2. Key Concepts and Skills
• Be able to compute the future value of multiple
cash flows
• Be able to compute the present value of multiple
cash flows
• Be able to compute loan payments
• Be able to find the interest rate on a loan
• Understand how interest rates are quoted
• Understand how loans are amortized or paid off
6C-2
3. Chapter Outline
• Future and Present Values of Multiple Cash
Flows
• Valuing Level Cash Flows: Annuities and
Perpetuities
• Comparing Rates: The Effect of Compounding
• Loan Types and Loan Amortization
6C-3
4. Multiple Cash Flows – FV
Example 1
• Suppose you invest $500 in a mutual fund today
and $600 in one year. If the fund pays 9% annually,
how much will you have in two years?
– FV2 = 500(1.09)2
+ 600(1.09)1
= 594.05 + 654.00
= 1,248.05
6C-4
5. Multiple Cash Flows – FV
Example 1 Continued
How much will you have in 5 years if you make no
further deposits?
• First way:
– FV5 = 500(1.09)5
+ 600(1.09)4
= 769.31 + 846.95 = 1,616.26
• Second way – use value at year 2:
– FV5 = FV2 (1.09)3
= 1,248.05(1.09)3
=1,616.26
6C-5
6. Multiple Cash Flows – FV
Example 2
• Suppose you plan to deposit $100 into an account
in one year and $300 into the account in three
years. How much will be in the account in five
years if the interest rate is 8%?
– FV5 = 100 (1.08)4
+ 300(1.08)2
= 136.05 + 349.92
= 485.97
6C-6
7. Multiple Cash Flows – PV
Example
• You are considering an investment that will pay
you $1,000 in one year, $2,000 in two years and
$3,000 in three years. If you want to earn 10% on
your money, how much would you be willing to
pay?
– PV = 1,000 / (1.10)1
+ 2,000 / (1.10)2
+3,000 / (1.10)3
= 4,815.93
6C-7
9. Decisions Based on Discounted
Cash Flows
• Your broker calls you and tells you that he has
this great investment opportunity. If you invest
$100 today, you will receive $40 in one year and
$75 in two years. If you require a 15% return on
investments of this risk, should you take the
investment?
– PV = -100 + 40 / (1.15)1
+ 75 / (1.15)2
= -8.51
– No, the investment costs more than what you would be
willing to pay.
6C-9
10. Saving For Retirement
• You are offered the opportunity to put some
money away for retirement. You will receive five
annual payments of $25,000 each beginning in 40
years. How much would you be willing to invest
today if you desire an interest rate of 12%?
– PV = 25,000 / (1.12)40
+ 25,000 / (1.12)41
+25,000 /
(1.12)42
+ 25,000 / (1.12)43
+25,000 / (1.12)44
= 1,084.71
6C-10
11. Saving For Retirement Timeline
0 1 2 … 39 40 41 42 43 44
0 0 0 … 0 25K 25K 25K 25K 25K
Notice that the year 0 cash flow = 0 (0 CF0)
The cash flows in years 1 – 39 are 0 (0 CF1; 39 N)
The cash flows in years 40 – 44 are 25,000 (25,000
CF2; 5 N)
6C-11
12. Annuities and Perpetuities Defined
• Annuity – finite series of equal payments that
occur at regular intervals
– If the first payment occurs at the end of the period, it is
called an ordinary annuity
– If the first payment occurs at the beginning of the
period, it is called an annuity due
• Perpetuity – infinite series of equal payments
6C-12
13. Annuities and Perpetuities –
Basic Formulas
Perpetuity with 1st
payment at end of the period
PV = C / r
Annuities with 1st
payment at end of the period
−+
=
+
−
=
r
1r)(1
CFV
r
r)(1
1
1
CPV
t
t
6C-13
14. Annuities and the Calculator
• You can use the PMT key on the calculator for the
equal payment
• The sign convention still holds
• Ordinary annuity versus annuity due
– You can switch your calculator between the two types by
using the 2nd
BGN 2nd
Set on the TI BA-II Plus
– If you see “BGN” or “Begin” in the display of your
calculator, it is set for an annuity due
– Most problems are ordinary annuities
6C-14
15. Annuity – Sweepstakes Example
• Suppose you win the $10 million sweepstakes.
The money is paid in equal annual end-of-year
installments of $333,333.33 over 30 years. If the
appropriate discount rate is 5%, how much is the
sweepstakes actually worth today?
Using financial calculator:
30 N; 5 I/Y; 333,333.33 PMT; CPT PV = 5,124,150.29
6C-15
29.150.124,5
0.05
)05.0(1
1
1
333,333.33
r
r)(1
1
1
CPV
30t
=
+
−
=
+
−
=
16. Buying a House
• You are ready to buy a house, and you have
$20,000 to cover both the down payment and
commission costs. Commission costs are
estimated to be 4% of the loan value. You have
an annual salary of $36,000, and the bank is
willing to allow your monthly mortgage payment
to be equal to 28% of your monthly income. The
interest rate on the loan is 6% per year with
monthly compounding (.5% per month) for a 30-
year fixed rate loan. How much money will the
bank lend you? How much can you offer for the
house?
6C-16
17. Buying a House - Continued
• Bank loan
– Monthly income = 36,000 / 12 = 3,000
– Maximum mortgage payment = .28(3,000) = 840
• Total Price
– Commission costs = .04(140,105) = 5,604
– Down payment = 20,000 – 5,604 = 14,396
– Total Price = 140,105 + 14,396 = 154,501
6C-17
140,105
0.06/12
0.06/12)(1
1
1
840
r
r)(1
1
1
CPV
12(30)t
=
+
−
=
+
−
=
18. Finding the Payment
• Suppose you want to borrow $20,000 for a new
car. You can borrow at 8% per year, compounded
monthly (8/12 = .66667% per month). If you wish
to make monthly installment payments over a 4-
year period, what is your monthly payment?
6C-18
19. Finding the Number of Payments –
Another Example
• Suppose you borrow $2,000 at 5%, and you are
going to make annual payments of $734.42. How
long will you take to pay off the loan?
6C-19
3t2,000
0.05
0.05)(1
1
1
734.42
r
r)(1
1
1
CPV
tt
=⇒=
+
−
=
+
−
=
20. Finding the Rate
• Suppose you borrow $10,000 from your parents to
buy a car. You agree to pay $207.58 per month
for 60 months. What is the monthly interest rate?
6C-20
9%r10,000
r/12
r/12)(1
1
1
207.58
r/12
r/12)(1
1
1
CPV
6012t
=⇒=
+
−
=
+
−
=
21. Future Values for Annuities
• Suppose you begin saving for your retirement by
depositing $2,000 per year in a savings account.
If the interest rate is 7.5%, how much will you
have in 40 years?
6C-21
454,513.04
0.075
10.075)(1
2,000
r
1r)(1
CFV
40t
=
−+
=
−+
=
22. Annuity Due
• You are saving for a new house and you put
$10,000 per year in an account paying 8%. The
first payment is made today. How much will you
have at the end of 3 years?
6C-22
35,061.12(1.08)
0.08
10.08)(1
10,000r)(1
r
1r)(1
CFV
3t
=
−+
=+
−+
=
25. Growing Annuity
A growing stream of cash flows with a fixed maturity
t
t
r
gC
r
gC
r
C
PV
)1(
)1(
)1(
)1(
)1(
1
2
+
+×
++
+
+×
+
+
=
−
+
+
−
−
=
t
r
g
gr
C
PV
)1(
)1(
1
6C-25
26. Growing Annuity: Example
A retirement plan offers to pay $20,000 per year
for 40 years and increase the annual payment by
three-percent each year. What is the present value
at retirement if the discount rate is 10 percent?
57.121,265$
10.1
03.1
1
03.10.
000,20$
40
=
−
−
=PV
6C-26
27. Growing Perpetuity
A growing stream of cash flows that lasts forever
+
+
+×
+
+
+×
+
+
= 3
2
2
)1(
)1(
)1(
)1(
)1( r
gC
r
gC
r
C
PV
gr
C
PV
−
=
6C-27
28. Growing Perpetuity - Example
The expected dividend next year is $1.30, and
dividends are expected to grow at 5% forever.
If the discount rate is 10%, what is the value of this
promised dividend stream?
00.26$
05.10.
30.1$
=
−
=PV
6C-28
29. Effective Annual Rate (EAR)
• This is the actual rate paid (or received) after
accounting for compounding that occurs during
the year.
• If you want to compare two alternative
investments with different compounding periods,
you need to compute the EAR and use that for
comparison.
6C-29
30. Annual Percentage Rate (APR)
• This is the annual rate that is quoted by law.
• By definition, APR = period rate times the number
of periods per year.
• Consequently, to get the period rate we rearrange
the APR equation:
– Period rate = APR / number of periods per year
• You should NEVER divide the effective rate by the
number of periods per year – it will NOT give you
the period rate.
6C-30
31. Computing APRs
• What is the APR if the monthly rate is 0.5%?
– 0.5(12) = 6%
• What is the APR if the semiannual rate is 0.5%?
– 0.5(2) = 1%
• What is the monthly rate if the APR is 12% with
monthly compounding?
– 12 / 12 = 1%
6C-31
32. Things to Remember
• You ALWAYS need to make sure that the interest
rate and the time period match.
– If you are looking at annual periods, you need
an annual rate.
– If you are looking at monthly periods, you need
a monthly rate.
• If you have an APR based on monthly
compounding, you have to use monthly periods for
lump sums, or adjust the interest rate appropriately
if you have payments other than monthly.
6C-32
33. Computing EARs - Example
• Suppose you can earn 1% per month on $1
invested today.
– What is the APR? 1(12) = 12%
– How much are you effectively earning?
• FV = 1(1.01)12
= 1.1268
• Rate = (1.1268 – 1) / 1 = .1268 = 12.68%
• Suppose you put it in another account and earn
3% per quarter.
– What is the APR? 3(4) = 12%
– How much are you effectively earning?
• FV = 1(1.03)4
= 1.1255
• Rate = (1.1255 – 1) / 1 = .1255 = 12.55%
6C-33
35. Making Decisions using EAR
• You are looking at two savings accounts. One
pays 5.25%, with daily compounding. The other
pays 5.3% with semiannual compounding. Which
account should you use?
– First account:
• EAR = (1 + .0525/365)365
– 1 = 5.39%
– Second account:
• EAR = (1 + .053/2)2
– 1 = 5.37%
• Which account should you choose and why?
6C-35
36. Making Decisions using EAR
Continued
• Let’s verify the choice. Suppose you invest
$100 in each account. How much will you
have in each account in one year?
– First Account:
FV = 100 (1 + .0525/365)365
= 105.39
– Second Account:
FV = 100 (1 + .053/2)2
= 105.37
• You have more money in the first account.
6C-36
37. Computing APRs from EARs
If you have an effective rate, how can you
compute the APR? Rearrange the EAR equation
and you get:
+= 1-EAR)(1mAPR m
1
6C-37
38. APR - Example
Suppose you want to earn an effective rate of 12%
and you are looking at an account that compounds
on a monthly basis. What APR must they pay?
[ ]
11.39%or
8655152113.1)12.1(12 12/1
=−+=APR
6C-38
39. Computing Payments with APRs
Suppose you want to buy a new computer system
and the store is willing to allow you to make
monthly payments. The entire computer system
costs $3,500. The loan period is for 2 years, and
the interest rate is 16.9% with monthly
compounding. What is your monthly payment?
6C-39
172.88C
0.169/12
0.169/12)(1
1
1
C
r
r)(1
1
1
C3,500
24t
=⇒
+
−
=
+
−
=
40. Future Value with Monthly
Compounding
Suppose you deposit $50 a month into an account
that has an APR of 9%, based on monthly
compounding. How much will you have in the
account in 35 years?
6C-40
147,089.22
0.09/12
10.09/12)(1
50
r
1r)(1
CFV
35(12)t
=
−+
=
−+
=
41. Present Value with Daily
Compounding
You need $15,000 in 3 years for a new car. If you
can deposit money into an account that pays an
APR of 5.5% based on daily compounding, how
much would you need to deposit?
FV = PV(1 + r)t
15,000 = PV (1+0.055/365)3(365)
PV = 12,718.56
6C-41
42. Continuous Compounding
• Sometimes investments or loans are figured
based on continuous compounding
EAR = eq
– 1
– The e is a special function on the calculator normally
denoted by ex
• Example: What is the effective annual rate of 7%
compounded continuously?
EAR = e.07
– 1 = .0725 or 7.25%
6C-42
43. Pure Discount Loans - Example
• Treasury bills are excellent examples of pure
discount loans. The principal amount is repaid at
some future date, without any periodic interest
payments.
• If a T-bill promises to repay $10,000 in 12 months
and the market interest rate is 7 percent, how
much will the bill sell for in the market?
FV = PV(1 + r)t
10,000 = PV (1+0.07)
PV = 9,345.79
6C-43
44. Interest-Only Loan - Example
• Consider a 5-year, interest-only loan with a 7%
interest rate. The principal amount is $10,000.
Interest is paid annually.
What would the stream of cash flows be?
• Years 1 – 4: Interest payments of .07(10,000) = 700
• Year 5: Interest + principal = 10,700
• This cash flow stream is similar to the cash flows
on corporate bonds, and we will talk about them
in greater detail later.
6C-44
46. Amortized Loan with Fixed
Payment – Another Example
• Consider a 4 year loan with annual payments. The interest
rate is 8%, and the principal amount is $5,000.
– What is the annual payment? This is an ordinary
annuity.
6C-46
47. APR may differ from EAR because of the different
basis of interest calculation
– flat basis
– annual rest basis
– reducing balance basis
Another Reason Why APR May
Differ from EAR
48. A borrower approached a bank for a loan. The bank
quoted a rate of 12% but did not tell him the basis of
interest calculation. Below are the details of the loan.
What is the EAR if flat basis applies? If annual rest
basis applies? If reducing balance basis applies?
Loan=$100,000
Repay over 3 years in 36 equal installments
EAR =?
Example
49. • For interest calculations, the principal of the
loan is not reduced by the installment payments
• Interest is charged on the full sum of the
principal over 3 yrs.
Monthly payments
= [100,000 + (100,000 X 0.12 X 3)] / 36
= 3777.78
Flat Basis
50. EAR =(1+ APR /12)12
- 1
=23.39%
The flat rate of 12% is equivalent to EAR of 23.39%.
21.2%r100,000
r/12
r/12)(1
1
1
3,777.78
r
r)(1
1
1
CPV
12(3)t
=⇒=
+
−
=
+
−
=
Flat Basis Continued
51. • For interest calculation, the principal of the loan
will only be reduced by the installment amount
at the end of every year.
• Interest is charged on balance at beginning of
the year.
Monthly PMT= 41,634.90 / 12
= 3469.58
41,634.90C100,000
0.12
0.12)(1
1
1
C
r
r)(1
1
1
CPV
3t
=⇒=
+
−
=
+
−
=
Annual Rest Basis
52. • Based on the monthly payment of 3,469.58,
what is the effective rate charged?
APR = 15.06%
EAR = (1+APR/12)12
- 1
= 16.1468%
15.06%r100,000
r/12
r/12)(1
1
1
3,469.58
r
r)(1
1
1
CPV
3(12)t
=⇒=
+
−
=
+
−
=
Annual Rest Basis (Continued)
53. • For interest rate calculations, the principal of the
loan will be reduced as and when the installment
payments are made.
APR = 12%
EAR = (1+APR/12)12
- 1
=12.6825%
This is the best basis for the borrower.
3,321.43C100,000
0.12/12
0.12/12)(1
1
1
3,777.78
r
r)(1
1
1
CPV
12(3)t
=⇒=
+
−
=
+
−
=
Reducing Balance Basis
You can also use this as an introduction to NPV by having the students put –100 in for CF0. When they compute the NPV, they will get –8.51. You can then discuss the NPV rule and point out that a negative NPV means that you do not earn your required return. You should also remind them that the sign convention on the regular TVM keys is NOT the same as getting a negative NPV.
Lecture Tip: The annuity factor approach is a short-cut approach in the process of calculating the present value of multiple cash flows and that it is only applicable to a finite series of level cash flows. Financial calculators have reduced the need for annuity factors, but it may still be useful from a conceptual standpoint to show that the PVIFA is just the sum of the PVIFs across the same time period.
Other calculators also have a key that allows you to switch between Beg/End.
Note that the outstanding balance on the loan at any point in time is simply the present value of the remaining payments.
The bank will lend you $140,105.
At most, you can offer $154,501.
Formula
PV = 840[1 – 1/1.005360] / .005 = 140,105
Lecture Tip: It should be emphasized that annuity factor tables (and the annuity factors in the formulas) assumes that the first payment occurs one period from the present, with the final payment at the end of the annuity’s life. If the first payment occurs at the beginning of the period, then FV’s have one additional period for compounding and PV’s have one less period to be discounted. Consequently, you can multiply both the future value and the present value by (1 + r) to account for the change in timing.
Note that the procedure for changing the calculator to an annuity due is similar on other calculators.
What if it were an ordinary annuity? FV = 32,464 (so you receive an additional 2,597.12 by starting to save today.)
If you use the regular annuity formula, the FV will occur at the same time as the last payment. To get the value at the end of the third period, you have to take it forward one more period.
Lecture Tip: To prepare students for the chapter on stock valuation, it may be helpful to include a discussion of equity as a growing perpetuity.
Point out to students that in this example the year 1 cash flow was given as $1.30.
If the current dividend were $1.30, then we would need to multiply it by (1+g) to estimate the year 1 cash flow.
Point out that the APR is the same in either case, but your effective rate is different. Ask them which account they should use.
Using the calculator:
The TI BA-II Plus has an I conversion key that allows for easy conversion between quoted rates and effective rates.
2nd I Conv NOM is the quoted rate; down arrow EFF is the effective rate; down arrow C/Y is compounding periods per year. You can compute either the NOM or the EFF by entering the other two pieces of information, then going to the one you wish to compute and pressing CPT.
Remind students that rates are quoted on an annual basis. The given numbers are APRs, not daily or semiannual rates.
Calculator:
2nd I conv 5.25 NOM Enter up arrow 365 C/Y Enter up arrow CPT EFF = 5.39%
5.3 NOM Enter up arrow 2 C/Y Enter up arrow CPT EFF = 5.37%
It is important to point out that the daily rate is NOT 0.014, it is 0.014383562
Lecture Tip: Here is a way to drive the point of this section home. Ask how many students have taken out a car loan. Now ask one of them what annual interest rate s/he is paying on the loan. Students will typically quote the loan in terms of the APR. Point out that, since payments are made monthly, the effective rate is actually more than the rate s/he just quoted, and demonstrate the calculation of the EAR.
On the calculator: 2nd I conv down arrow 12 EFF Enter down arrow 12 C/Y Enter down arrow CPT NOM
See the Lecture Tip in the IM for a discussion of situations that present a mismatch between the payment period and the compounding period.
Remind students that the value of an investment is the present value of expected future cash flows.
Note: You may wish to use this opportunity to compare the two types of loan amortizations. We see that the total interest is greater for the fixed payment case. It is equal to (10 x $7451.474)-$50,000=$24,514.74 versus $22,000 for the fixed principal payment case. The reason for this is that the loan is repaid more slowly early on, so the interest is somewhat higher. This doesn’t mean that one loan is better than the other; it simply means that one is effectively paid off faster. For example, the principal reduction in the first year is only $3,451.47 in the fixed payment case as compared to $5,000 in the fixed principal payment case.
Lecture Tip: Consider a $200,000, 30-year loan with monthly payments of $1330.60 (7% APR with monthly compounding). You would pay a total of $279,016 in interest over the life of the loan. Suppose instead, you cut the payment in half and pay $665.30 every two weeks (note that this entails paying an extra $1330.60 per year because there are 26 two week periods). You will cut your loan term to just 23.7 years and save almost $69,000 in interest over the life of the loan.Calculations on TI-BAII plusFirst: 200,000 PV; 360 N; 7 I/Y; 12 P/Y; CPT PMT = 1330.60 (interest = 1330.60*360 – 200,000=279,016)Second: 200,000 PV; -665.30 PMT; 7 I/Y; 26 P/Y; CPT N = 616.4 payments / 26 = 23.71 years (interest = 665.30*616.4 – 200,000=210,094.7)
