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How to compute the present value of a liability

timlee

Joined
8/31/11
Messages
29
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13
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!
 
\( PV = x\large(\frac{1-(1+r)^{-n}}{r}\right)\)

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!
Click to expand...
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.
 
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?
 
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?
 
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

\( \frac{10000}{(1+.07)^9} = 5439 \)
 
amanda: Thanks!

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

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
 
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?
Click to expand...

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.
 
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