How to compute the present value of a liability

timlee · 11/5/11

There is a liability which will owe $x$ dollars per year in $n$ years with interest rate $r$ (annual compounding). Is the present value of this liability $ \sum_{i=1}^n \frac{x}{(1+r)^i} $ or $ \sum_{i=1}^n x(1+r)^i ? $
Are there differences in calculating present value of a bond and of a liability or loan?
Thanks!

TRobinson · 11/6/11

$ PV = x\large(\frac{1-(1+r)^{-n}}{r}\right)$

x = yearly payment
n = number of payments
r = annual rate of interest

timlee said:
Are there differences in calculating present value of a bond and of a liability or loan?
Thanks!

Typically the bond will have a larger final payment to include in your calculations. For the final payment you would need to multiply it by $(1+r)^{-n}$ and add it to the PV of your annuity.

timlee · 11/6/11

Treefingers: Thanks!

Could you explain why the formula of present value of a liability is this?

Are you saying the PV formulas for a liability and for a bond are the same?

Alexei · 11/6/11

Here is why:
http://en.wikipedia.org/wiki/Time_value_of_money

All these formulae are for simple calculations since in real life one needs to account for interest rate risk, reinvestment risk, credit risk and other risks.

timlee · 11/6/11

Thanks!

But in this problem "The interest rate is 7% (annual compounding). A pension fund has a liability will owe
$10,000 per year in 9 years time. What is the present value of this liability?", the answer is $5439, but I got $65152.32 by the formula provided above as
$P=\sum_{t=1}^9 \frac{10000}{(1+0.07)^t} \approx 65152.32 $

I dont know how the answer is achieved. Am I wrong somewhere?

amanda.jayne · 11/6/11

A bond is a liability, as is every financial obligation.

You are not dealing with an annuity.

All the question is asking is that you solve for the present value of a future sum

$ \frac{10000}{(1+.07)^9} = 5439 $

timlee · 11/6/11

amanda: Thanks!

What does "future sum" mean? From "owe $10,000 per year in 9 years time", how is 10000 is the future sum?

amanda.jayne · 11/6/11

In 9 years you will owe 10000.

The problem is asking you what the value of this obligation is, today.

Think of it this way: If interest rates in a savings account are 7%, how much would you need to deposit into your bank account today such that in 9-years' time your account balance will be 10,000.

The problem, as stated is rather unclear:

"The interest rate is 7% (annual compounding). A pension fund has a liability will owe
$10,000 per year in 9 years time. What is the present value of this liability?"

I would understand this as a perpetuity starting in 9-years' time.

If you take out the 'per year', it is a simple time value of money problem with the 5439 solution

My guess is that the problem is either mis-stated or the answer is wrong

TRobinson · 11/6/11

timlee said:
Thanks!

But in this problem "The interest rate is 7% (annual compounding). A pension fund has a liability will owe
$10,000 per year in 9 years time. What is the present value of this liability?", the answer is $5439, but I got $65152.32 by the formula provided above as
$P=\sum_{t=1}^9 \frac{10000}{(1+0.07)^t} \approx 65152.32 $

I dont know how the answer is achieved. Am I wrong somewhere?

It seems there is no regular payment "per year" at all - as amanda pointed out. The reason you got 65152.32 is because you found the PV of 9 payments of 10,000 per year.

If your question is correct and your answer is wrong, I would solve it as a perpetuity; you can use the formula in my post. The n value would become -$ \infty $, making your (1+r) term 0 and you would just have your payment per year over your interest rate. You would then move the value of the perpetuity back 8 years with $(1.07)^{-8}$ because the formula leaves you one year before the payments begin.

Now that I know your question, I guess the difference between calculating a liability and the value of a bond could be described as follows: while the liability is just one payment in the future, the bond is a payment in the future combined with a regular and level interest payment (coupon) which forms an annuity. A loan repayment might fall somewhere in between as they are often amortised (interest and principal repayment combined) to create regular level payments over a period. Of course it's a simplification, but hopefully it helps.

Good luck.