**Walk through on S1 homework__** BY Jerry wu-I know solutions are already provided but I HOPE this will help all of you (please DO NOT edit, let me know if I made a mistake)

Chapter 4

25) **PV= C/(1+r)^t**

Both options have same C which is 50,000

t=5, plug into equation and solve for r and compare which option has a higher rate of return , which is the interest rate. In this case, option H has a higher interest rate.

Option G: 50,000=85,000/(1+r)^5

(1+r)^5=1.7

r= (1.7)^1/5-1=0.112=11.2%

Option H: 50,000=175,000/(1+r)^10

(1+r)^10=3.5

r=(3.5)^1/10-1=0.1335=13.35%

28) The key of this question is determing the correct t. The first annuity happens at year 3 and all the way up to year 22. Therefore, there are 20 annuity payments in total. (22-3+1=20, notice it's not 19 years since you get paid on year 3 also). The discount rate is 8% so r=0.08

Using equation **PVA= C({1-[1/(1+r)]^t}/r)**

PVA=2,000({1-[1/(1+0.08)]^20}/0.08

PVA=19,636.29 (This is the present of of the annuity before the first payment at year 3, so it's PVA of year 2)

To find out PV we have to discount it all the way back.

PV= FV/(1+r)^t

PV= 19,636.29/(1+0.08)^2 (Notice T=2 here because we have to discount 2 years back to period O , which is NOW)

PV= 16,834.96

29) Again, the key of this question is determining the correct t. First payment happens on year 6, starting from this point, there's a 15 year annuity to be paid out. Therefore T=15. Using the below Equation, we can find out the PVA for year 5.

C=500, r=0.15 (15% annuity)

**PVA= C({1-[1/(1+r)]^t}/r)**

PVA=500({1-[1/1+0.15)]^15/0.15)

PVA=2923.69 (This is PVA for year 5, so we have to discount it back to PV 0)

**PV= FV/(1+r)^t**

PV=2923.69/(1+.12)^5

PV=1658.98

32) if cost today= PV of future cash flow, then the company would be indifferent to invest in the project. So in order to solve for r, we plug in the cost of project =PV

**PV= C/r** Perpetuity formula.

240,000=21,000/r

r=21,000/240,000

r=0.0875 =8.75% (This means at this discount rate, company would be indifferent to invest in the project)

43) Again, same stuff, determining t is the tricky part. First payout is at year 9 and all the way up to year 25. This means t= 17 (25-9+1). 17 payments until maturity. r=0.10 (10% discount rate)

**PVA= C({1-[1/(1+r)]^t}/r)**

PVA= 2,000{[1-(1/1+0.1)^17]/0.10}

PVA=16,043.11 (this is PVA of year 8)

To discount it back to Period 0, we have to discount 8 periods, so t=8

PV=16,043.11/1.10^8=7484.23

52) This one is the big mama of the chapter, similar example is given on pg114 eg 4.24. I suggest everyone get familiarize with the time line concept if confused. The key is recognizing the fact that there are 4 seperate payouts involved in college tuitions. The kids are 2 years apart in age. The question tells you one of the kid will be going to college in 15 years, and the other in 17 years. We need to find out the present value (cost) the year before you would make the first college payment for each child. t=4 (four payments of tuition), Annual interest rate is 5.5%, r=0.055, C=23,000 (estimated expense per year)

**PVA= C({1-[1/(1+r)]^t}/r)**

PVA=23000({1-[1/(1+0.055)]^4}/0.055

PVA=80,618.45

Older kid will go to college in 15 years from now, so the above represents the cost one year prior to his first year in college. To get PV 0, we have to discount 14 periods.

**PV=FV/(1+r)^t**

PV=80618.45/(1+0.055)^14

PV=38097.81

Since they are 2 years apart, the younger kid will go to college 2 years after the first kid. Thus you will need to discount it 2 years more t= 16 (14+2) than the oldest kid to get the PV 0.

PV=80618.45/(1+0.055)^16

PV=34229.07

Total cost of both child's college expense today would be 38097.81+34229.07=72326.88

According to the question, you will make the first payment 1 year from today and the LAST payment until you first kid goes to college. That means for the annuity the t=15. So to calculate your C (how much you need to save each year)you would use the equation again, plugging in t=15 using PVA=72326.88.

72326.88= C({1-[1/(1+0.055)]^15/0.055)

solve for C you'll get 7205.31 (Money you could have spent on beer or new wardrobe every year in order to send your kids to college (hopefully not UCLA for all that sacrifice) in 15 years)

54) The key is to determine your after tax payout. (aftertax cash flow0

Since Tax rate is 28% (0.28 ), your after tax cash flow should be:

160,000 (1-0.28) or just 160000 x 0.72=115,200

There were some debates on how the annuity should be calculated for Option (using t= 30 vs t=31), I think it's best to ask the professor again and varify. For me peronally, I understood why the solution used t=31. Please allow me to explain my logic. It is mentioned that option A is a 31 years annuity. Since you get paid a sum of 115200 (after tax) immediately. This payment IS PART of your 31 years annuity and therefore is subject to the 10% (r= 0.10)

Thus:

option A- PVA= (1+0.10)115200({1-[1/(1+0.10)^31]}/0.10)

PVA=1,201,180.55

option B, it is mentioned it is a 30 year annuity rather compare to option A (31 years), thus the t= 30, and the lump sum payment you get immediately is both non-taxable and NOT part of your 30 years annuity payment. Therefore:

Option B- C= 72759.60 from ( 101,055 X 0.72), r= 0.10, t=30

PVB =72,759.60({1-[1/(1+0.10)^30]}/0.10) + 446,000

PVB= 1,131,898.53

PVA> PVB, therefore you should take option A since the present value is greater even on aftertax basis

Chapter 5

Before going into the first 3 questions, it is good to know that price of bonds is always in inverse relationship with the interest rate or in this case the YTM (yield to maturity), read page 133- top of 134, eg 5.2. If the bond is selling at premium you know that the coupon rate is higher than YTM, if bond is selling at a discount, you know that coupon rate is less than YTM. If bond is selling at par value, which is normally 1000, then coupon rate is equal to YTM.

**PV= F/(1+R)^t**

Pure discount bonds are normally paid out in semianual payments, this means you'll get paid twice a year (normally every 6 months). For a 10 year bond, the t then becomes 20. t=10x2=20.

1)

a) YTM=5%, your YTM for every 6 months is then 0.025 (0.05/2=0.025)

P= 1000/(1+0.025)^20= 610.27

b) YTM= 10%, r=0.05 (0.10/2=0.05)

p=1000/(1+0.05)^20= 376.89

c) YTM= 20%, r=0.10 (0.20/2=0.10)

P= 1000/(1+0.10)^20 =161.51

As you can see above, when the YTM goes up (interest rate increses), the price of the bond goes down.

2) The key of this question is determining the correct C to use when you value the bonds. When the coupon rate is 8%, that means you get paid 4% (0.08/2=0.04) of the bond's par value every 6 months (semiannual payments). Therefore your C would be 1000x 0.04= 40. t = 40 because time to maturity is 20 years but there are 40 payments in total.

Using equation P= C X Ar^t + 1000[1/(1+r)^t] on page 134 (Level coupon bonds)

you'll get (please refer to solution book, chapter 5 B-101 for full calulation)

a) P= 1000, Like mentioend above, When the YTM (interest rate) and the coupon rate are the same, then the bond is selling at Par vaulue

b) P=828.41 When YTM > coupon rate, bond is selling at discount

c) P=1231.15 When YTM < coupon rate, bond is selling at premium

3) 12 year bond 2 years ago, implies that the bond has 10 more years to maturity t=20 (10x2=20)

P= 970 (1000 X 97% of par value= 970)

C= 43 (1000 x 0.043) since 8.6% is the coupon rate (0.086/2=0.043)

We want to know the YTM, so we need to solve for r

970=43({1-[1/(1+r)]^20}/r) + 1000/(1+r)^20

r= 4.531%

YTM = 4.531%x 2=9.06% (remember it's semi annual payments)

12)

P=1095

r=? (you should know since P is selling at premium, the YTM must be smaller than my coupon rate based on what we've talked about above, this is a good way to varify your answers on the exam)

coupon rate is 8%, semiannual payments, so C is 1000 X 0.04= 40

t= 40 since it's 20 years to maturity, 40 payments (20x 2) in total.

1095=40({1-[1/(1+r)]^40}/r) + 1000/(1+r)^40

Solve for r=3.55% , YTM = 7.10% (3.55 x2)

32) Again, the big mama of chapter 5, a lot of concepts are involved.

Current Yield= C/current price of bond

Capital gains yield = (New price- Original price)/ original price. Please note New price of bond can be less than original price of bond, causing a loss in your capital gains yield.

You only know both bonds have 5 more years to maturity, but this time it tells

you it's on annual payments rather than semiannual payment.

Bond P:

Therefore t=5, YTM is 8% thus r= 0.08, C= 1000 X 0.10 (10% coupon rate)

bond price at period zero, P0= 100 ({1-[1/(1+0.08)]^5}/0.08) + 1000/(1+0.08)^5 =1079.85 (bond selling at premium)

We need to compare whether the bond is making money for us, the best way is to calculate the bond price for next period, which is when the bond has only 4 years left to maturity. when t= 4

P0= 100 ({1-[1/(1+0.08)]^4}/0.08) + 1000/(1+0.08)^4 =1066.24

Therefore your current yield is 100/1079.85= 9.26%

Capital gains yield is (1066.24-1079.85)/1079.85= -1.26% (bond P is at a loss)

Bond D:

t=5, r=0.08, C= 1000 X0.06= 60

P0= 60 ({1-[1/(1+0.08)]^5}/0.08) + 1000/(1+0.08)^5 =920.15

P1= 60 ({1-[1/(1+0.08)]^4}/0.08) + 1000/(1+0.08)^4= 933.76

Current yield 60/920.15= 6.52%

Capital gains yield= (933.76-920.15)/920.15= 1.48%

*The above proves that, all things being the same, the coupon rate can affect the prices of bonds and also whether your capital gains yield will be at loss or profit.

*It also indicates that, all things held constant, the premium bonds have high current income while having price depreciation as maturity nears. The discount bonds have low current income but have price appreciation as maturity nears. Even though both bonds have a total return of 8%, it is distributed differently between capital gains and current income.