Question

Consider these long-term investment data: • The price of a 10-year $100 par zero coupon inflation-indexed bond is $80.65. • A real-estate property is expected to yield 2% per quarter (nominal) with a SD of the (effective) quarterly rate of 10%. What is the probability of loss or shortfall after 10 years?

Answer 1

In this sum, we need to compare the quarterly yields of the zero coupon bond with the returns of the real estate property. and we will incur losses only when we get lower returns than the zero coupon bond return.

Step 1: Calculate Zero coupon Bond Quarterly YTM

Do this, let us assume the quarterly YTM to be X and the total term 40 quarters( each year has 4 quarter, hence 10 year will have 40 quarters)  therefore using compound interest formula, we get

80.65(1+X)^40 = 100, this gives us X of 0.539%.

Step 2: Calculate the probability of losses.

We assume that the returns follow, normal distribution , we have to calculate the Z Score of the distribution ( using Z score data table)

We will only incur losses when the real estate returns are lower than bond returns Therefore, we need to find Z score X (0.539%)  of a given mean of 2% and standard deviation of 10%

from the formula we know, Z= [ (0.539 - 2)/10] = -.146092. looking in the z table, we know that Probability of z (P(z)) is .4404 or 44.04% that is there is a 44.04% chance that we will bear losses.

reference table for finding P(z).