1. Managerial Finance Session 3
Nicole Hruban

2. TIME VALUE OF MONEY Future Value of a Single Sum:
Note: Compounding Interest Assumption
FVN=n=PV0 x(1+ i)n
Present Value of a Future Sum:
Future Value of a Ordinary Annuity:

3. TIME VALUE OF MONEY Future Value of an Annuity Due: (1+i)
Present Value of an Ordinary Annuity:

4. TIME VALUE OF MONEY Present Value of an Annuity Due:
Future Value of a Constant Growth Annuity:
Present Value of a Constant Growth Annuity:

5. TIME VALUE OF MONEY Present Value of a Perpetuity: PV=A/i
PV of constant growth perpetuity can be written as:
PVGAP=A/(i-g)
PV of annuity diminishing at a constant rate:
PVDAP=A/(i+g)
Effective Interest Rate:
Nominal Rate = Annual Percentage Rate(APR)
Effective Annual Rate = Annual Percentage Yield (APY)

6. Questions
Suppose you want to be able to withdraw $10,000 five years from and $20,000 six years from now from an account. How much do you have to deposit today if you can earn 5% annually to satisfy your withdrawal needs? (assume there is a zero balance after the withdrawals)

7. Questions
Suppose you want to be able to withdraw 10,000 five years from and 20,000 six years from now from an account. How much do you have to deposit today if you can earn 5% annually to satisfy your withdrawal needs? (assume there is a zero balance after the withdrawals) 1 5 6
$10,000
$20,000 X
i=5%

8. INVESTMENT TODAY = $7,835.26 + $14,924.31 = 22,759.57
Questions
Suppose you want to be able to withdraw 10,000 five years from and 20,000 six years from now from an account. How much do you have to deposit today if you can earn 5% annually to satisfy your withdrawal needs? (assume there is a zero balance after the withdrawals)
FV = $10,000; n=5, i=5%
PV = $7,835.26
FV = $20,000; n=6, i=5%
PV = = $14,924.31
INVESTMENT TODAY = $7, $14, = 22,759.57
Time Line:
i = 5%

9. Questions
You decided to invest a $16,000 bonus in a money market account that guarantees a 5.5% annual interest, compounded monthly for 7yrs. A one time fee of $30 is charged to set-up the account . In addition, there is an administrative charge of 1.25% of the balance in the account at the end of the year.
How much is in the account at the end of the first year?
How much at the end of the seventh year?

10. Questions
You decided to invest a $16,000 bonus in a money market account that guarantees a 5.5% annual interest, compounded monthly for 7yrs. A one time fee of $30 is charged to set-up the account . In addition, there is an administrative charge of 1.25% of the balance in the account at the end of the year.
How much is in the account at the end of the first year? 1 11 12
$16,000-$30 =
$15,970
Balance – 1.25% = X
i = 5.5% per annum or 5.5%/12 per month which compounds
charge: 1.25% of balance

11. Questions
You decided to invest a $16,000 bonus in a money market account that guarantees a 5.5% annual interest, compounded monthly for 7yrs. A one time fee of $30 is charged to set-up the account . In addition, there is an administrative charge of 1.25% of the balance in the account at the end of the year.
How much is in the account at the end of the first year?
5.5%/12
FVN=n=PV0 x(1+ i)n =
FV12=15,9700 x( /12)12 = $
Minus Admin Charge:
$ *0.9875= $16,659.95
OR: FV12 = PV0 x ( /12)12 *

12. Questions
You decided to invest a $16,000 bonus in a money market account that guarantees a 5.5% annual interest, compounded monthly for 7yrs. A one time fee of $30 is charged to set-up the account . In addition, there is an administrative charge of 1.25% of the balance in the account at the end of the year.
How much is in the account at the end of the first year?
How much at the end of the seventh year?

13. Questions
You decided to invest a $16,000 bonus in a money market account that guarantees a 5.5% annual interest, compounded monthly for 7yrs. A one time fee of $30 is charged to set-up the account . In addition, there is an administrative charge of 1.25% of the balance in the account at the end of the year.
How much at the end of the seventh year? 12 72 84
$15,970 X
i = 5.5% per annum or 5.5%/12 per month
charge: 1.25% of balance

14. Questions
You decided to invest a $16,000 bonus in a money market account that guarantees a 5.5% annual interest, compounded monthly for 7yrs. A one time fee of $30 is charged to set-up the account . In addition, there is an administrative charge of 1.25% of the balance in the account at the end of the year.
How much at the end of the seventh year?
For one year: FV12 = PV0 x ( /12)12 *
For seven years: FV12 = PV0 x ( /12)12*7 *
= 15,970 * ( /12)84 *
=$23,449.11* =$

15. Questions
On January 1st, 2002 Jack deposited $1,000 into Bank X to earn an interest rate of j per year compounded semi-annually. On January 1st, he transferred his account to Bank Y to earn an interest rate of k per annum compounded quarterly. On January 1st, 2010 the balance at Bank Y was $1,
If Jack could have earned the interest rate of k per annum compounded quarterly from January 1st, 2002 to January 1st, 2010 his balance would have been $2, What was ratio of k/j?

16. Questions
On January 1st, 2002 Jack deposited $1,000 into Bank X to earn an interest rate of j per year compounded semi-annually. On January 1st, he transferred his account to Bank Y to earn an interest rate of k per annum compounded quarterly. On January 1st, 2010 the balance at Bank Y was $1,
If Jack could have earned the interest rate of k per annum compounded quarterly from January 1st, 2002 to January 1st, 2010 his balance would have been $2, What was ratio of k/j?
2002
2007
2010
$1,000
Comp. semi-annually
Comp. quartely
$1,990.76
i = j
i = k
We also know: if k would be earned for the entire period, the balance would be: $2,203.76

17. Questions
On January 1st, 2002 Jack deposited $1,000 into Bank X to earn an interest rate of j per year compounded semi-annually. On January 1st, he transferred his account to Bank Y to earn an interest rate of k per annum compounded quarterly. On January 1st, 2010 the balance at Bank Y was $1,
If Jack could have earned the interest rate of k per annum compounded quarterly from January 1st, 2002 to January 1st, 2010 his balance would have been $2, What was ratio of k/j?
FVN=n=PV0 x(1+ k/4)4*N = > solve for k
2,203.76=1,000 x(1+ k/4)4*8 =
= (1+k/4)32 / ^1/32
^1/32 = 1+k/4
1.025 =1+k/4 => k=0.1

18. Questions We also know:
On January 1st, 2002 Jack deposited $1,000 into Bank X to earn an interest rate of j per year compounded semi-annually. On January 1st, he transferred his account to Bank Y to earn an interest rate of k per annum compounded quarterly. On January 1st, 2010 the balance at Bank Y was $1,
If Jack could have earned the interest rate of k per annum compounded quarterly from January 1st, 2002 to January 1st, 2010 his balance would have been $2, What was ratio of k/j?
We also know:
We have an initial investment, that earned j compounded semi-annually for five years
FVN=n=PV0 x(1+ j/2)2*N =
BALANCE07=1,000 x(1+ j/2)2*5
The BALANCE07 will earn a rate of 10% (k) per annum compounding quarterly

19. Questions We also know:
We have an initial investment, that earned j compounded semi-annually for five years
FVN=n=PV0 x(1+ j/2)2*N =
BALANCE07=1,000 x(1+ j/2)2*5
The BALANCE07 will earn a rate of 10% (k) per annum compounding quarterly for 3 years:
BALANCE10 = BALANCE07 x (1+0.1/4)3*4
1, = 1,000 x(1+ j/2)2*5 x (1+0.1/4)3*4
1, = 1,000x(1+j/2)10 x
= 1,000x(1+j/2)10 / ^1/10 then solve for j => j = 0.08
THEREFORE:
k/j = 0.1/0.08 = 1.25

20. Questions
The proceeds of a $10,000 death benefit are left on deposit with the insurer at an effective annual interest rate of 5%. The balance at the end of 7 years is paid to the beneficiaries in equal monthly payments of X for 10 years (120 payments). During the payout period, interest is credited at an annual rate of 3%. What is the value of X?

21. Questions
The proceeds of a $10,000 death benefit are left on deposit with the insurer at an effective annual interest rate of 5%. The balance at the end of 7 years is paid to the beneficiaries in equal monthly payments of X for 10 years (120 payments), with the first payment starting immediately. During the payout period, interest is credited at an annual rate of 3%. What is the value of X?
2002
2007
$10,000
i=3%
i=5%
120 X X

22. Questions
The proceeds of a $10,000 death benefit are left on deposit with the insurer at an effective annual interest rate of 5%. The balance at the end of 7 years is paid to the beneficiaries in equal monthly payments of X for 10 years (120 payments), with the first payment starting immediately. During the payout period, interest is credited at an annual rate of 3%. What is the value of X?
We can calculate the balance at the end of year 7:
FV7=PV0 x(1+ i)7 =
FV7=10,000 x( )7 = $14,071
We can then calculate the annuity payment by:
Solving for A
With
PVA = $14,071
i = 0.03/12 =
N=120 (i.e. 12*10)
ANNUITY DUE
ANNUITY: $119.01

23. Questions
Your father now has $1,000,000 invested in an account that pays 9.00%. He expects inflation to average 3%, and he wants to make annual constant dollar (real) end-of-year withdrawals over each of the next 20 years and end up with a zero balance after the 20th year. How large will his initial withdrawal (and thus constant dollar (real) withdrawals) be?

24. Questions
Your father now has $1,000,000 invested in an account that pays 9.00%. He expects inflation to average 3%, and he wants to make annual constant dollar (real) end-of-year withdrawals over each of the next 20 years and end up with a zero balance after the 20th year. How large will his initial withdrawal (and thus constant dollar (real) withdrawals) be? 1 19 20
$1,000,000
Constant $ X
Constant $ X
Constant $ X
i = 9%
g = 3% $0

25. Questions
Your father now has $1,000,000 invested in an account that pays 9.00%. He expects inflation to average 3%, and he wants to make annual constant dollar (real) end-of-year withdrawals over each of the next 20 years and end up with a zero balance after the 20th year. How large will his initial withdrawal (and thus constant dollar (real) withdrawals) be?
rNOM 9.00% Initial sum ,000,000
Inflation 3.00% Years 20
A=$88,530.31

26. Questions
Your father now has $1,000,000 invested in an account that pays 9.00%. He expects inflation to average 3%, and he wants to make annual constant dollar (real) end-of-year withdrawals over each of the next 20 years and end up with a zero balance after the 20th year. How large will his initial withdrawal (and thus constant dollar (real) withdrawals) be?

27. Question
You have a coin you wish to sell. A potential buyer offers to purchase the coin from you in exchange for a series of three annual payments of $50 starting one year from today. What is the current value of the offer if the prevailing rate of interest is 7% compounded annually? What is the current value of the offer if the prevailing rate of interest is 7% compounded monthly?

28. Question
You have a coin you wish to sell. A potential buyer offers to purchase the coin from you in exchange for a series of three annual payments of $50 starting one year from today.
What is the current value of the offer if the prevailing rate of interest is 7% compounded annually?
𝑃𝑉𝐴=𝐴 ∗( (1+𝑖) 𝑁 −1 𝑖 ( 1+𝑖) 𝑁 ) ) = 50 * ( (1+0.07) 3 − ( ) 3 )=
The difference is that section 1 we were increasing both the payment frequency and the compounding frequency. Here we are increasing only the compounding frequency. We aren't getting our payments any sooner but we are compounding more frequently so less front money (i.e., a smaller PV) is required in order to grow to the specified future value.

29. Question
You have a coin you wish to sell. A potential buyer offers to purchase the coin from you in exchange for a series of three annual payments of $50 starting one year from today.
What is the current value of the offer if the prevailing rate of interest is 7% compounded monthly?
Step1: Translate 7% rate into effective interest rate
EAR= (1+0.07/12)^12 – 1 =

30. Question
You have a coin you wish to sell. A potential buyer offers to purchase the coin from you in exchange for a series of three annual payments of $50 starting one year from today.
What is the current value of the offer if the prevailing rate of interest is 7% compounded monthly?
𝑃𝑉𝐴=𝐴 ∗( (1+𝑖) 𝑁 −1 𝑖 ( 1+𝑖) 𝑁 ) ) = 50 * ( ( ) 3 − ( ) 3 )=130.67
The difference is that section 1 we were increasing both the payment frequency and the compounding frequency. Here we are increasing only the compounding frequency. We aren't getting our payments any sooner but we are compounding more frequently so less front money (i.e., a smaller PV) is required in order to grow to the specified future value.