What's wrong with my PV calc?

[chatty06]

Hi All, I'm completing question 2.4 in the GARP book quantitative analysis.

The question is just asking to get the present value of a bond that is expected to be $85 after 1 year at a continuous interest rate of 5%.

In the answer it says it should be $80.85 but when I run my PV calcs I get 80.95. Any thoughts? I tried both on my calculator and in excel and I get the same price. This seems like a simple calculation.

Thanks for the help.

 

[David Harper CFA FRM]

Hi @chatty06 So in that case, 80.95 looks to be the price if the interest rate (used to discount) is 5.0% annual compounding/discounting rather than the requested continuous; it's the same 5.0% which is referred to the stated or nominal rate. So you appear to be getting $85/(1+5%)^1 = 85/1.05 = 80.95. The general form of discrete compounding is given by:

• PV = FV * (1 +r/k)^(k*T) where k is periods per year and T is years; or equivalently, PV*(1+r/k)^(k*T) = FV. You are using PV = 85.00/(1+5.0%/1)^(1*1), just to show you the complete general form (because it's applicable generally)

For continuous, because at a continuous rate, r, over T years, FV = PV*exp(rT), it's also true that PV = FV*exp(-rT). So you want PV = 85.00*exp(-5.0%*1) = 85*exp(-5.0%). On the TI BA II+, the exp(x) is above the LN as e^x because exp(x) is just another way of writing e^x and here is a key relationship: EXP(.) and LN(.) are inverse functions, so you'll notice in Excel that =EXP(LN(x)) will return X. Continuous compounding is elegant, Hull uses it throughout. For example, if today's price of 4.00 grows to 5.00 tomorrow, the continuously compounding rate is given by LN(5/4) = 22.3% and we can test with 4*exp(22.3%) = 5. I hope that's helpful!

 

[chatty06]

Thanks David!

Just to confirm I am understanding this correctly...

Using my BA II Plus calculator I would do the following.

.05 --> e^x which gives me 1.051271 and the continuous interest rate.

Then compute PV using the following inputs:
5.1271% --> I/Y
1 --> N
-85 --> PMT
CPT --> PV = 80.8545

This gets me to the correct answer. Let me know if I misinterpreted anything along the way. Thanks again.

 

[David Harper CFA FRM]

Hi @chatty06 It's cool you found that solution Let me just show you the quicker way. To compound PV at stated rate of (r) continuously over T years, we use FV=PV*exp(rT). I think of discounting as compounding but backwards, such that to discount the FV to the present value at a stated rate of (r) over T years with continuous discounting, we us PV = FV*exp(-rt) which is the same as PV = FV/exp(rt) because x^(-a) = x*1/a = x/a. So with respect to continuous discounting, the most efficient formula is PV = FV*exp(-rt) and so you can get the solution on the calculator with the following (where "##" indicates my comment):

.05 [+/-] [e^x] ## and display should read 0.9512, and not that you get e^x by [2nd][LN]. This is performing the exp(-rT) = exp(-5%*1 year) = exp(-0.05)
* 85 = ## and display should read 80.8545

Bear with me, because I am fond of these fundamentals (too many even financial professionals gloss over this without realizing it's a cause for later confusions), compare this to the answer to a slightly different version of the question: "what is the present value of a bond that is expected to be $85 after 1 year at a semi-annual interest rate of 5.0% ." And now we use the discrete (as opposed to continuous) version where because PV*(1+r/k)^(k*T) = FV, the PV = FV*(1+r/k)^(-k*T) = FV/(1+r/k)^(k*T). And in this case, the answer would be PV = 85.00/(1+0.05/2)^(2*1) = $80.94024; see second column below in the XLS i just created which is here https://www.dropbox.com/s/8gf9ma7zb6szj74/0610-present-value.xlsx?dl=0. Notice the PV is decreasing as the compound frequency is increasing.

What you did, from my perspective, is you converted 5.0% continuous rate to its annual (discrete) equivalent. This is reviewed in Hull Chapter 4, where discrete R_m = m*exp[exp(R_c/m)-1], or to match my symbols, R_k = k*exp[exp(R_c/k)-1] such that if the continuous rate R_c = 5.0% and we want the annual discrete equivalent rate (i.e., m or k = 1 period per year), then R_m = 1*[exp(0.050/1) - 1] = 5.127110%. Then your calculator discounted with this annual rate. Your solution is good! 5.127110% is also called the effective annual rate (EAR) such that the question asked has the same answer as the question "what is the present value of a bond that is expected to be $85.00 after 1 year if the effective annual rate is 5.12711%" which is shorthand for "what is the present value of a bond that is expected to be $85.00 after 1 year if the interest rate is 5.0% per annum with annual compounding." (the latter is how Hull tends to write it). I hope that's helpful, thanks!

 

[chatty06]

Thank you. This makes sense and it's a much simpler approach.

 

[Amierul]

Hi @David Harper CFA FRM, Good day.

I have a question here... regarding the example above. Do you know why if i did not specify the PV=0 i will get different value?

 

[David Harper CFA FRM]

Hi @Amierul No, sorry, I do not know: I am getting the same $3,769.35 when I omit the PV keystroke (because when reset/cleared the default for PV should already be zero). However, I never run the solution this way. I always input four variables, and solve for the 5th. I don't see the value in omitting/assuming the PV. The five keystrokes are in a row on the calculator; I prefer to "use the entire row" by keying four inputs and solving for the fifth. Hope that's helpful,

 

[Amierul]

okie, Thanks @David Harper CFA FRM for the answer. I will follow ur way then.