Financial Calculations

1. Present Value of an Investment

15 Years Case

Present Value of the Investment = $4,900 [ (1/0.08) − (1/0.08(1 + 0.08)^15)] = $41,941.45

2. Effective Annual Rate (EAR)

EAR = (1 + (APR/m))^m − 1 If m becomes infinite, EAR = e^APR − 1

7% APR and quarterly compounding: EAR = (1 + (0.07/4))^4 − 1 = 7.19%

16% APR and monthly compounding: EAR = (1 + (0.16/12))^12 − 1 = 17.23%

11% APR and daily compounding: EAR = (1 + (0.11/365))^365 − 1 = 11.63%

3. Value of a Perpetual Stream of Payments

Present Value of the Perpetuity = (Payment/Interest Rate) Payment = $2,500, Interest Rate = 0.061

PV of the perpetuity at date t=14 is $2,500/0.061 = $40,983.61. Discounting it back to date t=7, we should have PV7 = $40,983.61/(1+0.061)^7 = $27,077.12.

4. Bond Pricing

Bond Price = 35 * (1/YTM − 1/YTM(1 + YTM)^n) + Face Value/(1 + YTM)^n = $837.11

5. Bond Yield

If interest rate increases to 9% Bond Price = 35 * (1/ 0.045 − 1/0.045(1 + 0.045)^4) + $1,000/(1 + 0.045)^4 = $964.12

Percentage Change in the Price of the Bond = ($964.12 − $1,000)/$1,000 = −3.59%

6. Clean Price of a Bond

Clean Price = Invoice Price − Coupon Payment * (Days to Next Coupon Date/Days in Coupon Period) = $950 − $1,000 * 0.068/2 * 4/6 = $927.33

7. T-Bill

1) Price of the Bill

Price = $10,000 * (1 − 3.4% * 87/360) = $9,917.83

2) Bond Equivalent Yield

Bond Equivalent Yield = ($10,000 − $9,917.83/$9,917.83) * (365/87) = 3.48%

8. Stock Investment

Dividend Yield = $10/$540 = 1.85%

Capital Gain = ($500 − $540)/$540 = −7.41%

Total Return = 1.85% − 7.41% = −5.56%

9. Margin Account

Margin = ($400 * $25 − $400 * $40 * 50%)/($400 * $25) = 20%

If the maintenance margin requirement is 30%, she will receive a margin call.

Additional money required = $8,000 − $5,000 = $3,000

Assume she has to sell N shares Value of remaining stock = ($400 − N) * $25 = $10,000 − $25N Remaining loan = $8,000 − $25N Margin = (($10,000 − $25N) − ($8,000 − $25N))/(($10,000 − $25N) = 50% N = 240

Assume she can buy Y more shares Value of stock = ($400 + Y) * $50 = $20,000 + $50Y Maximum borrowing = $8,000 + $50Y Margin = (($20,000 + $50Y) − ($8,000 + $50Y))/(($20,000 + $50Y) = 50% Y = 80