Mathematics Assignment #75260

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Assignment_3_CF_09_16_2018.docx

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Corporate Finance

Assignment 3

September 17, 2018

Group 5

Problem 1:

Melody won a $30 million lottery today. The payout is the following: Melody gets $1 million today and $1 million each year for the next 29 years. Suppose that the discount rate is 6%, how much money worth today for the lottery assuming federal tax rate is 25% and state tax is 8%?

( …… …… )

PV Year 1-29

5-Key Method
N	I/Y	PV	PMT	FV
29	6%	$13,590,721.02	-1,000,000	0

Add CF0

1,000,000 + 13,590,721.02 = $14,590,721.02

Federal Tax: 145907.21 x 0.25 = $3,647,680.26

State Tax: 145907.21 x 0.08 = $1,167,257.68

14,590,721.02 – 3,647,680.26 – 1,167,257.68 = $9,775,783.08

Problem 2:

You are offered the opportunity to put some money away for retirement. You will receive 6 annual payments of $20,000 each beginning in 25 years. How much would you be willing to invest today if you desire an interest rate of 9%?

……

( …… )

……

20,000

Cash Flow Key Method
CF0	CF01	F01	CF02	F02	I	NPV
0	0	24	20,000	6	9%	11,340.85

Problem 3:

If you can afford a $500 monthly car payment, how much of a car can you afford if interest rate is 5% on 48-month loans?

Monthly interest rate = 5% 12 = 0.4167%

5-Key Method
N	I/Y	PV	PMT	FV
48	0.416%	21,714.89	-500	0

Problem 4:

You borrow a GPM of $120,000 with annual payments and 30-year term. The interest rate is 10% and the payment factors from year 1 to year 30 are: 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, 100%, ..., 100%.

Comment by Maira Alejandra Soto: Change the spreadsheet. I calculated the GPM with the same model that DR Lin had used in class. I kept the same idea of the colors.

1. What are the annual payments for years 1 to 30?

See highlighted blue

2. What is remaining balance at the end of each year?

See highlighted yellow

3. What are the interest payment and principal payment for years 1 to 30?

See highlighted green

Problem 5:

Sam Strother and Shawna Tibbs are vice-presidents of Mutual of Seattle Insurance Company and co-directors of the company's pension fund management division. A major new client, the Northwestern Municipal Alliance, has requested that Mutual of Seattle present an investment seminar to the mayors of the represented cities, and Strother and Tibbs, who will make the actual presentation, have asked you to help them by answering the following questions.

a. How is the value of a bond determined? What is the value of a 10-year, $1,000 par value bond with a 10 percent annual coupon if its required rate of return is 10 percent? Does the bond sell at par?

Value of a bond is determined by calculating the present value of the contractually promised principal plus interest payments (Cash flows) discounted back to the present using the market’s required yield to maturity on similar risk. Comment by Maira Alejandra Soto: Changed answer in order to give a more complete explanation.

Interest payment = 1000 x 10% = $100

( …… …… )

100

1,100

5-Key Method
N	I/Y	PV	PMT	FV
10	10%	-1000	100	1000

PV = par value The bond sells at par.

b. What would be the value of the bond described in Part a. if, just after it had been issued, the expected inflation rate rose by 1 percentage point, causing investors to require an 11 percent return? Would we now have a discount or a premium bond?

5-Key Method
N	I/Y	PV	PMT	FV
10	11%	-941.11	100	1000

PV < par value The bond sells at a discount.

c. What would be the value of the bond described in Part a. if, just after it had been issued, the expected inflation rate dropped by 1 percentage point, causing investors to require a 9 percent return? Would we now have a discount or a premium bond?

5-Key Method
N	I/Y	PV	PMT	FV
10	9%	-1064.18	100	1000

PV > par value The bond sells at a premium.

d. What would happen to the value of the 10-year bond over time if the required rate of return remained at 11 percent, or if it remained at 9 percent? Would we now have a premium or a discount bond in either situation?

5-Key Method @ 11%
	N	I/Y	PV	PMT	FV
After year 0	10	11%	-941.11	100	1000
After year 1	9	11%	-944.63	100	1000
After year 2	8	11%	-948.54	100	1000
After year 3	7	11%	-952.88	100	1000
After year 4	6	11%	-957.69	100	1000
After year 5	5	11%	-963.04	100	1000
After year 6	4	11%	-968.98	100	1000
After year 7	3	11%	-975.56	100	1000
After year 8	2	11%	-982.87	100	1000
After year 9	1	11%	-990.99	100	1000
After year 10	0	11%	-1000.00	100	1000

The initial value of a bond that pays 10% at a YTM of 11% is lower than its par value, therefore, its value increases over the time as its maturity approaches back to face value. At a rate of 11% we would have a discount bond. Comment by Maira Alejandra Soto: Changed. More explanantion.

5-Key Method @ 9%
	N	I/Y	PV	PMT	FV
After year 0	10	9%	-1064.18	100	1000
After year 1	9	9%	-1059.95	100	1000
After year 2	8	9%	-1055.35	100	1000
After year 3	7	9%	-1050.33	100	1000
After year 4	6	9%	-1044.86	100	1000
After year 5	5	9%	-1038.90	100	1000
After year 6	4	9%	-1032.40	100	1000
After year 7	3	9%	-1025.31	100	1000
After year 8	2	9%	-1017.59	100	1000
After year 9	1	9%	-1009.17	100	1000
After year 10	0	9%	-1000.00	100	1000

The initial value of a bond that pays 10% at a YTM of 9% is higher than its par value, therefore, its value decreases over the time as its maturity approaches back to face value. At a rate of 9% we would have a premium bond. Comment by Maira Alejandra Soto: Changed. More explananation.

e. What is the yield to maturity (YTM) on a 10-year, 9 percent annual coupon, $1,000 par value bond that sells for $887.00? How about it sells for $1,134.20? What does the fact that a bond sells at a discount or at a premium tell you about the relationship between investors’ YTM and the bond's coupon rate?

Interest payment = 1000 x 9% = 90

5-Key Method
N	I/Y	PV	PMT	FV
10	10.91%	-887.00	90	1000

5-Key Method
N	I/Y	PV	PMT	FV
10	7.08%	-1134.20	90	1000

Given the same period, coupon rate, and par value, when the bond sells at a discount, the YTM is greater than the coupon rate. When the bond sells at a premium, the YTM is less than the coupon rate.

f. How does the calculation for valuing a bond change if semiannual payments are made? Find the value of a 10-year, semiannual payment, a 10 percent coupon bond if investor’s required rate of return is 13%.

Interest payment = 1000 x (10% 2) = 50

R = 13% 2 = 6.5%

5-Key Method
N	I/Y	PV	PMT	FV
20	6.5%	-834.72	50	1000

g. Suppose a 10-year, 10 percent, semiannual coupon bond with a par value of $1,000 is currently selling for $1,135.90, producing a yield to maturity (YTM) of 8 percent. However, the bond can be called after 5 years for a price of $1,050.

1. What is the bond's yield to call (YTC)?

Interest payment = 1000 x (10% 2) = 50

5-Key Method
N	I/Y	PV	PMT	FV
10	3.765%	-1135.90	50	1050

YTC = 3.765% x 2 = 7.53%

2. If you bought this bond, do you think you would be more likely to earn the YTM or the YTC? Why? Comment by Maira Alejandra Soto: Since the question is based on probabilities, I presented both scenarios in order to show what scenario has more than 50% probability to occur.

50+50 50+50 50+50 50+50 50+50 50+50 50+50 50+50 50+50 50+50

-1135.95

( 7.53% YTM )

1050

( 8.7 % YTM )

1000

( YEAR /VALUE BOND TO BE CALLED )

If Bond runs the next 5 years left

Explanation:

Background:

· 10-year bond, semi-annual, 10%, with call option for company at $1050 at after year 5

· YTM at t0: 8.0%

· YTC at t0: 7.53%

· YTM at the end of t5 for the remaining 5 years of the 10-year bond given the 10% coupon if the price happens to be $1050: around 8.7%

Which is more likely? That that the investor earns the YTM or the YTC? This is equivalent to asking which of the two scenarios has a probability of more than 50%.

When would the company call the bond? It would call, i.e., to reimburse the current investors, and immediately issue a 5-year bond for the remaining 5 years, if it can save money (lower interest rates). It saves money, if it will pay less than having the bond not called. Thus, 8.7% is the threshold at the end of t5. If the market allows refining for 8.7% or less (and leaving transaction costs aside), the company will refinance and call the bond. Nothing will happen, if the interest stays above 8.7%.

Assuming that future interests are unknown to us, we need to make an assumption of whether rates will go or down or stay the same. As we don’t have any, we should make an unbiased estimate that there is an equal probability of rates to go up or down, i.e., stay the same. If that is our assumption, then we need to assume that at the end of t5, interest rates (and the yield demanded by investors) will be 8% as well. At 8%, the company will call the bond (less than 8.7%), thus as investors at t0, it will be more likely for us to only receive the lower YTC than the YTM.

120,000$

Periods30

Rate10%

Payment19,398.84$

MonthsBeginningpmt FactorpmtInterestPrincipalEnding Balance

Factor multipling

the interest rate

0120,000$

1120,000$ 10%$1,939.8812,000.00$ (10,060.12)$ 130,060.12$ 0.09091

2130,060.12$ 20%3,879.77$ $13,006.01(9,126.24)$ 139,186.36$ 0.16529

3139,186$ 30%5,819.65$ $13,918.64(8,098.99)$ 147,285.35$ 0.22539

4147,285.35$ 40%7,759.53$ $14,728.53(6,969.00)$ 154,254.35$ 0.27321

5154,254$ 50%9,699.42$ $15,425.43(5,726.02)$ 159,980.36$ 0.31046

6159,980.36$ 60%11,639.30$ $15,998.04(4,358.73)$ 164,339.10$ 0.33868

7164,339$ 70%13,579.18$ $16,433.91(2,854.72)$ 167,193.82$ 0.35921

8167,193.82$ 80%15,519.07$ $16,719.38(1,200.31)$ 168,394.14$ 0.37321

9168,394$ 90%17,458.95$ $16,839.41619.54$ 167,774.60$ 0.38169

10167,774.60$ 100%19,398.84$ $16,777.462,621.38$ 165,153.22$ 0.38554

11165,153.22$ 100%19,398.84$ $16,515.322,883.51$ 162,269.71$ 0.35049

12162,269.71$ 100%19,398.84$ $16,226.973,171.86$ 159,097.85$ 0.31863

13159,097.85$ 100%19,398.84$ $15,909.783,489.05$ 155,608.79$ 0.28966

14155,608.79$ 100%19,398.84$ $15,560.883,837.96$ 151,770.84$ 0.26333

15151,770.84$ 100%19,398.84$ $15,177.084,221.75$ 147,549.09$ 0.23939

16147,549.09$ 100%19,398.84$ $14,754.914,643.93$ 142,905.16$ 0.21763

17142,905.16$ 100%19,398.84$ $14,290.525,108.32$ 137,796.84$ 0.19784

18137,796.84$ 100%19,398.84$ $13,779.685,619.15$ 132,177.69$ 0.17986

19132,177.69$ 100%19,398.84$ $13,217.776,181.07$ 125,996.62$ 0.16351

20125,996.62$ 100%19,398.84$ $12,599.666,799.17$ 119,197.45$ 0.14864

21119,197.45$ 100%19,398.84$ $11,919.747,479.09$ 111,718.36$ 0.13513

22111,718.36$ 100%19,398.84$ $11,171.848,227.00$ 103,491.36$ 0.12285

23103,491.36$ 100%19,398.84$ $10,349.149,049.70$ 94,441.66$ 0.11168

2494,441.66$ 100%19,398.84$ $9,444.179,954.67$ 84,486.99$ 0.10153

2584,486.99$ 100%19,398.84$ $8,448.7010,950.14$ 73,536.85$ 0.09230

2673,536.85$ 100%19,398.84$ $7,353.6812,045.15$ 61,491.70$ 0.08391

2761,491.70$ 100%19,398.84$ $6,149.1713,249.67$ 48,242.03$ 0.07628

2848,242.03$ 100%19,398.84$ $4,824.2014,574.63$ 33,667.40$ 0.06934

2933,667.40$ 100%19,398.84$ $3,366.7416,032.10$ 17,635.31$ 0.06304

3017,635.31$ 100%19,398.84$ $1,763.5317,635.31$ (0.00)$ 0.05731

374,670.31$ $120,000.00

Factor dividing the

loan to get the

payment

6.18594

Amount Borrowed

GRADUATED PAYMENT MORTGAGE

TOTAL PAID:

10% coupon5050505050505050501050PV

9% pa 47.8545.7943.8141.9340.1238.3936.7435.1633.65676.121039.56

8% pa48.0846.2344.4542.7441.1039.5238.0036.5335.13709.341081.11

implied cost of letting bond run to the end

Then compare to 1050:

If PV of letting it run is lower, don't call the bond.

If PV of letting bond run is higher, call at 1050 the equivalent rate is 8.70%