Question

What is the present value of a perpetuity of $ 7,000 that provides the first payment in 6 years?   Your opportunity cost is 10%, continuously compounded.

Answer 1

Solution:

• An annuity of infinite duration is perpetuity. Present value of perpetuity=Constant Annual Payment* Present value interest factor for a perpetuity
• Present value of interest factor for perpetuity=1/K where K is interest rate
• But if interest is compounded continuously we take e^r-1 as Present value of interest factor for perpetuity Now here annuity payment starts at the end ( Assumed at end of the year payment)of 6 years. So first we calculate present value at beginning of year 6 using formula

  Present value of perpetuity=Constant Annual Payment* Present value interest factor for a perpetuity

=A/{(e^r)-1}

• Present value of perpetuity at beginning of year 6 =7000/{(e^0.10)-1}=66558.3236 $= Present value at end of year 5.
• Here e^10%=1.1051709
• Present value of perpetuity at beginning of year 1= FV at end of year 5/{(e^rt)}, where r is 10% and t=5 year.[ Here we have used formula for calculating present value basis given future value if the interest rate is compounded continuously for n no. of years. PV=FV/{e^rt)}]
• Present value of perpetuity at beginning of year 1=66558.3236 / e^(0.1*5)=40369.66 $.
• Here e^10%*5=1.648721271.

Present value of perpetuity at beginning of year 1 i.e at time 0( assuming annuity given at end of the year)=66558.3236 / e^(0.1*5)=40369.66 $.