Sep 06

Price an annuity and perpetuity. FRM 2007

by David Harper, CFA, FRM, CIPM

FRM |

annuityIntro2

Learning Outcome

  • LO 21.3: Calculate the price of an annuity and a perpetuity using a calculator with time value functions.

Level (equal), sequential cash flows

If an instrument pays level (equal) cash flows in a sequence, there are two formats (finite or infinite) and variations (pay now, pay later):

  • Annuity: the sequence of level cash flows is finite
  • Perpetuity: the sequence of level cash flows is never-ending
  • Annuity due: the first cash flow is due immediately, at time zero (t=0)
  • Ordinary annuity: the first cash flow is due at the end of the first period (t=1)

image

Perpetuity

Pricing a perpetuity is the essence of simplicity: we "capitalize" the payment stream. If A = the perpetuity payment (the coupon, in bond terms) and r = the interest rate (the yield to maturity, in bond terms), the price of a perpetuity = A/r.

For example, if we assume an annual interest rate of 6%, a perpetuity that pays $100 every year is worth $1,666.67 = $100/6%.

If the perpetuity instead pays semiannually, it is worth the same because ($100/2)/(6%/2) = $50/3% = $1,667

Annuity

An annuity is just a subclass of a perpetuity, where the number of periods is limited to (N) instead of unlimited...

infinite

...but the formula is less intuitive. Where A = the annuity amount (i.e., the coupon payment, in bond terms), N = the number of periods, and r = the interest (discount) rate, the present value (PV) of an annuity is given by:

 

annuityformula

Price an annuity with the TI BAII+

It is easier to use the time value of money (TVM) keystrokes on the TI BAII+. Assume an annuity pays $100 every six months for ten years and the interest rate is 5%. We therefore input the following (note the periodicity is six months):

  • N=20 (10 years x twice per year)
  • PMT = $100
  • I/Y = 2.5% (5% divided by two)
  • FV = 0 (but this step is optional; we can omit and the zero is assumed)

And then we solve for the present value:

  • CPT PV = -$1,559

Did you notice that this is the same as pricing a bond except the face value (principal) is zero?

And we get the same result if we use the formula:

annuityformulaexample

Comments

  1. jnakahara 06 Sep 2007

    Hi David,

    Under the section “Level (equal), sequential cash flows”, the first two bullet points should be switched: Perpetuity has unlimited cashflows and the annuity has finite cashflows.

    Keep up the great work. The videos and tutorials have been tremendously helpful.

    - Jay

  2. David Harper, CFA, FRM 06 Sep 2007

    Jay,

    Thank, I appreciate that help! Thanks for feedback, too...I am blogging out more of the notes content this year and I am thrilled if repeat exposure might be helpful...David

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