(Related to Checkpoint 5.6) (Solving for Springfeld Learning sold zero coupon bo

Question

(Related to Checkpoint 5.6) (Solving for Springfeld Learning sold zero coupon bonds (bonds thst don't pay any interest, instead the boncholder gets just one payment, coming when the bond matures, from the issuer) and reoeived $1,000 for each bord that wil pay $20,000 when it matures in 40 years. a. At what rate is Springfield Learning borrowing the money from investors? b, f Nancy Muntz purchased a bond at the offering fc 1,000 and sold it 10 years later for th6 market price of 53,200, what annual rate of return did she earn? c. If Barney Gumble purchased Mntz's bond at the market price (S3,200) and held it 3D years until maturity, what annual rate of returm wouls he have earne? a. Al whal rale is Springlield Leag borowinng the rey from inveslors? 7.78 % ¡Round to two decimal paces.) b if Nancy Muniz purchased a bond at tie 0 Bing fc$1 000 and sold it 10 ears later for he market price of FV hal armual rate of return dd she earn? 12.33 % (Round to two decimal places.) c. If Barney Gumble purchased Muntz's bond a the market price ($3,200) and held it 3D years until maturity, what annual rate of retum would he have earnec? % CRcund to two decimal plao

Explanation / Answer

1. GIven,
Springfield is selling a bond for price, P = $1000
Value of the bond at maturity, A = $20,000
Time period = 40 years
Interest rate of borrowing for Springfield = i

A = P(1+r)t
20000 = 1000(1+i)40
(1+i)i = 20
1+i = 201/40
1+i = 1.077769
i = 0.77769
i = 7.78%

The borrowing rate of interest for Springfield is 7.78%.

b. Nancy Muntz bought the bond for $1000 and sold it after 10 years for the market price of $3200

Here, A = 3200
      P = 1000
      t = 10

A = P(1+i)t
3200 = 1000(1+i)10
(1+i)10= 3.2
1+i = 1.123349
i = 0.123349
i = 12.33%

Nancy would earn an interest of 12.33%

c) Barney Gumbey bought the bond for $3200 abd held it for 30 years till maturity.
Amount recieved at the end of maturity,A = $20,000
A=P(1+i)t
20000 = 3200(1+i)30
(1+i)30 = 6.25
(1+i)= 1.06299
i = 0.06299
i = 6.30%

Barney Gambley would recieve an annual interest of 6.30%

2. Present value
Option 1 = $1800 in today's value
Option 2 = $5000 in 13 years. Assuming 11% is the interest rate. Present value of 5000 would be as follows

PV = FV/(1+r)13
PV = 5000/ (1+0.11)13
   = 5000/(1.11)13
= $1287.57

Option 2 will have a present value of $1287.57

Option 3: 29000 in 27 years
PV = 29000/(1+0.11)27
      = 29000/(1.11)27
= $1732.52
Option 3 will have a present value of $1732.52

We should chose option 1, $1800 today as the present value of other two options are less than this.

3. Finance Company:
Loan rate = 12% compounded semi-annually
Effective Annual rate = (1+i/n)n - 1
= (1+0.12/2)2-1
      = 1.062-1
= 0.1236
      = 12.36%
Finance company would offer EAR of 12.36%

Bank : 13% compounded daily

EAR = (1+i/n)n - 1
= (1+0.13/365)365-1
= (1.000356)365-1
      = 1.138802 - 1
      = 0.138802
= 13.88%
Bank would offer the loan at an EAR of 13.88%


Bank offering 13% compounded daily would be more attractive as the number of compounding period increases and offers 13.88% as an effective annual rate

4. For CD1 APR = 3.95% compounded semi-annually

EAR = (1+i/n)n - 1
= (1+0.0395/2)2-1
= (1+ 0.01975)2-1
      = (1.01975)2-1
      = 1.03989 -1
= 0.03989
      = 3.99%

For CD 2, at 4% APR compounded daily

EAR = (1+i/n)n - 1
= (1+ 0.04/365)365-1
= (1+0.00011)365-1
      = 1.040808-1
      = 4.08%

Choose CD2 for 4% compounded daily, as the EAR would be 4.08% instead of CD 1 offering 3.95% compounded semi-annually with an EAR of 3.99%.

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