PAGE 1
CHAPTER 5
TIME VALUE OF MONEY (TVM)
1.
2.
3.
4.
5.
6.
7.
8.
WHY TIME VALUE OF MONEY IS IMPORTANT?
LUMPSUM VS ANNUITY
ORDINARY ANNUITY VS ANNUITY DUE
SIMPLE INTEREST VS COMPOUND INTEREST
TIME LINE TO SHOW TVM PROBLEMS
FUTURE VALUE OF A LUMP SUM (FV)
PRESENT VALUE OF A LUMP SUM (PV)
FINDING INTEREST RATE GIVEN PV AND FV
OF A LUMPSUM
9. FUTURE VALUE OF AN ANNUITY (FVA)
10. PRESENT VALUE OF AN ANNUITY (PVA)
11. FINDING INTEREST RATES AND PAYMENTS
GIVEN FVA OR PVA
12. PRESENT VALUE OF A PERPETUITY
13. FV (PV) OF ANNUITY VS FV (PV) OF ANNUITY
DUE
14. INTRA-YEAR COMPOUNDING IN FV, PV, FVA,
PVA
15. NOMINAL VS EFFECTIVE INTEREST RATES
16. MIXED CASH FLOWS
PAGE 2
DISCUSSION OF TIME VALUE OF MONEY TOPICS
1. WHY TIME VALUE OF MONEY IS IMPORTANT
IN MAKING FINANCIAL DECISIONS, THE DECISION MAKER
ENCOUNTERS ALTERNATIVES WITH CASHFLOWS DIFFERING IN
AMOUNT, TIMING AND RISKINESS. TO BE ABLE TO COMPARE THESE
ALTERNATIVES, THERE IS A NEED TO BRING THE CASHFLOWS TO A
COMMON BASE FOR COMPARISON. TIME VALUE OF MONEY MODELS
ENABLE US TO ACCOMPLISH THIS.
IN FINANCE, THE RECEIPT OR PAYMENT OF MONEY IS REFERRED
TO AS CASHFLOWS. CASSH INFLOW IS MONEY COMING IN OR RECEIPT
OF MONEY (SHOWN WITH A + SIGN) AND CASH OUTFLOW IS IS MONEY
GOING OUT OR PAYMENT OF MONEY (SHOWN WITH A NEGATIVE SIGN)
2. LUMPSUM VS ANNUITY
CASHFLOWS IN REAL LIFE CAN BE VARYING. IN ADDITION TO
THEIR AMOUNTS, CASHFLOWS CAN DIFFER IN THE MANNER OF THEIR
OCCURRENCE.
A LUMPSUM IS A CASHFLOW OCCURRING AT A PARTICULAR
TIME ONLY IN THAT AMOUNT. E.G. DEPOSITING $100 IN A SAVINGS
ACCOUNT TODAY IS AN EXAMPLE OF A LUMPSUM (NOW, PRESENT
VALUE). EXPECTING TO RECEIVE $100 A YEAR FROM NOW IS AN
EXAMPLE OF A LUMPSUM (LATER COMPARED TO NOW, FUTURE
VALUE)
AN ANNUITY IS MORE THAN ONE LUMPSUM (EQUAL IN AMOUNT)
OCCURRING WITHOUT INTERRUPTION. E.G. EXPECTING TO
DEPOSIT OR RECEIVE $100 SUCCESSIVELY A CERTAIN NUMBER OF
TIMES WITHOUT INTERRUPTION.
PAGE 3
3. ORDINARY ANNUITY VS ANNUITY DUE
IF THE CASHFLOWS OF AN ANNUITY WERE TO OCCUR AT THE
END OF EACH PERIOD, IT IS CALLED AN ORDINARY ANNUITY OR
DEFERRED ANNUITY OR SIMPLY ANNUITY. E.G. $1,000 TO BE
RECEIVED AS SALARY AT THE END OF EACH MONTH FOR 12
MONTHS.
IF THE CASHFLOWS OF AN ANNUITY WERE TO OCCUR AT THE
BEGINNING OF EACH PERIOD, IT IS CALLED AN ANNUITY DUE. E.G.
PAYING $500 RENT AT THE BEGINNING OF EACH MONTH FOR 12
MONTHS.
4. SIMPLE INTEREST VS COMPOUNT INTEREST
WHENEVER AN INVESTMENT IS MADE (EITHER AS A LOAN OR
SOME OTHER TYPE) INTEREST OR RETURN IS REQUIRED AT SOME
PRESPECIFIED NOMINAL RATE CORRESPONDING TO THE LEVEL OF
RISK PERCEIVED IN THE INVESTMENT. IF INTEREST (RETURN)
WERE TO BE COMPUTED ONLY ON THE ORIGINAL INVESTMENT
EACH PERIOD, IT WOULD BE SIMPLE INTEREST. IF INTEREST
(RETURN) WERE TO BE COMPUTRD NOT ONLY ON THE ORIGINAL
INVESTMENT EACH PERIOD, BUT ALSO ON THE RETURN FOR EACH
PERIOD, IT WOULD BE COMPOUND INTEREST.
EXAMPLE: A LOAN OF $1,000 IS MADE FOR 2 YEARS. THE
INTEREST RATE IS 10% THE LOAN AND INTEREST WILL BE PAID AT
THE END OF 2YEARS.
IF THE INTEREST IS SIMPLE INTEREST, THE INTEREST ON $1,000
AT 10% FOR THE 1ST YEAR IS 1000*.1 OR $100. THE INTEREST ON
$1000 AT 10% FOR THE 2ND YEAR IS 1000*.1 OR $100. AT THE END OF 2
YEARS THE LOAN PLUS THE SIMPLE INTEREST WOULD BE $1,000 +
$100 + $100 OR $1,200
IF THE INTEREST IS COMPOUND INTEREST, COMPOUNDED
ANNUALLY, I.E., INTEREST IS PAID ON THE AMOUNT OUTSTANDING
AT THE BEGINNING OF A YEAR IS PAID AT THE END OF THE YEAR,
THE INTEREST ON $1,000 AT 10% FOR THE 1ST YEAR IS 1000*.1 OR $100.
THE AMOUNT OUTSTANDING FOR THE 2ND YEAR WILL BE THE
ORIGINAL $1,000 AND THE INTEREST $100 FOR THE 1ST YEAR OR
$1,100. FOR THE 2ND YEAR INTEREST HAS TO BE PAID ON $1,100, I.E.,
PAGE 4
ON THE ORIGINAL INVESTMENT OF $1,000 AND ON THE $100
INTEREST FOR THE 1ST YEAR. THE INTEREST FOR THE 2ND YEAR
WILL BE 1,100*.1 OR $110 ($100 ON THE LOAN AND $10 ON THE 1ST
YEAR’S INTEREST). THE TOTAL AMOUNT TO BE PAID AT THE END
OF 2 YEARS WOULD BE $1,210 ($1,000 + $100 + $100 + $10). IT IS SEN
THAT THE COMPOUND INTEREST OF $210 IS MORE THAN THE
SIMPLE INTEREST OF $200! SINCE INVESTORS PREFER TO RECEIVE
MORE INTEREST AS OPPOSED TO LESS INTEREST, MOST SITUATIONS
INVOLVE OMPOUND INTEREST. THEREFORE IN TIME VALUE OF
MONEY COMPUTATIONS COMPOUND INTEREST WILL BE ASSUMED,
UNLESS OTHERWISE STATED.
ANNUAL COMPOUNDING/DISCOUNTING IS WHEN INTEREST IS
PAID/ RECEIVED ONCE EVERY YEAR AT THE END OF THE YEAR ON
THE AMOUNT OUTSTANDING AT THE BEGINNING OF THE YEAR.
INTRA-YEAR COMPOUNDING/DISCOUNTING IS WHEN INTEREST
IS COMPOUNDED/DISCOUNTED WITHIN THE YEAR, I.E., EVERY SIX
MONTHS (SEMIANNUALLY), EVERY 3 MONTHS (QUARTERLY), EVERY
MONTH (MONTHLY) ETC.
WE WILL INITIALLY STUDY ANNUAL COMPOUNDING/
DISCOUNTING. ONCE WE HAVE UNDERSTOOD THIS, INTRAYEAR
COMPOUNDING / DISCOUNTING WILL FOLLOW EASILY (TOPIC 14).
5. TIME LINE TO SHOW TVM PROBLEMS
A TIME LINE IS A GRAPHICAL DESCRIPTION OF CASH FLOWS
AND THEIR TIMING. THE RISKINESS MAY BE CAPTURED BY
GIVING THE NOMINAL INTEREST RATE.
i OR k = 10%
_____ ._______._______.______.______.______.______._________.
0
1
2
3
4
5
6………t…N= 10YEARS
-$100
THE ABOVE TIME LINE SHOWS A LUMPSUM CASH OUTFLOW OF $100
(LOAN OR INVESTMENT) AT TIME ZERO (NOW) FOR N (10 YEARS) AT
10%, COMPOUNDED ANNUALLY
PAGE 5
i OR k = 10%
_____ ._______._______.______.______.______.______._________.
0
1
2
3
4
5
6………t…N= 10YEARS
+$1,000
THE ABOVE TIME LINE SHOWS A LUMPSUM CASH INTFLOW OF $1000
AT A FUTURE TIME N (10 YEARS) AT 10%, COMPOUNDED ANNUALLY.
-$1000 $-1000 $-1000…………………………. -$1000……..-$1000
_____ ._______._______.______.______.______.______._________.
0
1
2
3
4
5
6………t…N= 10YEARS
THE ABOVE TIME LINE SHOWS AN ORDINARY ANNUITY (OUTFLOW)
OF $1000 AT THE END OF EACH YEAR FOR 10 YEARS COMPOUNDED
ANNUALLY AT 10%,.
TOPICS 6 THRU 16 WILL BE DISCUSSED USING PROBLEMS ON THE
HANDOUT FOR CHAPTER 5 ON MY WEBSITE.
PAGE 6
TIME VALUE OF MONEY PROBLEMS
1. You currently have $1,250 to invest. You want to take risk that will correspond to an
annual return of 11%, compounded annually. How much will your investment be
worth at the end of 13 years?
2. How much should one invest now, at 12% annual return, to have $20,000 at the end
of 15 years from now, if interest is compounded/discounted annually?
3. What should be the annual rate of return on an investment of $372 made today, if it
were to grow to $749 at the end of 7 years from now, if interest is compounded/
discounted annually?
4. How much money one can expect to have in a retirement account 25 years from now,
if $2,250 is invested every year (at the end of the year) for 25 years? The expected
annual rate of return is 9%, compounded annually.
5. How much (same amount) must be invested at the end of each year for 20 years from
now,at 11% annual rturn (annual compounding), in order to have $125,000 at the end
of the 20th year?
6. If the annual, year-end investment were to be $1,500, for 20 years, what should be the
annual rate of return, if the investment were to be worth $130,000 at the end of the
20th year?
7. You take a loan from a bank which is charging you 10% interest ( compounded/
discounted annually). The loan will be amortized by equal annual payments of
$1,375 for 5 years. What is the loan amount?
8. What will be your equal, annual year-end payments, that would amortize at the end
of 7 years a loan of $7,500 taken currently, if the interest rate is 8.5%?
9. What is the annual interest rate a bank is charging you on a current loan of $12,000
to be amortized in 10 years by making equal, annual, year-end payments of $1,800?
10. What is the value of an investment of $2,730 made today at the end of 6 years from
now at an annual rate of return of 8%, if interest is compounded/discounted
semiannually?
11. How much same amount should you invest at the end of each month for 30 years to
end up with $580,000 at the end of the 30th year, if the annual rate of return is 12%
(remember that
compounding/discounting is at the same frequency as the cash
flow, in this case monthly)?
PAGE 7
12. What should be the equal, quarterly (end-of-quarter) payments that would pay off in
5 years a loan of $8,000 taken currently, if the interest rate is 9%? (Remember that
interest rates are annual, unless otherwise stated)
13. What annual interest rate is the bank charging you on a loan of $10,000 taken
currently and that needs to be paid off in 4 years by making equal, monthly (end-ofmonth) payments of $250?
14. What is the equivalent amount now of a fture cash flow of $1,000 per year (yearend) that is expected to last for ever, if the interest rate is 12%?
15. You are promised a cash flow of $250 per year at the beginning of each year for
ten years from now. The applicable rate of return is 7.5%.
A. How much would this cash flow be worth at the end of ten years?
B. How much is it worth now?
PAGE 8
TIME VALUE OF MONEY PROBLEMS
SOLUTION USING TI 83/83 PLUS
SET THE CALCULATOR TO TVM SOLVER SCREEN
#1
#2
#3
#4
P/Y 1
P/Y 1
P/Y 1
P/Y 1
C/Y 1
C/Y 1
C/Y 1
C/Y 1
PMT 0
PMT 0
PMT 0
PV 0
PV -1250
FV 20000
PV -372
PMT END PMT -2250
I% 11
I% 12
FV 749
I% 9
N 13
N 15
N7
N 25
POSITION THE CURSOR TO THE OUTPUT VARIABLE AND INVOKE SOLVE
FV 4854.1002
PV -3653.9252
I% 10.5146%
FV 190577.0165
#5
#6
#7
#8
P/Y 1
P/Y 1
P/Y 1
P/Y 1
C/Y 1
C/Y 1
C/Y 1
C/Y 1
PV 0
PV 0
FV 0
FV 0
FV 125000
FV 130000
PMT -1375
PV 7500
I% 11
PMT -1500
I%10
I% 8.5
N 20
N 20
N5
N7
POSITION THE CURSOR TO THE OUTPUT VARIABLE AND INVOKE SOLVE
PMT -1946.9546
I% 13.5828
PV 5212.3318
PMT -1465.2692
#9
#10
#11
#12
P/Y 1
P/Y 2
P/Y 12
P/Y 4
C/Y 1
C/Y 2
C/Y 12
C/Y 4
FV 0
PMT 0
PV 0
FV 0
PV 12000
PV -2730
FV 580000
PV 8000
PMT -1800
I% 8
I%12
I% 9
N 10
N 6*2=12
N 30*12=360
N 5*4=20
POSITION THE CURSOR TO THE OUTPUT VARIABLE AND INVOKE SOLVE
I% 8.1442
FV 4370.8180
PMT -165.9531 PMT -501.1366
#13
#14
# 15
P/Y 12
NO NEED FOR FINANCIAL
P/Y 1 A. FV -3802.0298
C/Y 12
FINANCIAL CALCULATOR.
C/Y 1
PV10000
INSTEAD USE FORMULA
PMTT BGN B. FV 0
PMT -250 PV OF PERPETUITY = PMT/k
PV 0
PV -1844.7218
N 4*12=48
= 1000/0.12
FV 0
I% 9.2418
= 8333.3333
PMT 250
I% 7.5
PAGE 9
MIXED (UNEVEN) CASH FLOWS
PROBLEM
You are given the following cash flows. The risk-appropriate required rate of
return is 10%
a.
b.
c.
d.
$275
$340
$620
$175
$400
_________________________________________________________
0
1
2
3
4
5
What is the equivalent lump sum now of the above cash flows?
What is the equivalent lump sum at the end of the 5th year of the above cash flows?
What same amount at the end of each year for 5 years would you be willing to take,
instead of the above cash flows?
What same amount at the beginning of each year for 5 years would you be willing to
take, instead of the above cash flows?
Solution:
P/Y 1 C/Y 1 PMT END
PMT 0 I% 10
a.
b.
b. (alternate)
FV 275
PV 275
PV 1364.7028
N 1
N4
N 5
PV -250.0000 .....(1)
FV -402.6275 .....(1)
FV -2197.8675
FV 340
PV 340
N 2
N3
PV -280.9917 .....(2)
FV -452.5400 ......(2)
FV 620
PV 620
N 3
N2
PV -465.8152 .....(3)
FV -750.2000 .....(3)
FV 175
PV 175
N 4
N1
PV -119.5274 .....(4)
FV -192.5000 ......(4)
FV 400
PV 400
N 5
N0
PV 248.3685 .....(5)
CPT FV -400.0000 ......(5)
Add (1) THRU (5): -1364.7028 Add (1) THRU (5) -2197.8675
c.
N 5
PV 1364.7028 OR FV 2197.8675
FV
0
OR PV 0
PMT -360.0052
d. PMT BGN
PMT -327.2774
PAGE 10
TIME VALUE OF MONEY PROBLEMS
SOLUTION USING TI BAII PLUS
#1
CLR TVM
P/Y 1
-1250 PV
11 I/Y
13 N
CPT FV 4854.1002
#2
CLR TVM
P/Y 1
20000 FV
12 I/Y
15 N
CPT PV -3653.9252
#5
CLR TVM
P/Y 1
125000 FV
11 I/Y
20 N
CPT PMT -1946.9546
#9
CLR TVM
P/Y 1
12000 PV
-1800 PMT
10 N
CPT I/Y 8.1442%
#6
#7
#8
CLR TVM
CLR TVM
CLR TVM
P/Y 1
P/Y 1
P/Y 1
130000 FV
-1375 PMT
7500 PV
-1500 PMT
10 I/Y
8.5 I/Y
20 N
5N
7N
CPT I/Y 13.5828% CPT PV 5212.3318 CPT PMT -1465.2692
#10
#11
CLR TVM
CLR TVM
P/Y 2
P/Y 12
-2730 PV
580000 FV
8 I/Y
12 I/Y
6*2=12 N
30*12=360 N
CPT FV 4370.8180 CPT PMT -165.9531
#13
P/Y 12
10000 PV
-250 PMT
4*12=48 N
CPT I/Y 9.2418%
#3
#4
CLR TVM
CLR TVM
P/Y 1
P/Y 1
-372 PV
-2250 PMT
749 FV
9 I/Y
7N
25 N
CPT I/Y 10.5146% CPT FV 190577.0165
#12
CLR TVM
P/Y 4
8000 PV
9 I/Y
5*4=20 N
CPT PMT -501.1366
#14
No Need for Financial Calculator. Instead Use Formula:
PV of Perpetuity = PMT/k
= 1000/0.12
= 8333.3333
#15
CLR TVM
P/Y 1
BGN
250 PMT
10 N
7.5 I/Y
a. CPT FV -3802.0298
b. 0 FV
CPT PV -1844.7218
PAGE 11
MIXED (UNEVEN) CASH FLOWS
PROBLEM
You are given the following cash flows. The risk-appropriate required rate of
return is 10%
a.
b.
c.
d.
$275
$340
$620
$175
$400
_________________________________________________________
0
1
2
3
4
5
What is the equivalent lump sum now of the above cash flows?
What is the equivalent lump sum at the end of the 5th year of the above cash flows?
What same amount at the end of each year for 5 years would you be willing to take,
instead of the above cash flows?
What same amount at the beginning of each year for 5 years would you be willing to
take, instead of the above cash flows?
Solution:
a.
CLR TVM
b.
CLR TVM
b. (alternate) CLR TVM
P/Y 1
P/Y 1
P/Y 1
10 I/Y
10 I/Y
10 I/Y
275 FV
275 PV
1364.7028 PV
1N
4N
5 N
CPT PV -250.0000 .....(1) CPT FV -402.6275 .....(1) CPT FV -2197.8675
340 FV
340 PV
2N
3N
CPT PV -280.9917 .....(2)
CPT FV -452.5400 ......(2)
620 FV
620 PV
3N
2N
CPT PV -465.8152 .....(3)
CPT FV -750.2000 .....(3)
175 FV
175 PV
4N
1N
CPT PV -119.5274 .....(4)
CPT FV -192.5000 ......(4)
400 FV
400 PV
5N
0N
CPT PV 248.3685 .....(5)
CPT FV -400.0000 ......(5)
Add (1) THRU (5): -1364.7028
c.
Add (1) THRU (5) -2197.8675
1364.7028 PV OR 2197.8675 FV
5N
CPT PMT -360.0052
d. SET BGN
CPT PMT -327.2774