# Using Forward or Spot rates for NPV?

I have to calculate the NPV for Capital Budgeting in a project with annual cash flows discounted by a risk - free interest rates

1.Instead of using a constant interest rate, should it better to use different interest rate for each period? 2. Should I use the Forward interest rate or the Spot interest rate?

eg. $CF_3$ should be discounted by $(1+ f(3,4))^3$ where f(3,4) is the forward rate in period=3 for the period=4 ?

• NPV for a project? What is the project? Please provide more details. – Gordon Dec 9 '15 at 13:49
• Project of Capital Budgeting with annual cash flows – sparkle Dec 9 '15 at 14:14
• Generally, the interest rates, fixed or floating, apply to those cash flows are contractually specified. – Gordon Dec 9 '15 at 14:18
• I mean I have to valuate if the NPV > 0 using a discount rate that is a risk free + \beta (rm -rf). Now I'am at the "risk free part" – sparkle Dec 9 '15 at 14:44
• @sparkle: it's common practice to use forward rates for accuracy purposes. Just that sometimes people do not want to bother and quickly use spot curves for simplicity reasons. Thus, I would go for forward curves in particular if you do not experience any difficulty to extract rates as long as the period gets bigger. Hope it helps – owner Dec 9 '15 at 16:33

If you want to discount the CF3 from 3 years in the future to today you should use (1 + 3yr spot rate)^3. There's no reason to use forward rates for that purpose. The forward rates should only be used for period-by-period discounting - for example, if you wanted to find the value after 3 years of a CF4 which occurs after 4 years, you would use (1+ 1yr Forward rate from 3yr to 4yr).

(addressing the comment): I really cant think why you would need forward rates in a capital budgeting type problem where everything is discounted from year n to today.

• Thanks. So what are some cases when I have to use the forward rates? – sparkle Dec 10 '15 at 10:30
• dm63 is right. It might be because you are confusing forward rates with rates varying with their maturity (along a term structure of interest rate) – MarinD Dec 10 '15 at 17:39

This is all theoretical and real life will diverge from the theory

The spot rates and forward rates are linked.

Spot rate for the nth period should equal the product of all the forward rates up to that period.

i.e

Let Spot{n} = spot rate for nth period

Let Forw{k,j} = forward rate to period j at period k

Let X_m be the m'th period.

Then (1+Spot{n})^n = (1 + Forw{0,X_1}) * (1 + Forw{X_1,X_2}) * ... * (1 + Forw{X_n-1,X_n})

Of course in real life, there will be slight or not so slight differences which gives rise to arbitrage opportunities.

So the answer to your question is, you can use either. If you use spot rates, just take it to the correct power. If you use forward rates, you take the product of the forward rates until the correct period. Neverthess, it is good practice to take these calculations with a grain of salt. It is unlikely that you got your cash flows correct anyway so never mind a bit of error in the rates that you use.