Accounting

FNAN 303

Solutions to lecture problems – time value of money, part 2

Lecture Problem 1-a

How much is a building worth that is expected to produce cash flows of $200,000 in 1 year and $500,000 in 2 years if the cost of capital is 10.0 percent? Since there’s no information on expected cash flows at times other than in 1 and 2 years, assume those are the only times with non-zero expected cash flows.

Time	0	1	2
Expected cash flow	0	$200,000	$500,000
Present value	?

PV = C0 + [C1/(1+r)1] + [C2/(1+r)2]

C0 = 0 C1 = 200,000 C2 = 500,000 r = .100

PV = C0 + [C1/(1+r)1] + [C2/(1+r)2]

= 0 + [200,000/(1.100)] + [500,000/(1.100)2]

= 0 + 181,818 + 413,223

= 595,041

Answer: $595,041

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Lecture Problem 1-b

A building that is worth $500,000 and has a cost of capital of 10.0% is expected to produce cash flows of $200,000 in 1 year and X in 3 years. What is X?

Time	0	1	2	3
Expected cash flow	0	$200,000	$0	X
Present value	$500,000

PV = C0 + [C1/(1+r)1] + [C2/(1+r)2]

PV = 500,000 C0 = 0 C1 = 200,000 C2 = 0 C3 = X r = .100

PV = C0 + [C1/(1+r)1] + [C2/(1+r)2] + [C3/(1+r)3]

500,000 = 0 + [200,000/(1.10)] + [0/(1.100)2] + [X/(1.100)3]

500,000 = 0 + [200,000/(1.10)] + [X/(1.100)3]

500,000 = 0 + 181,818 + [X/(1.100)3]

So 500,000 – 181,818 = [X/(1.100)3]

So 318,182 = [X/(1.100)3]

The present value of the cash flow expected in year 3 is $318,182

The value of the cash flow expected in year 3 is X where 318,182 = [X/(1.100)3]

So X = [318,182 × (1.100)3] = $423,500

Confirm:

PV = C0 + [C1/(1+r)1] + [C2/(1+r)2] + [C3/(1+r)3]

PV = 0 + [200,000/(1.100)1] + [0/(1.100)2] + [423,500/(1.100)3]

= 0 + 181,818 + 0 + 318,182

= 500,000 ☺

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Lecture Problem 2

You own a building that is expected to pay annual cash flows forever. What is the value of the building if the cost of capital is 8.0% and annual cash flows of $500,000 are expected with the first one in 1 year?

Time	0	1	2	3	4	…
Cash flow	$0	$500,000	$500,000	$500,000	$500,000	…
Present value	?

The cash flows reflect a fixed perpetuity

PV = C / r

C = $500,000 r = .080

PV = $500,000 / .080 = $6,250,000

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Lecture Problem 3-a

You own a building that is expected to pay annual cash flows forever. What is the amount of the annual cash flow produced by a building expected to be if the building is worth $2,300,000, the cost of capital is 5.0%, and annual fixed cash flows are expected with the first one due in one year?

Time	0	1	2	3	4	…
Cash flow	$0	C	C	C	C	…
Present value	$2,300,000

The cash flows reflect a fixed perpetuity

C = PV × r

PV = $2,300,000 r = .050

C = $2,300,000 × .050

= $115,000

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Lecture Problem 3-b

What is the present value (as of today) of the expected cash flow produced by a building in 3 years if the building is worth $2,300,000, the cost of capital is 5.0%, and annual fixed cash flows are expected with the first one due in one year?

Approach

1) Find the cash flow expected in 3 years

2) Find the present value of the cash flow expected in 3 years

1) Find the cash flow expected in 3 years

Time	0	1	2	3	4	…
Cash flow	$0	C	C	C	C	…
Present value	$2,300,000

The cash flows reflect a fixed perpetuity

C = PV × r

PV = $2,300,000

r = .050

C = $2,300,000 × .050

= $115,000

2) Find the present value of the cash flow expected in 3 years

PV0 = C3 / (1+r)3

C3 = $115,000 (note that all cash flows are $115,000)

r = .050

PV0 = 115,000 / 1.0503

= $99,341

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Lecture Problem 3-c

What is the cost of capital for the building if its value is $1,700,000 and annual cash flows of $100,000 are expected with the first one in 1 year?

Time	0	1	2	3	4	…
Cash flow	$0	$100,000	$100,000	$100,000	$100,000	…
Present value	1,700,000

The cash flows reflect a fixed perpetuity

r = C / PV

C = $100,000 PV = $1,700,000

r = $100,000 / $1,700,000 = .0588 = 5.88%

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Lecture Problem 4

An investment is expected to generate annual cash flows forever. The first annual cash flow is expected in 1 year and all subsequent annual cash flows are expected to grow at a constant rate annually. We know that the cash flow expected in 2 years from today is expected to be $50.00 and the cash flow expected in 5 years from today is expected to be $55.62. What is the cash flow expected to be in 4 years from today?

Approach:

1) Find the annual growth rate

2) Find the cash flow expected in 4 years

Time	0	1	2	3	4	5	…
CF	0	C1	C2	C3	C4	C5	…
CF	0	C1	50.00	C3	C4	55.62	…
CF	0	C1	C2	C2 × (1+g)	C2 × (1+g)2	C2 × (1+g)3	…
CF	0	C1	C2	C3	C4	C4 × (1+g)	…
CF	0	C1	C2	C3	C4	C5	…

1) Find the annual growth rate

We know that Cb = Ca × (1+g)b-a so C5 = C2 × (1+g)3

C2 = 50.00 C5 = 55.62

So 55.62 = 50.00 × (1+g)3

(55.62/50.00) = (1+g)3

(55.62/50.00)(1/3) = [(1+g)3](1/3) = 1 + g

= 1.0361

So g = 1.0361 – 1 = .0361

2) Find the cash flow expected in 4 years

We know that Cb = Ca × (1+g)b-a so C4 = C2 × (1+g)4-2 = C2 × (1+g)2

C2 = 50.00 and g = .0361

So C4 = C2 × (1+g)4-2 = 50.00 × (1.0361)2 = $53.68

We also know that Cb = Ca × (1+g)b-a so C5 = C4 × (1+g)5-4 = C4 × (1+g)1

So 55.62 = C4 × 1.0361

C5 = 55.62 / 1.0361 = $53.68

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Lecture Problem 5-a

You own a building that is expected to pay annual cash flows forever. What is the value of the building if the cost of capital is 8.3% and annual cash flows are expected to grow by 2.3% per year forever with the first one expected to be $500,000 in 1 year?

Time	0	1	2	3	4	…
Cash flow	$0	$500,000	$500,000 × 1.023	$500,000 × 1.0232	$500,000 × 1.0233	…
Cash flow	$0	$500,000	$511,500	$523,265	$535,300	…
Present value	?

The cash flows reflect a growing perpetuity

PV = C1 / (r – g)

C1 = $500,000 r = .083 g = .023

PV = $500,000 / (.083 – .023) = $500,000 / .060 = $8,333,333

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Lecture Problem 5-b

You own a building that is expected to pay annual cash flows forever. If the building is worth $2,300,000, the cost of capital is 5.0%, and annual cash flows are expected with the first one due in one year and all subsequent ones growing annually by 2.2%, then what is the amount of the cash flow produced by the building in 1 year expected to be?

Time	0	1	2	3	4	…
Cash flow	$0	?	? × (1.022)	? × (1.022)2	? × (1.022)3	…
Present value	$2,300,000

The cash flows reflect a growing perpetuity

C1 = PV × (r – g)

PV = $2,300,000 r = .050 g = .022

C1 = $2,300,000 × (.050 – .022)

= $2,300,000 × .028

= $64,400

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Lecture Problem 5-c

You own a building that is expected to pay annual cash flows forever. If the building is worth $2,300,000, the cost of capital is 5.0%, and annual cash flows are expected with the first one due in one year and all subsequent ones growing annually by 2.2%, then what is the amount of the cash flow produced by the building in 3 years expected to be?

We know that C1 = $64,400

Therefore, C2 = C1 × (1.022)

And C3 = C2 × (1.022) = C1 × (1.022)2 = 64,400 × (1.022)2 = 67,265

Time	1	2	3
Cash flow	C1	C2	C3
Cash flow	C1	C1 × (1.022)	C2 × (1.022)
Cash flow	C1	C1 × (1.022)	C1 × (1.022)2
Cash flow	64,400	65,816.80	67,264.77

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Lecture Problem 5-d

You own a building that is expected to pay annual cash flows forever. If the building is worth $850,000, the cost of capital is 8.30%, annual cash flows are expected with the first one due in one year and equal to $62,000, and all subsequent cash flows are expected to grow annually by a constant rate, then what is the expected annual growth rate of the expected cash flows?

Time	0	1	2	3	4	…
Cash flow	$0	62,000	62,000 × (1+g)	62,000 × (1+g)2	62,000 × (1+g)3	…
Present value	850,000

The cash flows reflect a growing perpetuity

PV = C1 / (r – g), so g = r – (C1 / PV)

PV = $850,000 r = .0830 C1 = $62,000

g = r – (C1 / PV)

= .0830 – (62,000 / 850,000)

= .0830 – .0729

= .0101

= 1.01%

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Lecture Problem 6

What is the price of an investment that pays you $1,000 per month with the first payment in 1 month and the last payment in 6 months if the expected return for the investment is 1.0% per month?

From the timeline, we can see that the cash flows reflect a 6-period ordinary annuity

Time	0m	1m	2m	3m	4m	5m	6m
Pmt #		1	2	3	4	5	6
CF	0	1000	1000	1000	1000	1000	1000

Time reflects the number of periods from time 0

Pmt # is a count of the number of payments

END Mode

Enter 6 1.0 1,000 0

N I% PV PMT FV

Solve for -5,795.48

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Lecture Problem 7-a

What is the price of a car if the loan from the bank to buy it involves annual payments of $7,300 to the bank for 5 years at an annual interest rate of 9.4 percent with the first annual payment made to the bank in 1 year and a special payment of $5,000 to the bank in 5 years?

The car price equals the opposite of the present value of the loan payments, including the special payment, discounted by the interest rate of the loan.

Time	0	1	2	3	4	5
CF from regular payments	0	-7,300	-7,300	-7,300	-7,300	-7,300
CF from special payment						-5,000
Present value	?

The present value of the loan payments equals:

1) The present value of a 5-period annuity with regular cash flows of -$7,300 and a discount rate of 9.4%

2) A cash flow of -$5,000 in 5 years with a discount rate of 9.4%

1) The PV of the annuity

END mode

Enter 5 9.4 -7,300 0

N I% PMT PV FV

Solve for 28,102

The present value of the cash flows associated with the regular payments is -$28,102

2) PV of -$5,000 in 5 years

Note that mode is not relevant

Enter 5 9.4 0 -5,000

N I% PMT PV FV

Solve for 3,191

The present value of the cash flows associated with the special payment is -$3,191

Combine the 2 pieces

The present value of the loan payments = -$28,102 + (-$3,191) = -$31,293

So the price of the car is $31,293

(Answers may differ slightly due to rounding)

Note: can be done in 1 step, since N = 5 for annuity and N = 5 for special payment

END mode

Enter 5 9.4 -7,300 -5,000

N I% PMT PV FV

Solve for 31,293

The present value of the cash flows associated with the payments is -$31,293

So the price of the car is $31,293

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Lecture Problem 7-b

What is the price of a car if the loan from the bank to buy it involves annual payments of $6,200 to the bank for 5 years at an annual interest rate of 11.7 percent with the first annual payment made to the bank in 1 year and a special payment of $5,000 to the bank in 3 years?

The car price equals the opposite of the present value of the loan payments, including the special payment, discounted by the interest rate of the loan.

Time	0	1	2	3	4	5
CF from regular payments	0	-6,200	-6,200	-6,200	-6,200	-6,200
CF from special payment				-5,000
Present value	?

The present value of the loan payments equals:

1) The present value of a 5-period annuity with regular cash flows of -$6,200 and a discount rate of 11.7%

2) A cash flow of -$5,000 in 3 years with a discount rate of 11.7%

1) The PV of the annuity

END mode

Enter 5 11.7 -6,200 0

N I% PMT PV FV

Solve for 22,517

The present value of the cash flows associated with the regular payments is -$22,517

2) PV of -$5,000 in 3 years

Note that mode is not relevant

Enter 3 11.7 0 -5,000

N I% PMT PV FV

Solve for 3,588

The present value of the cash flows associated with the special payment is -$3,588

Combine the 2 pieces

The present value of the loan payments = -$22,517 + (-$3,588) = -$26,105

So the price of the car is $26,105

(Answers may differ slightly due to rounding)

Note: must be done in 2 steps, since N = 5 for annuity and N = 3 for special payment

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Lecture Problem 8-a

What is the value of an investment that pays you $1,000 a month with the first payment today and the last payment in 6 months if the discount rate is 1.0% per month?

From the timeline, we can see that the cash flows reflect a 7-period annuity due

Time	0m	1m	2m	3m	4m	5m	6m
Pmt #	1	2	3	4	5	6	7
CF	1000	1000	1000	1000	1000	1000	1000

Time reflects the number of periods from time 0

Pmt # is a count of the number of payments

N = 7 (number of payments, not necessarily when last payment is expected to be made)

BEGIN Mode

Enter 7 1.0 1,000 0

N I% PV PMT FV

Solve for -6,795.48

==============================

Lecture Problem 8-b

What is the value of an investment that pays you $1,000 per month for 7 months if the first payment is today and the discount rate is 1.0% per month?

Monthly payments “for 7 months” tells us that there are 7 payments

From the timeline, we can see that the cash flows reflect a 7-period annuity due

Time	0m	1m	2m	3m	4m	5m	6m
Pmt #	1	2	3	4	5	6	7
CF	1000	1000	1000	1000	1000	1000	1000

Time reflects the number of periods from time 0

Pmt # is a count of the number of payments

N = 7 (number of payments, not necessarily when last payment is expected to be made)

BEGIN Mode

Enter 7 1.0 1,000 0

N I% PV PMT FV

Solve for -6,795.48

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Lecture Problem 8-c

What is the value of an investment that pays you $2,000 in 5 months plus $1,000 per month with the first $1,000 payment today and the last $1,000 payment in 5 months if the discount rate is 1.0% per month?

From the timeline, we can see that the cash flows reflect a 6-period annuity due plus a cash flow of $2,000 in 5 months

Time	0	1	2	3	4	5	6
Pmt #	1	2	3	4	5	6
CF	1000	1000	1000	1000	1000	1000 + 2000	0

Value of investment = PV of the6-period annuity due + PV of the cash flow of $2,000 in 5 months

PV of the 6-period annuity due

BEGIN Mode

Enter 6 1.0 1,000 0

N I% PV PMT FV

Solve for -5,853.43

PV of the cash flow of $2,000 in 5 months

PV = C5 / (1+r)5

r = .010

C5 = 2,000 (note: this CF reflects the cash flow not associated with the annuity due)

PV = 2,000 / (1.010)5 = 1,902.93

Mode is not relevant, since PMT = 0

Enter 5 1.0 0 2000

N I% PV PMT FV

Solve for -1,902.93

Aggregate the two sources of value

The value of the investment = 5,853.43 + 1,902.93 = $7,756.36

Important note

You can not do this problem in one calculator step: if you input in $2,000 as FV, the calculator will treat that cash flow as taking place in N periods, which would be in 6 periods and you would produce the following:

BEGIN Mode

Enter 6 1.0 1,000 2,000

N I% PV PMT FV

Solve for -7,737.52

Time line reflected by the preceding:

Time	0	1	2	3	4	5	6
CF	1000	1000	1000	1000	1000	1000	2000

========================================

Lecture Problem 8-d

What is the value of an investment that pays you $2,000 in 6 months plus $1,000 a month with the first $1,000 payment today and the last $1,000 payment in 11 months if the discount rate is 1.0% per month?

From the timeline, we can see that the cash flows reflect a 12-period annuity due plus a cash flow of $2,000 in 6 months

Time	0	1	2	3	4	5	6	7	8	9	10	11	12
Pmt #	1	2	3	4	5	6	7	8	9	10	11	12
CF	1000	1000	1000	1000	1000	1000	1000 + 2000	1000	1000	1000	1000	1000	0

Time reflects the number of periods from time 0

Pmt # is a count of the number of payments

Value of investment = PV of the 12-period annuity due + PV of the cash flow of $2,000 in 6 months

PV of the 12-period annuity due

BEGIN Mode

Enter 12 1.0 1,000 0

N I% PV PMT FV

Solve for -11,367.63

PV of the cash flow of $2,000 in 6 months

PV = C6 / (1+r)6

r = .010

C6 = 2,000 (note: this CF reflects the cash flow not associated with the annuity due)

PV = 2,000 / (1.010)6 = 1,884.09

Mode is not relevant, since PMT = 0

Enter 6 1.0 0 2000

N I% PV PMT FV

Solve for -1,884.09

Aggregate the two sources of value

The value of the investment = 11,367.63 + 1,884.09 = $13,251.72

Note that the investment is worth more when the $2,000 comes sooner (in 6 months vs. 11 months in the previous problem).

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Lecture Problem 9

If Arturo has $50,000 to invest, how much can he expect to receive each year as a fixed annual payment if he receives his first fixed annual payment today, he receives his last payment in 7 years, and he expects to earn 1.0% per year?

The cash flows that Arturo expects to receive reflect an annuity due with 8 annual payments. We want to find the size of the payments such that the annuity due would have a present value of $50,000.

A timeline shows that the number of payments is 8 (even though the last payment occurs in 7 years from now):

Time	0	1	2	3	4	5	6	7
Pmt #	1	2	3	4	5	6	7	8
CF	X	X	X	X	X	X	X	X

BEGIN Mode

Enter 8 1.0 -50,000 0

N I% PV PMT FV

Solve for 6,469.82

Arturo would receive annual payments of $6,469.82

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Lecture Problem 10

If Arturo has $50,000 to invest in an investment that makes fixed monthly payments of $1,000, how many monthly payments will he receive if he receives his first fixed monthly payment today and his return is 1.0% per month?

Time	0	1	2	…	N – 2	N – 1	N
	0y0m	0y1m	0y2m	…
Pmt #	1	2	3	…	N – 1	N
CF	1000	1000	1000	…	1000	1000
Investment value	50,000

The cash flows that Arturo expects to receive reflect an annuity due with $1,000 monthly payments. We want to find the number of annuity due payments that are necessary for the annuity due to have a present value of $50,000.

BEGIN Mode

Enter 1.0 -50,000 1,000 0

N I% PV PMT FV

Solve for 68.67

Arturo would receive 68.67 payments

==============================

Lecture Problem 11

What is the quarterly interest rate for Tony’s loan if he borrowed $5,400 today, he must make equal quarterly payments of $830, with the first quarterly payment due later today and the last quarterly payment due in 6 quarters?

Timeline tip for FNAN 303: the cash flows occur quarterly so the timeline period is a quarter

Time	0	1	2	3	4	5	6	7
Payment #	1	2	3	4	5	6	7
Regular payments	-$830	-$830	-$830	-$830	-$830	-$830	-$830	0
Present value	-$5,400

The payments associated with the loan reflect an annuity due with 7 regular cash flows of -$830 and a present value of -$5,400. Therefore, the quarterly rate can be found as the rate such that a 7-period annuity due with regular cash flows of -$830 for 7 periods has a present value of -$5,400.

BEGIN mode

Enter 7 5,400 -830 0

N I% PV PMT FV

Solve for 2.51

The interest rate for the loan is 2.51%

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Lecture Problem 12

In 5 years, Arturo expects to have $40,000 to invest in a security that will make fixed payments. What annual expected return does his investment need in order for him to receive $9,100 per year with the first fixed annual payment received in 6 years from today and the last annual payment in 12 years from today?

Timeline

Time	0	1	2	3	4	5	6	7	8	9	10	11	12
Re-time						0	1	2	3	4	5	6	7
Pmt #							1	2	3	4	5	6	7
Pmt amt							9,100	9,100	9,100	9,100	9,100	9,100	9,100
Inv val						40,000

We want to find the annual discount rate such that the present value of a 7-period ordinary annuity of 9,100 per year is $40,000. This is the appropriate approach because the annual payments start in 1 year (after the “re-timed” time 0) and end in 7 years (after the “re-timed” time 0), so there are 7 payments.

END Mode

Enter 7 -40,000 9,100 0

N I% PV PMT FV

Solve for 13.20

The investment must have an expected annual return of 13.20%

=================================

Lecture Problem 13-a

I Scream Ice Cream Company is analyzing several of its upcoming flavors. What is the value of peanut chocolate crunch if the discount rate is 6.2 percent and the flavor is expected to generate annual fixed cash flows forever with the first cash flow of $100,000 in 4 years?

The value can be found in 2 steps.

1) Find the value of the flavor as of one year before the first payment is made

2) Find the present value (as of today) of the flavor

Time	0	1	2	3	4	5	6	…
CF	0	0	0	0	100k	100k	100k	…

1) The first of an infinite series of fixed payments occurs in 4 years, so we can find the value of the flavor as of 3 years from now as a fixed perpetuity

PV3 = C4 / r

C4 = $100,000 r = .062

PV3 = 100,000 / .062 = 1,612,903

2) The value today of something worth 1,612,903 in 3 years can be found as

PV0 = PV3 / (1+r)3

PV3 = 1,612,903 r = .062

PV0 = 1,612,903 / (1.062)3 = $1,346,588

Mode is not relevant, since PMT = 0

Enter 3 6.2 0 1,612,903

N I% PV PMT FV

Solve for -1,346,588

The value of the flavor is $1,346,588

================================

Lecture Problem 13-b

What is the value of nougat swirl if the cost of capital is 7.2 percent and the flavor is expected to generate annual cash flows forever with the first cash flow of $90,000 in 5 years and all subsequent cash flows growing annually by 1.5 percent?

The value can be found in 2 steps.

1) Find the value of the flavor as of one year before the first payment is made

2) Find the present value (as of today) of the flavor

Time	0	1	2	3	4	5	6	7	…
CF	0	0	0	0	0	90k	90k × 1.015	90k × 1.0152	…
CF	0	0	0	0	0	90,000	91,350	92,720	…

1) The first of an infinite series of constantly growing payments occurs in 5 years, so we can find the value of the flavor as of 4 years from now as a perpetuity with constant growth

PV4 = C5 / (r – g)

C5 = $90,000 r = .072 g = .015

PV4 = 90,000 / (.072 – .015) = 90,000 / (.057) = 1,578,947

2) The value today of something worth 1,578,947 in 4 years can be found as

PV0 = PV4 / (1+r)4

PV4 = 1,578,947 r = .072

PV0 = 1,578,947 / (1.072)4 = $1,195,607

Mode is not relevant, since PMT = 0

Enter 4 7.2 0 1,578,947

N I% PV PMT FV

Solve for - 1,195,607

The value of the flavor is $1,195,607

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Lecture Problem 13-c

I Scream Ice Cream Company is analyzing several of its upcoming flavors. What is the value of coffee toffee if the cost of capital is 8.0 percent and the flavor is expected to generate fixed annual cash flows of $250,000 with the first cash flow in 6 years and the last cash flow in 14 years?

The value can be found in 2 steps.

1) Find the value of the flavor as of one year before the first payment is made

2) Find the present value (as of today) of the flavor

Time	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14
Pmt #							1	2	3	4	5	6	7	8	9
CF	0	0	0	0	0	0	250k	250k	250k	250k	250k	250k	250k	250k	250k

1) The first of a series of 9 fixed payments occurs in 6 years, so we can find the value of the flavor as of 5 years from now as an ordinary annuity with 9 payments of $250,000. The nine fixed payments occur in 6, 7, 8, 9, 10, 11, 12, 13, and 14 years from now.

END Mode

Enter 9 8 250,000 0

N I% PV PMT FV

Solve for -1,561,722

2) The value today of something worth 1,561,722 in 5 years can be found as

PV0 = PV5 / (1+r)5

PV5 = 1,561,722 r = .08

PV0 = 1,561,722 / (1.08)5 = $1,062,882

Mode is not relevant, since PMT = 0

Enter 5 8 0 1,561,722

N I% PV PMT FV

Solve for -1,062,882

The value of the flavor is $1,062,882

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