1. The Time Value of
Money
Prepared & Presented By:
Md. Nafeez-Al-Tarik
Professional Finance Studies (PFS)
2. Time Value of Money
• The value of $1 today is worth more than the certain $1 in future.
• We can earn interest on $1 that can be invested immediately.
• Time value of money reflects the relationship between present value, future
value, time and interest rate.
3. Some Useful Terms
Opportunity Cost
• Next best value that investors forgo by choosing a particular course of action
Real Risk free rate
• Single period interest rate for a completely risk-free security if no inflation were expected
• Reflects time preferences of individuals for current versus future real consumptions.
Default Risk Premium
• Compensates investors for the possibility that the borrower will fail to make promised payment
Inflation Premium
• Compensates investors for expected inflation and reflects average inflation rate expected over the maturity
of debt.
4. Some Useful Terms
Required Rate of Return
Real risk free rate + Inflation premium + Default risk premium + Liquidity Premium + Maturity Premium
Nominal Interest Rate
• Real Risk free rate + Inflation Premium
Liquidity Premium
• Compensates investors for the risk of loss relative to an investment's fair value if the investment needs to be
covered to cash quickly.
Maturity Premium
Compensates investors for the increased sensitivity of the market value of debt to a change in market interest rate
as the maturity is extended.
5. Future Value
• Amount of money that an investment made today will grow to by some future date.
**Remember:
• For a given interest rate, the future value increases with the number of periods.
• For a given number of periods, the future value increases with the interest rate.
Continuous Compounding
• Instead of compounding for discrete periods, if the number of compounding periods per
year becomes infinite, then interest rate is said to be compounded continuously.
6. Example 1
• You invested $500,000 in a local bank that promises to pay you 8% per year
compounded annually. How much will you have if your money remains
invested at 8% for the next 10 years with no withdrawals?
7. Texas Instrument
BA II Plus (Professional)
• Clear TVM:
Press 2ND FV CE/C
Type 500,000 Press PV
PMT = 0
Type 8 Press I/Y
Type 10 Press N
Press CPT Press FV
Result: FV -1,079,462.499
8. Example 2
• You took a loan of $500,000 from a local bank that will charge you 8% per
year compounded quarterly. How much will you have to pay at the end of 10
year if you are charged 8% compounded quarterly?
9. Texas Instrument
BA II Plus (Professional)
• Clear TVM:
Press 2ND FV CE/C
Type 500,000 Press PV
PMT = 0
Type 2 Press I/Y
Type 40 Press N
Press CPT Press FV
Result: FV -1,104,019.832
10. Simple Vs Effective Interest Rate
Stated Interest Rate
• Annual interest rate stated for an investment.
Effective Interest Rate
• The actual interest rate that accrues after taking into consideration the effects
of compounding.
11. Continuous Compounding
• Instead of compounding for discrete periods, if the number of
compounding periods per year becomes infinite, then interest rate is said to
be compounded continuously.
• If the number of compounding periods per year becomes infinite, then
interest is said to compound continuously.
12. Example 3
• You invest $50,000 today that will earn 7 percent compounded continuously
for two years. What will be the Future value at the end of year two?
Ans: First multiply .07 by 2 and press equal sign Press 2ND Press LN
You will get FV of $1. Now you need to multiply by $50,000 to get the PV.
Ans: 57,513.6899
13. Example 4
Continuing with the CD example, suppose your bank offers you a CD with a
two- year maturity, a stated annual interest rate of 8 percent compounded
quarterly, and a feature allowing reinvestment of the interest at the same
interest rate. You decide to invest $10,000. What will the CD be worth at
maturity?
Ans: N = 8; I/Y=2; PV=-10,000; PMT = 0; Press CPT and then FV
14. Example 5
An Australian bank offers to pay you 6 percent compounded monthly. You
decide to invest A$1 million for one year. What is the future value of your
investment if interest payments are reinvested at 6 percent?
Ans: N = 12; I/Y=0.5; PV=-1,000,000; PMT = 0; Press CPT and then FV
15. Example 6
Suppose a $10,000 investment will earn 8 percent compounded continuously
for two years.
Ans: Multiply .08 with 2, Then press 2nd button and then ln button.
Multiply it with 10,000
16. Example 7
Suppose your company’s defined contribution retirement plan allows you to
invest up to €20,000 per year. You plan to invest €20,000 per year in a stock
index fund for the next 30 years. Historically, this fund has earned 9 percent per
year on average. Assuming that you actually earn 9 percent a year, how much
money will you have available for retirement after making the last payment?
Ans: N = 30; I/Y=9; PV=0; PMT = -20,000; Press CPT and then FV
17. Example 8
An insurance company has issued a Guaranteed Investment Contract (GIC)
that promises to pay $100,000 in six years with an 8 percent return rate. What
amount of money must the insurer invest today at 8 percent for six years to
make the promised payment?
Ans: N = 6; I/Y=8; FV=100,000; PMT = 0; Press CPT and then PV
18. Example 9
Suppose you own a liquid financial asset that will pay you $100,000 in 10 years
from today. Your daughter plans to attend college four years from today, and
you want to know what the asset’s present value will be at that time. Given an 8
percent discount rate, what will the asset be worth four years from today?
Ans: N = 6; I/Y=8; FV=100,000; PMT = 0; Press CPT and then PV
19. Example 10
The manager of a Canadian pension fund knows that the fund must make a
lump-sum payment of C$5 million 10 years from now. She wants to invest an
amount today in a GIC so that it will grow to the required amount. The current
interest rate on GICs is 6 percent a year, compounded monthly. How much
should she invest today in the GIC?
Ans: N = 120; I/Y=0.5; FV=5000,000; PMT = 0; Press CPT and then PV
20. Example 11
Suppose you are considering purchasing a financial asset that promises to pay
€1,000 per year for five years, with the first payment one year from now. The
required rate of return is 12 percent per year. How much should you pay for
this asset?
Ans: N = 5; I/Y=12; FV=0; PMT = 1,000; Press CPT and then PV
21. Example 12
You are retiring today and must choose to take your retirement benefits either as a lump sum or as
an annuity. Your company’s benefits officer presents you with two alternatives: an immediate lump
sum of $2 million or an annuity with 20 payments of $200,000 a year with the first payment starting
today. The interest rate at your bank is 7 percent per year compounded annually. Which option has
the greater present value? (Ignore any tax differences between the two options.)
Ans: N = 20; I/Y=7; FV=0; PMT = 200,000;
*This is an example of annuity due. For this press 2nd PMT 2ND Enter 2nd CPT
You will be able to see BGN written on the top of your screen
Press CPT and then PV
22. Example 13
A German pension fund manager anticipates that benefits of €1 million per year must be paid
to retirees. Retirements will not occur until 10 years from now at time t = 10. Once benefits
begin to be paid, they will extend until t = 39 for a total of 30 payments. What is the present
value of the pension liability if the appropriate annual discount rate for plan liabilities is 5
percent compounded annually?
Ans: N = 30; I/Y=5; FV=0; PMT = 1,000,000; Press CPT and then PV
This is PV at Time-9; The value is 15,372,451.03
Now, N = 9; I/Y=5; FV=15,372,451.03; PMT = 0; Press CPT and then PV
23. Example 14
The British government once issued a type of security called a consol bond,
which promised to pay a level cash flow indefinitely. If a consol bond paid
£100 per year in perpetuity, what would it be worth today if the required rate
of return were 5 percent?
Ans: 100/.05
24. Example 15
Consider a level perpetuity of £100 per year with its first payment beginning at
t = 5. What is its present value today (at t = 0), given a 5 percent discount rate?
Ans: 100/.05
Then, N = 4; I/Y=5; FV=2,000; PMT = 0; Press CPT and then PV
Example 16: Homework
25. Example 17
For 1998, Limited Brands, Inc., recorded net sales of $8,436 million. For 2002,
Limited Brands recorded net sales of $8,445 million, only slightly higher than in
1998. Over the four-year period from the end of 1998 to the end of 2002, what
was the rate of growth of Limited Brands’ net sales?
Ans: 2nd 5; OLD = 8,436 Enter NEW = 8,445 Enter; #PD = 4 Enter;
Press and Press CPT
Homework: Example 18
26. Example 19
You are interested in determining how long it will take an investment of
€10,000,000 to double in value. The current interest rate is 7 percent
compounded annually. How many years will it take €10,000,000 to double to
€20,000,000?
Ans: PV = - 10,000,000; I/Y=7; FV= 20,000,000; PMT = 0; Press CPT and
then N
27. Example 20
You are planning to purchase a $120,000 house by making a down payment of
$20,000 and borrowing the remainder with a 30-year fixed-rate mortgage with
monthly payments. The first payment is due at t = 1. Current mortgage interest
rates are quoted at 8 percent with monthly compounding. What will your
monthly mortgage payments be?
Ans: N = 360; I/Y=8/12; FV=0; PV =100,000; Press CPT and then PMT
28. Example 21
Jill Grant is 22 years old (at t = 0) and is planning for her retirement at age 63
(at t = 41). She plans to save $2,000 per year for the next 15 years (t = 1 to t =
15). She wants to have retirement income of $100,000 per year for 20 years,
with the first retirement payment starting at t = 41. How much must Grant
save each year from t = 16 to t = 40 in order to achieve her retirement goal?
Assume she plans to invest in a diversified stock-and-bond mutual fund that
will earn 8 percent per year on average.
29. Example 21(Ans)
Ans: The problem should be done in 3 steps
Step 1: N = 15; I/Y=8; PV=0; PMT = 2,000; Press CPT and then FV (FV=54,304.22)
Step 2: N = 20; I/Y=8; FV=0; PMT = 100,000; Press CPT and then PV
(PV=981,814.74)
Step 3: N = 25; I/Y=8; FV=981,814.74; PV =-54,304.22; Press CPT and then PMT
