Ch 5 & Ch 6 Time Value of Money & Discounted Cash Flow Valuation
Present Value (PV) – is the beginning amount, also calls as the amount of money you have with
Future Value (FV) – is the end amount, also called as the amount of money you will have at a
Cash Flows (CF) – is the amount of money you receive or pay at regular intervals. The CF for a
particular period is often given as a subscript, CFt where t is the time period. Hence,
CF0 = PV = the cash flow at time 0 where as CF3 is the cash flow at the end of period 3.
Interest Rate (I) – is the interest rate earned (received) or paid at the end of each period.
Interest Rate in Dollars (INT) – is the dollars of interest rate earned during the year.
Number of Periods (N) – is the number of periods involved in the analysis.
The first step in time value analysis is to set up a time line, which will help visualize what is
happening in a particular problem. For example consider the following diagram, where PV
represents $100 that is in hand today and FV is the value that you will get in future after 5 years.
PV = $100
Intervals from 0 to 1, 1 to 2, 2 to 3 etc are time periods such as years or months. Time 0 is today
and it is the beginning of period 1, time 1 is one period from today and it is end of period 1 and
beginning of period 2, etc. Cash flows are directly shown below the tick marks and the relevant
interest rate is shown just above the time line.
Find Future Value
A dollar in hand today is worth more than a dollar to be received in the future because if you had
it now, you could invest it, earn interest and end up with more than a dollar in the future. The
process of going from present value to future is called compounding.
Formula: FV N = PV (1 + I ) N
Q1: If you deposit $10,000 in a bank account that pays 10% interest annually, how much will be
in your account after 5 years? (FV = $16,105.100)
Q2: It is now Jan 1, 2009. Today you will deposit $1,000 into a savings account that pays 8%.
How much will you have in your account on Jan 1, 2012? (fv = $1259.712)
Q3: If you invest $ 5,000 today at a compound interest of 9%, what will be its future value after
75 years? (fv = $ 3,205,954)
Q4: Calculate the value 5 years hence of a deposit of Rs 1,000 made today if the interest rate is
a)8%, b)10%, c)12% and d)15%. (a. $1,469.328 : b. $1,610.510 : c. $1,762.342 : d. $2,011.357)
Q5: An initial $500 compounded for 2 years at 6%. What is its FV? (fv = $561.8)
We can also find the interest rate “i” given PV, FV and No. of years “N” and we can find N
given PV, FV and i.
Find Present Value
Finding present value is the reverse of finding future value. The process of going from future
value to present value is called discounting.
Formula: PV =
(1 + I )
Q6: What is the PV of a security that will pay $5,000 in 20 years if securities of equal risk pay
7% annually? (pv = $1,292.095)
Q7: Find the present value of $500 due in 1 year at a discount rate of 6% (pv = $471.698)
We can also find i and n.
Q8: If you deposit $5,000 today at 12% rate of interest in how many years will this amount grow
to $1, 60,000? (n = 30.581 y)
Q9: A finance company offers to give $8,000 after 12 years in return for $ 1,000 deposited today.
What is i? (I = 18.921%)
Till now, we have dealt with single payment or lump sum amounts. But many assets provide a
series of cash flows. When the payments are equal and are made in fixed intervals, it’s called an
annuity. So when $100 is paid at the end of the year for 3 years, it’s called ordinary annuity.
If the payments are made at the beginning of the year, it’s called annuity due.
NOTE: FVA(due) = FVA(ordinary) x (1 + i)
Future value of ordinary annuity
Q10: Fifteen annual payments of $ 5,000 are made into a deposit account that pays 14% interest
per year. What is the future value of this annuity at the end of 15 years? (fv = $219212.071)
Q11: A finance co. advertises that it will pay a lump sum of $ 10,000 at the end of 6years to
investors who deposit annually $ 1000. What is i? (I = 20.279%)
Q12: Mr X deposits $ 100,000 in a bank which pays 10% interest. How much can he withdraw
annually for a period for 30 years? Assume that at the end of 30 Years the amount deposited will
whittle down to 0. (pmt = $10,607.925)
Future value of annuity due
Q13: Fifteen annual payments of $ 5,000 are made into a deposit account that pays 14% interest
per year. What is the future value of this annuity due at the beginning of 15 years? (Fv =
Present value of ordinary annuity
Q14: Find the PV of these ordinary annuities
a. $400 per year for 10 years at 10%($2,457.827)
b. $200 per year for 5 years at 5%($865,895)
c. $400 per year for 5 years at 0%($2000)
Present value of annuity due
Q15: Rework a, b and c assuming they are annuity due.
Finding Annuity Payments, PMT
Q16. Suppose we need to accumulate $10,000 and have it available 5 years from now. Suppose
further that we can earn a return of 6% on our savings, which are currently 0. Find how large our
deposits must be each year? (pmt = $1,773.964)
Some securities promise to make payments for ever. This is called perpetuity.
PV of a perpetuity = PMT/i
Q17: What is the present value of $100 perpetuity if the interest rate is 7%? If the rate doubles to
14%, what would be its present value? (pv = $1,428.571)
Uneven Cash Flows
Present Value of uneven cash flows
Annuity plus additional final payments
Q18: Suppose you are about to receive $100 each year for 5 years and at the end of the fifth year
along with $100, you will also receive $ 1000 onetime payment. What is the present value given
that i=10%? (pv = $ 1000)
Q19: What is the PV of a 5 year ordinary annuity of $100 plus an additional $500 at the end of 5
years if the interest rate is 5%? What is the PV if the $100 payments occur in years 1 through 10
and the $500 comes at the end of year 10? (pv = $824.711: $1079.130)
Irregular cash flows
Q20: What’s the PV of the following uneven cash flow stream: $0 at time 0, $100 in year 1,
$200 in year 2, $0 in year 3 and $400 in year 4 if the interest rate is 8% (npv = $558.072)
Q21: What is the future value of this cash flow stream: $100 at the end of 1 year, $ 150 due after
2 years, and $300 due after 3 years if the appropriate interest rate is 15%?(fv= $604.750)
Q22: An investment costs $465 and is expected to produce cash flows of $100 at the end of each
of the next 4 years then a lump sum payment of $500 at the end of forth year. What is the
expected rate of return on this investment? (I = 22.851%)
Q23: An investment costs $465 and is expected to produce cash flows of $100 at the end of year
1, $200 at the end of year 2 and $300 at the end of year 3. What is the expected rate of return on
this investment? (irr = 11.709%)
Semiannual and other compounding periods
In all our previous questions we assumed annual compounding. But in reality when you deposit
$100 in a bank that pays you 5% annual interest, it credits interest each 6 months i.e.
semiannually. Hence of the bank it is called semiannual compounding.
Whenever payments occur more than once a year, you must make two conversions:
1. Convert the stated interest rate into a periodic rate.
Periodic rate = stated annual rate/number of payments per year
With the stated annual rate of 5% compounded semiannually, the periodic rate is 5%/2 =
2. Convert the number of years into number of periods
Number of periods = (no. of years)(periods per year)
With 10 years and semiannual compounding, there are 10(2) = 20 periods.
Q24: Would you rather invest in an account that pays 7% with annual compounding or 7%with
monthly compounding? Would you rather borrow at 7% and make annual or monthly payments?
Q25: What’s the FV of $100 after 3 years if the appropriate interest rate is 8% compounded
annually? Compounded monthly?(fv = $125.971 : 127.024)
Q26: What’s the PV of $100 due in 3 years it the appropriate interest rate is 8% compounded
annually? Compounded monthly?
Comparing Interest Rates
Different compounding periods are used for different types of investments. Example: bank
accounts generally pay interest daily, most bonds pay interest semiannually, stocks pay dividend
quarterly etc. if we want to compare investments of loans with different compounding periods
properly, we need to put them on a common basis.
The nominal interest rate ( I NOM ) is also called annual percentage rate (APR) is the rate that
credit card companies, student loan officers, auto dealers tell you they are charging on loans. If
two banks offer loans with a stated rate of 8% but one requires monthly payment and the other
requires quarterly payment, they are not charging the same true rate. The one that requires
monthly payments is charging more because it will get your money sooner.
The effective annual rate (EFF) is also called the equivalent annual rate (EAR) is the rate that
would produce the same FV under annual compounding as would more frequent compounding at
a given nominal rate.
If a loan or an investment uses annual compounding its nominal rate is also its effective rate.
However if compounding occurs more than once a year, the EFF is higher than I NOM .
Effective annual rate (EFF) = 1 + I NOM − 1
Q27: By law, credit card issuers must print their annual percentage rate on their monthly
statements. A common APR is 18% with interest paid monthly. What is the EFF on such loan?
Q28: Suppose you deposited $100 in a bank that pays rate of 10% but adds interest daily based
on a 365 day year. How much would you have after 9 months?
An important application of compound interest involves loans that are paid off in installments
over time. Eg automobile loans, Mortgages etc.
Q29: Suppose you borrowed $ 30,000 on a student loan at a rate of 8% and must repay it in three
years at the end of each of the next 3 years. How large will be your payments be, how much of
the first payment represents interest, how much would be principal and what would your ending
balance be after 1st year?
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