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quidni
Topic Author
Posts: 27
Joined: November 5th, 2008, 7:32 pm

### Mean-variance portfolio optimization: bond returns

Hi, excuse me, in the classical (Markwowitz) mean-variance portfolio optimization, I'm trying to implement in MATLAB by the portfolio object and related functions... if my portfolio contains also bonds, how should I compute the (total) returns for these, to give them as input in MATLAB?

I mean, I need a series of historical returns to give as input in MATLAB, so it can compute the mean of these (if I want to assume the expected returns and volatilities are simply historical), and the related correlation matrix.

But may I simply download the historical price series for the bonds, then compute the historical returns by ln(price_t/price_(t-1)), and then give them to MATLAB?

I was thinking to give to MATLAB instead directly the historical series of the ytm of the bonds, but I think I may be wrong...

The "total return" for a bond may be thought as its "price return"? If I think to the theoretical price of a bond, as its discounted cash flows, these last should contain also its coupon, and so this would be the end of the story? Or am I missing something?

Thanks a lot

Alan
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Joined: December 19th, 2001, 4:01 am
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### Re: Mean-variance portfolio optimization: bond returns

Way too much trouble, IMO. Just pick a bond mutual fund or etf that suits (or various if you must) and collect their total returns.

bearish
Posts: 6453
Joined: February 3rd, 2011, 2:19 pm

### Re: Mean-variance portfolio optimization: bond returns

As a practical matter, I think Alan is right. But, in the event you were to do it at the bond level, I’d say you can use the price return to estimate the covariance with your other assets, but should make sure to include the coupon contribution when calculating the average (expected) return. The YTM time series by itself is not sufficient for either purpose, since it’s missing the duration scale.

quidni
Topic Author
Posts: 27
Joined: November 5th, 2008, 7:32 pm

### Re: Mean-variance portfolio optimization: bond returns

Ok thanks @Alan and @bearish, but as @bearish supposed, unfortunately I can't substitute my bond with mutual bond funds, so I have to deal with them...

Thanks a lot @bearish for your advise, only a final couple of doubts:

1) when includind the coupon for the computation of the average (expected) return, I can use it (the coupon) in the log return formula, by simply computing ln( (price_t + coupon_t)/price_(t-1) ), or I should forget the log return and simply compute [ (price_t + coupon_t)/price_(t-1) - 1] ?

2) what do you mean with the "duration scale"? In some way indeed I would guess if the (historical) average YTM could be useful to estimate the average (expected) return...

Thanks a lot

bearish
Posts: 6453
Joined: February 3rd, 2011, 2:19 pm

### Re: Mean-variance portfolio optimization: bond returns

For 1) it shouldn’t really matter, as long as with coupon_t you mean the coupon that accrued over the relevant time period, not whether you happened to receive a coupon payment during the period or not. For 2), to a first order approximation, the price return is equal to minus the change in YTM times the duration of the bond.

quidni
Topic Author
Posts: 27
Joined: November 5th, 2008, 7:32 pm

### Re: Mean-variance portfolio optimization: bond returns

Ok, perfect! Thank you very much @bearish!