Question

25. Today (T=0), an investor purchased a 20 year bond with a 5.00% coupon and a face value of $100,000 for $106,550. In six months (T=0.5) interest rates have decreased by 0.50% and the investor decides to sell the bond immediately after receiving the first coupon payment. What is the investor’s total gain (loss) on the bond? HINT: Total Gain (Loss) = Price Change in Bond + Coupon
A. ($6,548)
B. ($6,048)
C. $7,130
D. $7,602
E. $7,630

Answer 1

→ yield to Maturity-(YTM) Cat T-o) As the tenn o bond is 2o years and he coupon pasment is eey 6monts, periods 20 x 2 = 40 2500 is hat yean'g rate Now ice Solving bo we get, ie 2.25% = 4.50ナ

YTM at T = 0 is the Interest rate prevailing for similar bonds of same maturity at T=0, which is 4.5%.

The interest rate reduced by 0.5% at T=0.5 to 4%.

Using the same formula as mentioned on the handwritten page,

where i = 4% / 2 = 2 % = 0.02

Coupon Payment = 2500

M = Redemption amount = 100,000

n = 39 (as the first coupon payment has already been done. So 39 coupon periods are remaining)

Solving the equation, we get the market value of the bond at T=0.5 equal to $113,451.82

It is given that the investor has sold the bond immediately after receiving the first coupon payment.

Gain to Investor = (Selling price - Purchase Price ) + Coupon Payment received

= [ (Market value of bond at T=0.5) - (Market Value of Bond at T=0) ] + $ 2,500

= $ (113,451.82 - 106,550) + $ 2500

= $ 6,901.82 + $ 2,500

= $ 9,401.82

As none of the options match with our answer, it would be wise to select the option which is nearest to our answer. Hence, E. $ 7,630 is the answer.

Dear learner, I can assure you that the calculations made above are accurate, and the options given in the question are not.

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If you want to use the shortcut formula of calculating Market price of bond using YTM,

CLE-P 2 Approx YTM = F+P C = Coupon/ Interest Payment F Face Value P Price n years to maturity

You will get less accurate, but similar answer, which won't match with the given options either.

While using this formula for calculating Approx YTM at T=0,

C = 2500 ; F = 100,000 ; P= 106,550 ; n = 40 ---------- (As the coupon payments are semi-annually)

Don't forget to multiply the approx YTM calculated above by 2, as we are making half year's calculations.

While using this formula to calculate P at T= 0.5,

C = 2500 ; F = 100,000 ; T = Approx YTM calculated above minus 0.5% ;

n = 39 (as one coupon payment out of 40 has already been made)

I hope the learner has gained much clarity from this solution.