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Financial Economics Midterm Exam ü Question 1. Suppose that you live next to the hot spring. You have a plan to construct "heating facilities" for the green houses by taking advantage of the hot spring. You use hot water from hot spring to warm the green houses. You need to buy and install water pumps and pipes etc. You need to borrow 10 million yen. (1) You will pay back an equal amount each month 24 times, starting next month. Let c be each payment. Suppose interest rate per month is given by x. Monthly compounding applies. Explain how to calculate monthly payment c, using related equation. Be aware that symbol x is per month, not APR. (2) If you use the heating facilities, you can harvest melons and flowers throughout the year, from the 3rd month to the 24th month. Each month your profit will be 1 million yen. How do you calculate net present value of your project of "heating facilities." Use the same symbol x to denote interest rate per month. (3) How do you calculate internal rate of return of the project of "heating facilities." Show an equation and explain. 1 10 106 c 24

t1

2 NPV 24

In[928]:=

3

t1

1 xt 1

1 xt 1

10 106

Clearans, y; ans NSolve 24

1

1 yt

10 106 0, y;

y y . ans3; Print"IRR ", y t1

IRR 0.086278

As for solutions for (3), we have only 3 solutions, although the equation becomes 24 degree polynomial.

ü Question 2 . Suppose that Mr. Otaru owns solar energy company. He thinks this year is the best time to expand his business. He is considering to issue 3-year coupon bond. He wonders what should coupon rate be to sell it at par? Mr. Otaru asked you about it. You made a research and calculated spot rates for those borrowers which have the same credit standing as Mr. Otaru’s company. year 0.5 1 1.5 2 2.5 3 spot rate as APR 1.0 2.0 2.5 3.0 3.0 3.0 Explain how you determine coupon rate which makes the bond par. Show an equation to solve, using the numbers given in the above table. Use symbols to denote variables, if necessary. Be sure to define these variables. In[930]:=

Clearans, b ans NSolve 100

100 b 1 0.05

b;

100 b

1 0.01

2

100 b

1 0.0125

3

100 b

1 0.015

4

100 b

1 0.015

5

100 b 100

1 0.0156

b b . ans1; Print"b ", b b 0.0150425

Coupon rate which makes coupon bond being par, with existing zero rates, is called “par bond yield.”

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Question 3. Suppose Mr.Green is wondering to invest in bond market. He observes the following prices for zero coupon bonds ( pure discount bonds ) that have no risk of default. Semiannual (biannual ) compounding applies. Interest rates are expressed as APR.

name of bond

maturity year

A B C D

0.5 1 1.5 2

price per Yield to zero $100 Maturity rate as APR Face Value Pa y1 1 97 y2 s2 96 y3 s3 95 y4 s4

(1) What is yield to maturity of bond A? 1 %. In case of pure discount bond, zero rate is the same as yield to maturity. (2) Show equations to determine the one and half -year zero rate S3. If necessary, use symbols given in the table.

1

100

Zero rate for 1.5 years satisfies the following equation; 96

Clearans, S3; ans NSolve96

100

1

S3 3 2

S3 2

3

, S3; S3 S3 . ans3;

Print"Zero rate expressed as APR is equal to ", 100 S3 , "." Zero rate expressed as APR is equal to 2.74007.

(3) Suppose that there is coupon bond named Bond E. Bond E has 6% coupon rate and remaining years of 2 years to maturity. Coupon payments are made twice a year starting a half year from now. What should be the price of Bond E. Show an equation to calculate. To do so use numbers given in the above table as much as possible.

¤ 95

We know that , 97

100

1

S2 2

2

100

1

, 96

S3 3 2

,

100

1

S4 4 2

. We use these relationships.

Print"price ",

3

3

1.005

97 100

3

96

103

100

95 100

price 106.625

Question 4. You collected data about zero rates from various bonds traded in the market. Semiannual compounding applies. ∏ maturity in years zero rate as APR

current situation 0.5 1 1.5 2 2.5 S1

S2

S3

S4

S5

zero rate in decimal 0.03 0.05 0.07 0.09 0.11 notation, APR

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(1) Draw “zero rate yield curve” . ClearA;

We draw a graph of relationship between maturity and zero rates. In the following, we read 2nd and 3ird row from 2nd column to 6th column. Then we plot these pairs. By “ Table[ { ... }, { k, 2, 6 } ]” , we make list of pairs. ∏ " maturity in years"

A

"current situation" 0.5 1 1.5 2 2.5 ;

"zero rate in decimal 0.03 0.05 0.07 0.09 0.11 notation, APR" Clearpoint; point ListPlotTableA2, k, A3, k, k, 2, 6, AxesLabel " year", "", ImageSize 180 % 0.10 0.08 0.06 0.04 0.02 year 1.0

1.5

2.0

2.5

We have only points so far. We want to draw a curve of “yield curve. ” Let’s draw line and, at the same time, make points visible. Clearline; line ListLinePlot TableA2, k, A3, k, k, 2, 6, AxesLabel " year", "", ImageSize 180 % 0.10 0.08 0.06 0.04 0.02 year 1.0

1.5

2.0

2.5

Showpoint, line, PlotLabel "Zero's Yield Curve" Zero's Yield Curve % 0.10 0.08 0.06 0.04 0.02 year 1.0

1.5

2.0

2.5

(2) What should price of 2.5-year bond with 4.8 % coupons be? Let P0 be price. Explain how to calculate bond price and its yield to maturity, using related equations. Use numbers or related symbols shown in the table. ClearS, P0 S

0.03 0.05 0.07 0.09 0.11 ;

S is treated as 15 matrix. So 0.03 is 1st row and 1st column. S1,1 0.03 Similarly,... S1,50.11 5

P0

t1

2.4

1 S1, tt

100

1 S1, 55

; Print"Price of 4.8 coupon bond ", P0

Price of 4.8 coupon bond 68.9357

(3) Suppose there is 2 year coupon bond. Let’s call it “Bond J”. You bought 2 year bond with coupon rate 4.8% today. You think you may have to sell this bond 6 months later. You wonder how much Exam20120606Ans.nb

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holding period rate of return you would have. You consider the case where zero rate yield curve remains the same 6 months later. Let P0 and P1 be bond prices today and 6 months later. Explain how to calculate rate of return for the 6 months, using related equations. Use numbers or related symbols shown in the table. Yield curve remains the same. It means zero rate applicable 6 months later, say on the first 6 month, is the same as today; 0.03. ClearP1 4

P1

t1

2.4

1 S1, t

t

100

1 S1, 44

; Print"P1 ", P1 P1 P0

Print" holding period rate of return:

",

P0

P1 P0 P0

P1 79.0088 P1 P0 holding period rate of return:

0.146123

P0

(4) There is another coupon bond. Let’s call it “Bond K”. It has 2 years to maturity and its coupon rate is 3.6%. Bond J and K have the different coupon rates but have the same maturity. Could their yields to maturity be the same? Explain it, using related equations with symbols and numbers given in the above table. If zero rates are different for different maturities, yield to maturity of bonds which have different coupon rates are different, even if they have the same maturity. If zero rates are the same, these two coupon bonds have the same yield to maturity. Their yield to maturity is the same as zero rates. Yield to maturity y is a kind of weighted average of zero rates. As for Bond K, its yield to maturity is also weighted average of zero rates. However coupon rates are different so weights are different. As a result weighted average has different value. Yield to maturity is different from Bond J. If zero rate yield curve is flat, it implies that S1~S4 are the same. In this case, yield to maturity which is solution y for (1) is the same as zero rate. The same is true with (2). Two bonds have the same yield to maturity.

Pj

2.4 1S1

1S22 2.4

1S33 2.4

2.4 1S44

100 1S44

Pk

1.8 1S1

1S22

1S33

1.8 1S44

100 1S44

1.8

1.8

... 1 ... 2

Suppose y is yield to maturity of Bond J. It means that y solves equation (1). Same with (2). Pj 4

t1

Pk 4

t1

2.4

1 yt 1.8

1 y

t

100

... 3

100

... 4

1 y4 1 y4

Let’s draw graphs of (3) and (4) . Bond J is dashed line in the figure. Bond J always comes above Bond K. Par bond yields are values of points shown as J and K in the figure. (1) If Bond J and K have the same yield to maturity, their prices must correspond to the intersections with vertical line. Our example is at y=0.01 in the figure. If prices of J and K satisfy such relationship that stay on the same vertical line, then two bonds have the same yield to maturity. (2) Meanwhile zero rates St’s can take various values. Only by chance, prices of J and K satisfy the relationship so that they stay on the same vertical line. (3) Two coupon bonds with different coupon rates do not have the same yield to maturity, even if they have the same maturity.

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(4) If zero rates are the same for all maturity, it means S1=S2=S3=S4. In this case zero rate and y cannot be different by graph. Equation (1) and (3) have the same price. If price is the same, in the figure, the corresponding values of St’s and y must be the same. Zero rates and y cannot be different. Zero rates are common to Bond J and K. So yield to maturity is the same for J and K.

Clearpj, pk, y, g1, g2, vertical, g3, g4 pjy_ : 4

t1

pky_ : 4

t1

2.4

1 yt 1.8

1 yt

100

1 y4

100

1 y4

phy_ : 100 g1 Plotpjy, y, 0, 0.05, PlotStyle Dashed; g2 Plotpky, y, 0, 0.05; g3 Plotphy, y, 0, 0.05, PlotRange 0, 0.05, 50, 120; vertical ListLinePlot0.01, 0, 0.01, 110; g4 ListPlot 0.024, 100, 0.018, 100 , PlotMarkers "J", "K"; Showg1, g2, g3, vertical, g4, PlotLabel "Price of Bond as function of y" Price of Bond as function of y 110

105

100

K

J

95

0.01

Exam20120606Ans.nb

0.02

0.03

0.04

0.05

5