Question

Assume a 10 year zero-coupon bond with a face value of $1,000. The interest rate has increased form 10% to 20%. What is the capital gain?

Answer 1

The face value of the bond is the value that the buyer of the bond gets at maturity of bond. So the face value of $1,000 means that the buyer of that bond will get $1,000 after 10 years. Now to calculate the capital gains we need to know the buying price of bond.

Let the price of bond be $X given that the interest rate was 10% which means after 10 years the buyer of bond gets 10% more of buying price. Mathematically this will be written as,

X + 10/100X = $1,000

X + 0.1X = $1,000

Which simply says that buying price plus the 10% on buying price must be equal to the face value.

Now we can easily solve for X.

But the question says that the interest rate has increased from 10% to 20% so now we need to calculate the price of bond such that the 20% on that price leads to the face value of $1,000.

X + 20/100 X = $1,000

X + 0.2X = $1,000

Now we can easily solve for X,

1.2X = $1,000

X = $1,000/1.2

X = $833.33

So at the interest rate of 20% the buying price is $833.33.

Now the capital gains is simply equal to the,

Capital gains = face value - buying price

Capital gains = $1,000 - $833.33

Capital gains = $166.67

So the capital gains is when interest rate increases to 20% is equal to $166.67