Actuarial Concept: Forward Rates (using Spot rates)
Subtopics: Actuarial inflating and discounting of cashflows
Use: Used in valuations
Purpose: Just to explain as best as I can what I’ve learnt, to my current understanding, as an actuarial analyst. Hopefully will be good documentation to refer back to, and easy enough to help anyone new to the concept to understand it.
Background: Inflating cashflows for outstanding cashflows and premium liabilities using actuarial methods such as PCE, PPCF, CPP, ICL, BF, we compare neutral and actual yield curve results, after inflating and discounting. So knowing what forward rates are and how they are calculated really helps.
Just my forwards rate documentation. I think what is going to help most people is the little diagram in the left column.
If you want a colour version, definitely visit Bionic Turtle (I've made a copy of it on this post, see below and see my notes on nominal interest rates...)
http://www.bionicturtle.com/how-to/article/forward_rates_and_spot_rates/
(For below image, if can't see clearly, can download it then open to full size. Thanks!)
Xuan 4:17 PM
hey A
can you pls clarify this for me
in the yield curve
for say time 0.375
there is a spot rate on the yield curve
so .... in english terms... that is the rate now (t=0) for buying that
bond with a time to maturity
in 0.375yrs?
BobA 4:20 PM
yep
Xuan 4:21 PM
so... if i went (1+ that rate)^0.375
what is that in english terms
BobA 4:22 PM
that is how many dollars you wud have a t=0.375 for each $1 invested
now
*at
Xuan 4:23 PM
if i divide that (1+ that rate)^0.375 by the average 6 month spot rate
what does that mean in english
woops
i meant the average 3 month spot rate
not 6mth
BobA 4:27 PM
that is how many dollars you wud have at t=0.375 for each $1 invested
at t=0.25
so its what we expect the 0.125 spot rate to look like in 3 months time
Xuan 4:29 PM
aka the 3 month forward rate for the 0.125 spot?
BobA 4:31 PM
yep
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*Note, Bionic Turtle uses the nominal interest rate::
Xuan 11:43 AM
mm
can i ask u another interest rate question
if the one year spot rate is 5%
is it definitely 2.5% for half a year?
BobA 11:44 AM
nope
it cud b 1% for the first half-year then 4% forward for the next
half-year
Xuan 11:45 AM
no no no
just talking about one year spot rate
not forward rates
cuz i get some forward rate formulas
that divide the spot rate /2
they're like
(1+spot/2)^2 for 1 year....
which I'm a little puzzled as to... why not just (1+spot)^1??
BobA 11:47 AM
ah
their 'spot' rate is really the nominal annual rate compounded
semi-annually
so.. its really a 2.5% per half-year rate
so youd get a little bit more than 5% over the full year
(1+0.05/2)^2
Xuan 11:49 AM
but that's not our spot rate
ours is the ... what the word for non-nominal
BobA 11:51 AM
effective
Xuan 12:06 PM
yep
that makes sense, thanks! ^_^
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Copy of Bionic Turtle's Explanation of Fwd Rates:
http://www.bionicturtle.com/how-to/article/forward_rates_and_spot_rates/
http://www.bionicturtle.com/how-to/article/forward_rates_and_spot_rates/
Forward rates and spot rates
FRM Learning Outcome (2007)
- LO 22.3: Calculate forward rates from a series of spot rates.
The forward rate is a prediction about future spot rates
Say we want to know what the market expects for the six month interest rate, but six months into the future. That's the six month forward rate six months from today. Then notation for this is 0.5 f 0.5or 0.5 f m where 'm' refers to how many years forward. So, 0.5 f 2 refers to the six month forward rate starting in two years.
Assume the following:
- The six month spot rate is 4.3%
- The one year spot rate is 4.55%
- We want to solve for the six month forward rate, in six months
We get the answer in the same way we bootstrapped the theoretical spot rate curve (term structure of interest rates): by assuming a no-arbitrage condition.
No arbitrage means you are indifferent to holding for one year or rolling over at the forward
Compare two scenarios:
- You invest for one year, at the one year spot rate, or
- You invest for six months, at the six month spot rate. Then you "roll over" into the the six month forward rate (for another six months).
At the start of the year, you should be indifferent because both should produce the same value at the end of the year (not really, there are two subtle differences: first, your rollover gives you a choice at the end of six months, it is a more liquid alternatives. Second, you have reinvestment risk on your rollover; e.g., rates could go down in the next six months. But we'll ignore these differences).
The six month forward rate six months from today must make you indifferent to the choice:
Just remember we tend to deal in bond-equivalent yields; i.e., a 4.55% one year spot rate corresponds to 2.275% every six months, a 4.3% six month spot rate corresponds to 2.15% over six months. The solution in one variable:
Under these assumptions, one-half of the the six month forward rate solves to 2.4%. We double to translate into its bond-equivalent yield of 4.8%. Therefore, the spot rate curve implies a six month forward rate six months from today (the notation is 0.5 f 0.5) of 4.8%. That is the rate that would make is indifferent between investing at the one year spot or rolling over at six months. Here is the solution and the generic form for any six month forward, given the spot rate at (m) and (m+1):
This EditGrid spreadsheet (which can be downloaded into MS Excel or other formats. Select File > Save As...) also performs the same calculation. Except the inputs are the zero coupon bond prices; i.e., the six month bond price is $97.90 and the one year bond price is $95.60.
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