What's wrong with my PV calc?

chatty06

New Member
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

David Harper CFA FRM
Staff member
Subscriber
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

New Member
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

David Harper CFA FRM
Staff member
Subscriber
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.
0610-present-value.png

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!
 
Last edited:

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
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,
 
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