being earned, taking into account compounding. It is the EAR (equivalent annual rate) when the stated period is a year
rEAR=(1+rNOM/m)m-1

Question 26

Applications: Example 1

What is the future value of an $100 3-year ordinary annuity with an APR of 5%, compounded semiannually?

Answer 26

Solution 2: find the rEAR and treat the problem as annuity

0—1(100)—2(100)—-3(100)

rEAR=(1+rNOM/m)^m - 1=(1+0.05/2)^2 - 1= 5.0625%

Now you can solve for FV using (PV=0, PMT=-100, n=3, r=5.0625%), to get FV=315.44

Question 27

Applications: Example 2

You need to borrow $100 from the bank. Bank A offers APR of 12.5% annually compounded, while Bank B offers APR of 12% compounded quarterly. Which one do you choose?
Note: You prefer lower interest rates.

Answer 27

Need to compare effective rate (rEAR)
rEAR=rNOM=12.5%, because m=1
Bank B:
rNOM=12%, but m=4. Thus:

rEAR=(1+rNOM/m)^m - 1= (1+0.12/4)^4 - 1= 12.55%

hus actual interest rate charged by Bank A (12.5%) is smaller than the rate charged by B (12.55%)

Question 28

Applications: Example 3

Would you rather have a savings account that pays 5% interest compounded semiannually or one that pays 5% compounded daily?

Answer 28

I prefer the savings account compounded daily. It generates interest more often.

Compounded semiannually:
n=2, rPER=2.5%,PV=-1,PMT=0, thus FV1=1.050625
That is, you earn 5.0625 for every $100 dollar invested
Compounded daily:
n=365, rPER=5%/365,PV=-1,PMT=0, thus FV1=1.051267
That is, you earn 5.1267 for every $100 dollar invested